diff --git "a/BdE1T4oBgHgl3EQfVgTd/content/tmp_files/load_file.txt" "b/BdE1T4oBgHgl3EQfVgTd/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/BdE1T4oBgHgl3EQfVgTd/content/tmp_files/load_file.txt" @@ -0,0 +1,2012 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf,len=2011 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='03104v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='AG] 8 Jan 2023 ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE ANGELO FELICE LOPEZ* AND DEBADITYA RAYCHAUDHURY** Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We study varieties X ⊆ PN of dimension n such that TX(k) is an Ulrich vector bundle for some k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' First we give a sharp bound for k in the case of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then we show that k ≤ n + 1 if 2 ≤ n ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We classify the pairs (X, OX(1)) for k = 1 and we show that, for n ≥ 4, the case k = 2 does not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Introduction Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' As is well known, the study of vector bundles on X can give important geometrical information about X itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Regarding this, one of the most interesting family of vector bundles associated to X and its embedding, that received a lot of attention lately, is that of Ulrich vector bundles, that is bundles E such that Hi(E(−p)) = 0 for all i ≥ 0 and 1 ≤ p ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The study of such bundles is closely related with several areas of commutative algebra and algebraic geometry, and often gives interesting consequences on the geometry of X and on the cohomology of sheaves on X (see for example in [ES, Be1, CMRPL] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Perhaps the most challenging question in these matters is whether every X ⊆ PN carries an Ulrich vector bundle (see for example [ES, page 543]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' It comes therefore very natural to ask if usual vector bundles associated to X can be Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, since Ulrich vector bundles are globally generated, it is better to consider twisted versions of the usual bundles associated to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The cases of the (twisted) normal, cotangent, restricted tangent and cotangent bundles have been dealt with in [Lop], with an essentially complete classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this paper we study the more delicate question: for which integers k one has that TX(k) is an Ulrich vector bundle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Ulrich vector bundles have special cohomological features, but also numerical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This makes the above question rather tricky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' It is easy to show that k ≥ 0 unless (X, OX(1), k) = (P1, OP1(1), −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In the case k = 0, a recent result [BMPT, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5] gives a classification: (X, OX(1)) = (P1, OP1(3)), (P2, OP2(2)) (we will give a new and simple proof in section 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' another proof is given in [C2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, for k ≥ 1, the question is more subtle as we will see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In the case of curves, one sees that k = 1 is not possible (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i)), while the cases k = 2, 3 can be dealt with on any curve (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1 and Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, the following sharp bound holds, showing that for curves k can be as large as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible curve of genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(k) is an Ulrich line bundle, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) k ≤ √8g + 1 − 1 2 and equality holds if and only if k is even and either X is one of the curves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) lying on a smooth cubic or X is a curve of type (k 2 +1, k +2) on a smooth quadric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, in both cases, TX(k) is an Ulrich line bundle, hence the bound is sharp for every even k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover, if X has general moduli, then k ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' As far as we know, only curves show this kind of behavior, meaning that k is not bounded in terms of the dimension (a somewhat bad bound can also be given in terms of the degree, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' As supporting evidence, we prove the following Research partially supported by PRIN “Advances in Moduli Theory and Birational Classification”, GNSAGA-INdAM and the MIUR grant Dipartimenti di Eccellenza 2018-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' ** Research partially supported by a Simons Postdoctoral Fellowship from the Fields Institute for Research in Mathe- matical Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Mathematics Subject Classification : Primary 14J60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Secondary 14J35, 14J40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n such that 2 ≤ n ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(k) is an Ulrich vector bundle, then k ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We should point out that, for n ≥ 2, we know no examples with k ≥ 2 and only one example with k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' As a matter of fact, the case k = 1 can be completely characterized, as follows Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX(1) is an Ulrich vector bundle if and only if (X, OX(1)) = (S5, −2KS5), where S5 is a Del Pezzo surface of degree 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, for k = 2, we have Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX(2) is not an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We do not know what happens for k = 2, n = 3, even though some evidence suggests that it might not be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, for surfaces, the cases k = 2, 3 point out to the possible existence, that needs to be further investigated, of some minimal surfaces of general type, as shown in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Finally, in any dimension, another interesting case is the one in which ωX and OX(1) are numerically proportional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This is dealt with in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='10, Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='11 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Notation Throughout the paper we work over the complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover we henceforth establish the following Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' X is a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' H is a very ample divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' For any sheaf G on X we set G(l) = G(lH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' d = Hn is the degree of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' C is a general curve section of X under the embedding given by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' S is a general surface section of X under the embedding given by H, when n ≥ 2 g = g(C) = 1 2[KXHn−1 + (n − 1)d] + 1 is the sectional genus of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' For 1 ≤ i ≤ n − 1, let Hi ∈ |H| be general divisors and set Xn := X and Xi = H1 ∩ · · · ∩ Hn−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Generalities on Ulrich bundles We collect some well-known facts, to be used sometimes later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let E be a vector bundle on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We say that E is an Ulrich vector bundle for (X, H) if Hi(E(−p)) = 0 for all i ≥ 0 and 1 ≤ p ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let E be a rank r Ulrich vector bundle for (X, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then (i) c1(E)Hn−1 = r 2[KX + (n + 1)H]Hn−1, (ii) If n ≥ 2, then c2(E)Hn−2 = 1 2[c1(E)2 − c1(E)KX]Hn−2 + r 12[K2 X + c2(X) − 3n2+5n+2 2 H2]Hn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) χ(E(m)) = rd n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (m + 1) · · · (m + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iv) Hn(E(m)) = 0 if and only if m ≥ −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (v) E∗(KX + (n + 1)H) is also an Ulrich vector bundle for (X, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (vi) E is globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (vii) h0(E) = rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (viii) E is arithmetically Cohen-Macaulay (aCM), that is Hi(E(j)) = 0 for 0 < i < n and all j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ix) E|Y is Ulrich on a smooth hyperplane section Y of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) KXi = (KX + (n − i)H)|Xi , 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' By [CH, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4(iii)] we have that c1(E)Hn−1 = deg(E|C) = r(d + g − 1) and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) on C = X1 we have KXHn−1 = 2(g − 1) − (n − 1)d thus giving (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see (ii) observe that the exact sequences, for 1 ≤ i ≤ n − 1, 0 → TXi → (TXi+1)|Xi → H|Xi → 0 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) give by induction that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) c2(S) = c2(X2) = c2(X)Hn−2 + (n − 2)KXHn−1 + �n − 1 2 � d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' It follows from [C1, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2)], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1), and Noether’s formula 12χ(OS) − K2 S = c2(S) that c2(E)Hn−2 = 1 2[c1(E)2 − c1(E)(KX + (n − 2)H)]Hn−2 − r � Hn − [KX+(n−2)]2Hn−2+c2(S) 12 � = = 1 2[c1(E)2 − c1(E)KX]Hn−2 − n−2 2 c1(E)Hn−1 − r � Hn − [KX+(n−2)H]2Hn−2+c2(S) 12 � Now (ii) follows from the above equation by using (i) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Next, (iii) is [CH, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see (iv) observe that E is 0-regular, hence it is q-regular for every q ≥ 0 and therefore Hn(E(q − n)) = 0, that is (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, (v) follows by definition and Serre duality, while (vi) follows by definition, since E is 0-regular, and [Laz, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' For (vii), (viii) and (ix) see [ES, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1] (or [Be1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1)]) and [Be1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' TX(k) Ulrich in any dimension We start by drawing some consequences on (X, H, k), of cohomological and numerical type, when TX(k) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let (X, H) = (Pn, OPn(1)), n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX(k) is an Ulrich vector bundle if and only if n = 1 and k = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The assertion is obvious if (X, H, k) = (P1, OP1(1), −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Vice versa suppose that TX(k) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If (X, H) = (Pn, OPn(1)), it follows by [ES, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1] (or [Be1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3]) that TPn(k) ∼= O⊕n Pn , hence 0 = det(TPn(k)) = OPn(nk + n + 1), so that 1 = −n(k + 1), giving n = 1, k = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (cohomological conditions) Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(k) is an Ulrich vector bundle we have: (i) Either (X, H, k) = (P1, OP1(1), −2), or k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) If n ≥ 2, then TX is aCM, that is Hi(TX(j)) = 0 for 1 ≤ i ≤ n − 1 and for every j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In particular Hi(TX) = 0 for 1 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) If k ≥ 1, then H0(TX) = 0, hence X has discrete automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iv) If n ≥ 2, then X is infinitesimally rigid, that is H1(TX) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (v) H0(KX + (n − k − 2)H) = 0 and, if n ≥ 2, also H0(KX + (n − k − 1)H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (vi) If q(X) ̸= 0 then H0(KX + (n − k)H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (vii) If k ≤ n − 1, then pg(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (viii) Let a(X, H) = min{l ∈ Z : lH − KX ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then k ≤ a(X,H)(n+2) 2n + n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover H0((⌈n(2k−n−1) n+2 ⌉ − 1)H − KX) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ix) KX − kH is not big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since TX(k) is an Ulrich vector bundle, it is globally generated by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now if k ≤ −1 we would have that 0 ̸= H0(TX(k)) ⊆ H0(TX(−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But then the Mori-Sumihiro-Wahl’s theorem [MS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 8], [W, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1] implies that (X, H) = (P1, OP1(2)), (Pn, OPn(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In the first case we have that 0 = Hi(TP1(k − 1)) = Hi(OP1(2k)) = 0 for i ≥ 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In the second case apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This proves (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now (ii) follows by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(viii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If k ≥ 1 we have that H0(TX) ⊆ H0(TX(k − 1)) = 0, hence (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iv) is implied by (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' As for (v), recall that, as is well known, Ω1 X(2) is globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now if H0(KX + (n − k − 2)H) ̸= 0 then we get the contradiction 0 ̸= H0(Ω1 X(2)) ⊆ H0(Ω1 X(KX + (n − k)H)) = Hn(TX(k − n))∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This gives the first part of (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Similarly, if q(X) ̸= 0 and H0(KX + (n − k)H) ̸= 0 then we get the contradiction 0 ̸= H0(Ω1 X) ⊆ H0(Ω1 X(KX + (n − k)H)) = Hn(TX(k − n))∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This gives (vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now if n ≥ 2, consider Y ∈ |H| smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX(k)|Y is an Ulrich vector bundle on Y by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(ix), hence Hn−1(TX(k − n + 1)|Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now the exact sequence 0 → TY (k − n + 1) → TX(k − n + 1)|Y → OY (k − n + 2) → 0 implies that Hn−1(OY (k − n + 2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence, setting L = KX + (n − k − 1)H, we get by Serre’s duality that H0(L|Y ) = H0(KY + (n − k − 2)H|Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore H0(L(−l)|Y ) = 0 for every l ≥ 0 and the exact sequences 0 → L(−l − 1) → L(−l) → L(−l)|Y → 0 show that h0(L(−l − 1)) = h0(L(−l)) for every l ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since this is zero for l ≫ 0, we get that they are all zero, hence H0(KX + (n − k − 1)H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This proves the second part of (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now, to see (vii), suppose that k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n ≥ 2, we see that (v) gives H0(KX) ⊆ H0(KX + (n − k − 1)H) = 0, hence (vii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n = 1 we have that k ≤ 0, hence X = P1 by (i) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Observe that a(X, H)H − KX ≥ 0, hence (a(X, H)H − KX)Hn−1 ≥ 0 and using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii), we get a(X, H) ≥ n(2k − n − 1) n + 2 This gives (viii) since, by its own definition, H0((a(X, H) − 1)H − KX) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Finally assume that KX − kH is big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then Serre’s duality gives H0(TX(k)) = Hn(Ω1 X(KX − kH))∗ = 0 by Bogomolov- Sommese vanishing [Bo, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4], contradicting Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (numerical conditions) Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(k) is an Ulrich vector bundle we have: (i) d = (n+2)(g−1) nk−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In particular either (X, H, k) = (P1, OP1(1), −2), or g = k = 0, or g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) k = n+1 2 + � n+2 2nd � KXHn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' equivalently KXHn−1 = n(2k−n−1) n+2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) If k < n+1 2 , then X is rationally connected and Hi(OX) = 0 for every i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iv) If k > n+1 2 , then −KX is not pseff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (v) TX is semistable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (vi) If n ≥ 2, then K2 XHn−2 ≤ 2n n−1c2(X)Hn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (vii) If n ≥ 2, then (12kn − 12k2 + 12k − 3n2 − 5n − 2)nd + 2(n + 12)K2 XHn−2 + 2(n − 12)c2(X)Hn−2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since c1(TX(k)) = −KX + nkH, we get by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(i) that (−KX + nkH)Hn−1 = n 2 � KXHn−1 + (n + 1)d � and this gives (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, using KXHn−1 = 2(g − 1) − (n − 1)d, we get that (nk − 1)d = (n + 2)(g − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now if nk − 1 = 0 then n = k = g = 1, but then TX(k) = OX(1) is not Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore nk − 1 ̸= 0 and d = (n+2)(g−1) nk−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence g ̸= 1 and if g = 0 then either k = 0 or k ̸= 0 and in the latter case we ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 5 have that nk < 1, hence (X, H, k) = (P1, OP1(1), −2) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This proves (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Next, (v) follows since Ulrich vector bundles are semistable by [CH, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9], hence Bogomolov’s inequality gives (vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see (iii), suppose that k < n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n ≥ 2, then (ii) gives that KXHn−1 < 0, hence X is rationally connected by (v) and [BMQ, Main Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='] (see also [CP, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence, as is well known, Hi(OX) = 0 for every i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n = 1 then k ≤ 0 and X = P1 by (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus we get (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If k > n+1 2 , then either n = 1 and g ≥ 2 by (i), so that −KX is not pseff, or n ≥ 2 and (ii) gives that KXHn−1 > 0, hence again −KX is not pseff and we get (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see (vii), observe that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) c2(TX(k))Hn−2 = c2(X)Hn−2 − k(n − 1)KXHn−1 + �n 2 � k2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(ii), we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) c2(TX(k))Hn−2 = �n2k2 2 − n 24 � 3n2 + 5n + 2 �� d−3nk 2 KXHn−1+ � 1 + n 12 � K2 XHn−2+ n 12c2(X)Hn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) and (ii), we obtain (vii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' For n ≥ 1 we denote by Qn a smooth quadric in Pn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let (X, H) = (Qn, OQn(1)), n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX(k) is not an Ulrich vector bundle for any integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since g = 0, it follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) that k = 0 and 2 = d = n + 2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ We will use the nef value of (X, H): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) τ(X, H) = min{t ∈ R : KX + tH is nef}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We observe that in [BS, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3] the nef value is defined only when KX is not nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, it makes sense and it will be used, throughout this paper, also when KX is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' A very useful observation is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(k) is an Ulrich vector bundle, then Ω1 Y (KX |Y +(n+1−k)H|Y ) is globally generated for any smooth subvariety Y ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover: (i) If ±(KX + n(n+1−2k) n+2 H) is pseff, then KX ≡ n(2k−n−1) n+2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) τ(X, H) ≥ n(n+1−2k) n+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) τ(X, H) ≤ n − nk n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In particular, if KX is not nef, then k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iv) If k ≥ n + 1, then KX is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that Ω1 X(KX + (n + 1 − k)H) is Ulrich and globally generated by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v) and (vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since Ω1 X(KX + (n + 1 − k)H) surjects onto Ω1 Y (KX |Y + (n + 1 − k)H|Y ), the latter is also globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover so is det(Ω1 X(KX+(n+1−k)H) = (n+1)KX+n(n+1−k)H, hence we get (iii) and, if k ≥ n+2, we also deduce that KX is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, if k = n+1, then Ω1 X(KX) is Ulrich, and we claim that det(Ω1 X(KX)) = (n + 1)KX is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In fact, if not, then [LS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1] implies that there is a line L ⊂ X such that Ω1 X(KX)|L is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence (n + 1)KX · L = deg(Ω1 X(KX)|L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But then we have a surjection Ω1 X(KX)|L → Ω1 L, contradicting the fact that Ω1 L is not globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This proves (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' As for (i) and (ii), set q = n(n+1−2k) n+2 , so that (KX + qH)Hn−1 = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now if ±(KX + qH) is pseff, then KX + qH ≡ 0 by [FL2, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='15] (see also [FL1, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7]), thus proving (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, (i) implies that either KX ≡ −qH and then τ(X, H) = q, or KX + qH is not pseff, hence not nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore, in the latter case, τ(X, H) > q, proving (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(k) is an Ulrich vector bundle we have that k ≤ (n + 2)(d − 4) + 4 4n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have X ⊂ PH0(H) = PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If N = n then (X, H) = (Pn, OPn(1)) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1 gives that n = 1 and k = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since d = 1 we have that k = −2 ≤ − 5 4 = (n+2)(d−4)+4 4n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY We now show that it cannot be that N = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume that N = n + 1, so that d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n = 1 we have that KX = (d − 3)H, g = �d−1 2 � and k = 3(d−3) 2 + 1 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now 0 = H0(TX(k − 1)) = H0((−d + 2 + k)H) and therefore −d + 2 + k ≤ −1, giving the contradiction d ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence n ≥ 2 and since C ⊂ P2 we have that g − 1 = d(d−3) 2 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) implies that d = 2(nk−1) n+2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v) gives that 0 = H0(KX + (n − k − 1)H) = H0((d − k − 3)H) and therefore 2(nk − 1) n + 2 − k ≤ −1 that is k(n − 2) + n ≤ 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore N ≥ n + 2 and C ⊂ PN−n+1 can be projected isomorphically to a non-degenerate smooth irreducible curve in P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then Castelnuovo’s bound gives that g − 1 ≤ d(d−4) 4 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii) implies the required bound on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ A nice consequence of the above lemmas is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' There does not exist any (X, H, k) with TX(k) an Ulrich vector bundle, when: (i) KX ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) ±KX is pseff and k = n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Under hypothesis (ii), we get from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6(i) that KX ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus we will be done if we prove (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume next that KX ≡ 0, so that k = n+1 2 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii) and n ≥ 3 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since H − KX is ample, it follows by Kodaira vanishing that Hi(H) = Hi(KX + H − KX) = 0 for i > 0, hence h0(KX + H) = χ(KX + H) = χ(H) = h0(H) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v) gives that h0(KX + n−3 2 H) = 0, whence, if n ≥ 5, we get the contradiction h0(KX + H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' It remains to consider the case n = 3, k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that pg(X) = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(vii) and q(X) = 0, for otherwise Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(vi) gives that h0(KX + H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore χ(OX) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand χ(OX) = 1 24c1(X)c2(X) = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ We now prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' By the Hodge index theorem we have that H2 |SK2 S ≤ (H|SKS)2, that is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) dK2 XHn−2 ≤ (KXHn−1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(vi), (vii) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) we obtain that 0 = (12kn − 12k2 + 12k − 3n2 − 5n − 2)nd + 2(n + 12)K2 XHn−2 + 2(n − 12)c2(X)Hn−2 ≤ ≤ (12kn − 12k2 + 12k − 3n2 − 5n − 2)nd + 3n2 + 11n + 12 n K2 XHn−2 ≤ ≤ (12kn − 12k2 + 12k − 3n2 − 5n − 2)nd + 3n2 + 11n + 12 nd (KXHn−1)2 which, using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii) becomes 4nk2 − 4n(n + 1)k − 3n2 − 7n − 4 ≤ 0 giving k ≤ n2 + n + √ n4 + 5n3 + 8n2 + 4n 2n < n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ The case k = 0 is known: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' ([BMPT, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5]) Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX is an Ulrich vector bundle if and only if (X, H) = (P1, OP1(3)), (P2, OP2(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 7 We will give a quick alternative proof in section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Next we study the case when KX and H are proportional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that the numerical classes of H and KX are proportional and that either (i) 1 ≤ n ≤ 11 and either k ≤ n+1 2 or k ≥ n + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' or (ii) n = 12, or (iii) n ≥ 13 and k ̸∈ {n + 2, n + 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX(k) is Ulrich if and only if (X, H, k) is one of the following: (1) (P1, OP1(1), −2), (2) (P1, OP1(3), 0), (3) (P2, OP2(2), 0), (4) (S5, −2KS5, 1), where S5 is a Del Pezzo surface of degree 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In the cases (1)-(4) we have that TX(k) is Ulrich by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i), Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Vice versa, suppose that the numerical classes of H and KX are proportional and that we are under one of hypotheses (i), (ii) or (iii) and that TX(k) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Observe that, since N 1(X) is a torsion free finitely generated abelian group, we can find an ample primitive divisor A and some r, s ∈ Z such that s > 0, H ≡ sA and KX ≡ −rA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In particular, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii) gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5) r(n + 2) = n(n + 1 − 2k)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If k ≤ 0 we are in cases (1)-(3) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(i) and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence assume that k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8 shows that k ̸= n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If (ii) or (iii) holds, since the numerical classes of H and KX are proportional, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(vi), (ii) and (vii) imply that 4nk2 − 4n(n + 1)k − 3n2 − 7n − 4 ≥ 0 so that k > n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This is a contradiction under hypothesis (ii) by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Under hypothesis (iii), we get that k ≥ n + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then it follows by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5) that −r − ks = (n − 2)k − n2 − n n + 2 s ≥ n − 8 n + 2s > 0 hence KX − kH = (−r − ks)A is ample, contradicting Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(ix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus it remains to consider hypothesis (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now assume (i), so that n ≥ 2 and Theorem 2 implies that it cannot be that k ≥ n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence k < n+1 2 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii) implies that X is Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently, the numerical and linear equivalence for divisors coincide on X and then KX = −rA and H = sA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We can also assume that r ≤ n − 1, for otherwise, as is well known, (X, H) = (Pn, OPn(1)), (Qn, OQn(1)), contradicting Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Next, set PA(t) := χ(KX + tA), so that PA(t) = h0(KX + tA) whenever t ≥ 1 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' By Riemann-Roch (see for example [Ho, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (1), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2], we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6) PA(t) = An n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' tn + An−1KX 2(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='tn−1 + An−2(K2 X + c2(X)) 12(n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' tn−2 + · · · + (−1)nχ(OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that n is even, for otherwise n and n + 2 are coprime, and consequently n divides r by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5), hence r ≥ n, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Set n = 2m, where 1 ≤ m ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If m = 1 we have that n = 2, k = 1 and we are in case (4) by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We will now exclude the remaining cases for m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that we can rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5) as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7) (m + 1)r = m(2m − 2k + 1)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Case 1: m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case k ≤ 2 and r ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (1a): k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that r = 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus (r, s) = (2, 1), contradicting Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (1b): k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 2s = 3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus (r, s) = (2, 3) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v) shows that h0(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This contradicts [A, Lemma 2] or [K, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY Case 2: m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case k ≤ 3 and r ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (2a): k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 4r = 15s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, 15 divides r which is clearly impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (2b): k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 9s = 4r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, 9 divides r which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (2c): k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 3s = 4r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus (r, s) = (3, 4) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v) shows that h0(KX+8A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This is a contradiction by [GL, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2], as KX + 8A is base-point-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Case 3: m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case k ≤ 4 and r ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (3a): k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 5r = 28s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, 28 divides r which is clearly impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (3b): k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 4s = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus (r, s) = (4, 1) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v) shows h0(KX + 5A) = h0(H) = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (3c): k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 12s = 5r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, 12 divides r which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (3d): k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 4s = 5r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, (r, s) = (4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have KX = −4A and H = 5A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence H0(KX + 15A) = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then PA(1) = PA(2) = PA(3) = PA(5) = PA(10) = PA(15) = 0, PA(0) = PA(4) = 1 and PA(t) = A8 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (t − 1)(t − 2)(t − 3)(t − 5)(t − 10)(t − 15)(t2 + at + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8) 1 = PA(0) = A8 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (4500b) =⇒ A8 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' b = 1 4500 and calculating the coefficient of t7 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6) we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9) A8 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (a − 36) = A7KX 2(7!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=') =⇒ a = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We also know that PA(4) = 1 and that gives us −A8 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (396)(16 + 4a + b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We simplify the above using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9) to obtain −38016A8 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' = 1 + 396 4500 which is clearly absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Case 4: m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case k ≤ 5 and r ≤ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (4a): k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 2r = 15s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, 15 divides r which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (4b): k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 35s = 6r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, 35 divides r which is also impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (4c): k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 25s = 6r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, 25 divides r which is also impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (4d): k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 5s = 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus (r, s) = (5, 2) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v) shows that h0(KX +10A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then PA(1) = PA(2) = PA(3) = PA(4) = PA(6) = PA(8) = PA(10) = 0, PA(0) = PA(5) = 1 so that we obtain PA(t) = A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (t − 1)(t − 2)(t − 3)(t − 4)(t − 6)(t − 8)(t − 10)(t3 + at2 + bt + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='10) 1 = PA(0) = −A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (11520c) = 1 =⇒ A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' c = − 1 11520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' and calculating the coefficient of t9 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6) we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='11) A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (a − 34) = A9KX 2(9!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=') =⇒ a = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We also know that PA(5) = 1 and that gives us −A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (360)(125 + 25a + 5b + c) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 9 We simplify the above using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='11) to obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='12) A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' b = 1 5(11520) − 1 5(360) − 70A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Finally, calculating the coefficient of t8 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6) we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='13) A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (b − 34a + 463) = A8(K2 X + c2(X)) 12(8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We simplify (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='13) using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='12) to obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='14) 67A10 + 5A8c2(X) + 1302 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(vii) shows that 115A10 = A8c2(X) and combining with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='14), we get that A10 is negative, which is clearly impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (4e): k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We see that 5s = 6r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus (r, s) = (5, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have KX = −5A and H = 6A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Again H0(KX + 24A) = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then PA(1) = PA(2) = PA(3) = PA(4) = PA(6) = PA(12) = PA(18) = PA(24) = 0, PA(0) = PA(5) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, we obtain PA(t) = A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (t − 1)(t − 2)(t − 3)(t − 4)(t − 6)(t − 12)(t − 18)(t − 24)(t2 + at + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='15) 1 = PA(0) = A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (746496b) =⇒ A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' b = 1 746496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' calculating the coefficient of t9 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6) we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='16) A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (a − 70) = A9KX 2(9!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=') =⇒ a = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We also know that PA(5) = 1 and that gives us A10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (41496)(25 + 5a + b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We simplify the above using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='15) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='16) to obtain A10 = �705000 746496 � (10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=') which is clearly absurd since A10 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that KX = eH, e ∈ Z (hence, in particular, if Pic(X) = ZH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX(k) is Ulrich if and only if (X, H, k) = (P1, OP1(1), −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This follows by [Lop, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1(i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We give another proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have that e = n(2k−n−1) n+2 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If k ≤ n+1 2 it follows by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='10 that (X, H, k) = (P1, OP1(1), −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now assume that k > n+1 2 , so that e ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n ≥ 2, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v) gives that k ≥ n + e, hence k(n − 2) + n ≤ 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then n = 1 and e = 2(k−1) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But 0 = H0(TX(k − 1)) = H0((k − 1 − e)H), hence e ≥ k, so that k ≤ −2, contradicting k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 2 with TX(k) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that there is an ample line bundle A on X such that KX = rA, H = sA for some r, s ∈ Z (hence, in particular, if Pic(X) ∼= Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let m(H, A) := min{m ≥ 0 : H0(mH + qA) ̸= 0 for all q ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then: (i) m(H, A) > (n−2)k−2 n+2 and, if n ≥ 3, then k < (n+2)m(H,A)+2 n−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) If A is effective, then n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) If m(H, A) ≤ n − 3, then n ≤ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Observe first that, if n ≥ 3, then (n−2)k−2 ≥ k−2 ≥ 0: In fact if k ≤ 1 we have a contradiction by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii) implies that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='17) r(n + 2) = n(2k − n − 1)s and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(v) gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='18) 0 = H0(KX + (n − k − 1)H) = H0((nk − 2k − 2)s n + 2 A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see (i), notice that it is obvious for n = 2, for m(H, A) ≥ 0 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n ≥ 3 we see by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='18) that (nk−2k−2)s n+2 ∈ Z and we can write (nk−2k−2)s n+2 = as+b for some a, b ∈ Z with a ≥ 0, 0 ≤ b < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since H0(aH + bA) = 0 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='18), we get that (n − 2)k − 2 n + 2 − 1 < a ≤ m(H, A) − 1 giving (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now suppose that A is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n ≥ 3 we know that (n − 2)k − 2 ≥ 0, contradicting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This proves (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see (iii), notice that if n ≥ 12, then k ≥ n + 2 by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='10(ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence (n−2)k−2 n+2 > n − 3 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='18) gives that 0 = H0((nk − 2k − 2)s n + 2 A) = H0((n − 3)H + qA) for some q ≥ 1, contradicting the hypothesis m(H, A) ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Curves Throughout this section we will have that X ⊆ PN is a smooth irreducible curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' It follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9 that if n = 1 and TX(k) is an Ulrich line bundle, then (X, H, k) = (P1, OP1(1), −2), (P1, OP1(3), 0) or k ≥ 2 and g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We will give below examples with k = 2, 3, essentially on any curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then we will give a sharp bound on k depending on the genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The case k = 2 can be characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that g ≥ 3 when k = 2 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX(2) is Ulrich if and only if there exists M ∈ Pic(X) such that Hi(M) = 0 for i ≥ 0 and H = KX + M is very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This occurs if and only if g ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(2) is Ulrich, set M = H −KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then Hi(M) = Hi(TX(1)) = 0 for i ≥ 0 and KX +M = H is very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Vice versa let M ∈ Pic(X) be such that Hi(M) = 0 for i ≥ 0 and H = KX + M is very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then Hi(TX(1)) = Hi(M) = 0 for i ≥ 0, so that TX(2) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that g ≥ 3 and let M ∈ Pic(X) be such that Hi(M) = 0 for i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We claim that H := KX + M is very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In fact deg M = g − 1 by Riemann-Roch, hence deg H = 3g − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If g ≥ 4, then deg H ≥ 2g + 1, hence H is very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If g = 3 we have that deg H = 2g and, as is well known, H is very ample unless H = KX + P + Q for two points P, Q ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But then M = P + Q is effective, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Instead, if g = 2 we have that deg(KX + M) = 3 > 2g − 2, hence h0(KX + M) = 2 by Riemann-Roch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let P + Q + R be an effective divisor linearly equivalent to KX + M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then KX + M − P − Q ∼ R, hence h0(KX + M − P − Q) = 1 and therefore KX + M is not very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The case k = 3 occurs on any curve X with (necessarily) odd genus g ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This was suggested to us by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Sernesi, whom we thank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let d = 3(g−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We claim that a general H ∈ Picd(X) is very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In fact, first observe that H1(H) = 0, for otherwise KX − H ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But KX − H is a general line bundle of degree g−1 2 ≤ g − 1, hence h0(KX − H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now, if H were not very ample, there will be two points p, q ∈ X such that h0(H − p − q) ≥ h0(H) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But this can be rewritten, by Riemann-Roch, as h1(H − p − q) ≥ 1, that is KX − H + p + q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence there are some points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' , p g+3 2 ∈ X such that KX − H + p + q ∼ p1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' + p g+3 2 that is H ∼ KX − p1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' − p g+3 2 + p + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 11 This means that H is in the image of the morphism h : X g+7 2 → Picd(X) sending (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' , p g+3 2 , p, q) to KX − p1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' − p g+3 2 + p + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But dim Imh ≤ g+7 2 < g, contradicting that H is general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This proves that there is a non-empty open subset W of Picd(X) such that any H ∈ W is very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consider the surjective morphism ψ : Picd(X) → Pic3g−3(X) given by ψ(L) = 2L and the isomor- phism ϕ : Pic3g−3(X) → Picg−1(X) given by ϕ(L) = L − KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let U be the non-empty open subset of Picg−1(X) such that Hi(M) = 0 for i ≥ 0 for any M ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now let H ∈ ψ−1(ϕ−1(U)) ∩ V ∩ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then H is very ample and 2H = KX + M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In the embedding given by H we have that Hi(TX(2)) = Hi(−KX + 2H) = Hi(M) = 0 for i ≥ 0, hence TX(3) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If k ≥ 1 and g − 1 is a prime number, then k ∈ {2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) we get that (k − 1)d = 3(g − 1) and g ≥ 2, hence d ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If 3 does not divide k − 1 we get that 3 divides d and (k − 1)d 3 = g − 1, so that k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If 3 divides k − 1 we get that k−1 3 d = g − 1, so that k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Every odd k ≥ 3 occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let E be an elliptic curve, let D be a divisor of degree 3 on E and let S = E × P1 with two projections π1 : S → E, π2 : S → P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Set C0 = π∗ 2(OP1(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then H = C0 + π∗ 1D is very ample on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let M ∈ Pic0(E) be not 2-torsion and let B = k−1 2 D + M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Again H1 := (k + 2)C0 + π∗ 1B is very ample on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Set L = −KS + (k − 1)H = (k + 1)C0 + π∗ 1(2B − 2M) so that L − H1 = −C0 + π∗ 1(B − 2M) while L − 2H1 = −(k + 3)C0 + π∗ 1(−2M) and it is easily seen by the K¨unneth formula, that Hi(L − pH1) = 0 for i ≥ 0, 1 ≤ p ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence, if X ∈ |H1| is a smooth irreducible curve, the exact sequence 0 → L − 2H1 → L − H1 → TX(k − 1) → 0 shows that Hi(TX(k − 1)) = 0 for i ≥ 0, that is TX(k) is an Ulrich line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ We now give a bound for k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We first analyze a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We use the notation (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' b1, b2, b3, b4, b5, b6) ∈ Z7 for the divisor aε∗L − �6 i=1 biEi on a smooth cubic W ⊂ P3, where ε : W → P2 is the blow up in six points, no three collinear and not on a conic, with exceptional divisors Ei and L is a line in P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊂ P3 be a smooth irreducible curve of genus 3 and degree 6 lying on a smooth cubic W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX(2) is Ulrich if and only if X is linearly equivalent to one of the following divisors on W: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) (4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 1, 1, 1, 1, 1), (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 2, 2, 1, 1, 1), (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 2, 2, 2, 2, 1), (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 2, 2, 2), (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 3, 3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let D = −KW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have that D · X = 6 and X2 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus Riemann-Roch gives that χ(2D − X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Further, D(2D − X) = 0, whence H0(2D − X) = 0, for otherwise X ∼ 2D and then X2 = 12, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, D(−3D + X) = −3, whence H0(−3D + X) = 0, which, be Serre’s duality, is H2(2D − X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, also H1(2D − X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now the exact sequence 0 → 2D − 2X → 2D − X → TX(1) → 0 gives that h1(TX(1)) = h2(2D − 2X) = h0(3KW + 2X) by Serre’s duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since deg TX(1) = 2, we deduce by Riemann-Roch, that TX(2) is Ulrich if and only if H1(TX(1)) = 0, hence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) TX(2) is Ulrich if and only if H0(3KW + 2X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' b1, · · · , b6) with b1 ≥ b2 ≥ · · · ≥ b6 ≥ 0 be the class of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' It follows from the assumption on degree and genus that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7) holds, whence X is as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2), it remains to show that H0(3KW + 2X) = 0 in all of these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY In case (4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 1, 1, 1, 1, 1), we have that 3KW + 2X = (−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 1, 1, 1, 1, 1) is clearly not effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In case (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 2, 2, 1, 1, 1), we have that 3KW + 2X = (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' −1, −1, −1, 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If it were effective, then so would be (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 0, 0, 0, 1, 1, 1), a contradiction since no three blown-up points are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In case (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 2, 2, 2, 2, 1), we have that 3KW + 2X = (3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 1, 1, 1, 1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume it is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In- tersecting with (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 1, 0, 0, 0, 0) we see that (2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 0, 1, 1, 1, −1) must be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Intersecting the latter with (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 0, 1, 0, 0, 0) we conclude that (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 0, 0, 1, 1, −1) must be effective, hence also (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 0, 0, 1, 1, 0), a contradiction since no three blown-up points are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In case (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 2, 2, 2), we have that 3KW + 2X = (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume it is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In- tersecting with (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 1, 0, 0, 0, 0) we see that (4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 2, 3, 1, 1, 1) must be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Intersecting the latter with (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 0, 1, 0, 0, 0) we conclude that (3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 2, 2, 1, 1, 1) must be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Finally, intersecting with (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 0, 1, 1, 0, 0, 0) we conclude that (2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 1, 1, 1, 1, 1) must be effective, a contradiction since the blown- up points do not lie on a conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In case (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 3, 3, 3), we have that 3KW +2X = (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 3, 3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Observe that D(3KW +2X) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, if 3KW + 2X were effective, it would contain a divisor Γ that is either irreducible, or is a union of three lines, or is a union of a line and a conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now pa(Γ) = −3, hence Γ is not irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, since the first coefficient of a line in W is at most 2 and of a conic at most 3, we see that Γ, whose first coefficient is 7, is not a union of three lines nor of a line and a conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This contradiction shows that 3KW + 2X is not effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Now we give the sharp bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' First assume that g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) gives that k ≤ 0, that is the required bound, and if equality holds, then Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9 shows that X is a curve of type (1, 2) on a smooth quadric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i), we can now assume that g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Observe that h0(H) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In fact, the only possibility remaining is that h0(H) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But then KX = (d−3)H, g = �d−1 2 � and k = 3(d−3) 2 +1 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now 0 = H0(TX(k −1)) = H0((−d+2+k)H) and therefore −d + 2 + k ≤ −1, giving the contradiction d ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now, if X has general moduli, since it has a g3 d, the Brill-Noether theorem implies that ρ(g, 3, d) ≥ 0, that is d ≥ 3g+12 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) we get that 3(g−1) k−1 ≥ 3g+12 4 , that gives k ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This proves the last assertion of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Turning to the first assertion, let X ⊆ PN be a smooth irreducible curve of genus g ≥ 2 such that TX(k) is an Ulrich line bundle and assume that k ≥ √8g + 1 − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i), the above inequality can be rephrased as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) g ≥ 2 9d2 − d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consider a general projection X′ of X to P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that X′ ∼= X, hence TX′(k) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We first observe that X′ cannot be a complete intersection (hence, in particular, X′ is nondegenerate), for otherwise TX′(k) = lH for some l ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now TX′(k), being Ulrich, is globally generated by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(vi), hence l ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also 0 = H0(TX′(k − 1)) = H0((l − 1)H) and therefore l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(vii) gives that d = h0(TX′(k)) = 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) and Castelnuovo’s bound, we get that either (d, g, k) = (6, 3, 2) or d ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that d ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We aim to show that X′ must lie on a smooth quadric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To this end, observe that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) imply that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) g > � 1 6d(d − 3) + 1 if d ≡ 0 (mod 3) 1 6d(d − 3) + 1 3 if d ≡ 1, 2 (mod 3) unless d = 9 and g = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But in the latter case it is easy to show that if X′ does not lie on a quadric, then it is a complete intersection of two cubics, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) and [Ha2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2] give that X′ lies on a quadric Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover Q is smooth, for otherwise it must be a cone, d = 2b + 1 is ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 13 odd and g = b2 − b by [Ha1, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) gives that 4(k − 1) = 6b − 9 − 3 2b+1, and therefore b = 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus X′ is a curve of type (a, b) on Q, with 2 ≤ a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In particular X′ is linearly normal, hence X = X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In the exact sequence 0 → OQ(k + 1 − 2a, k + 1 − 2b) → OQ(k + 1 − a, k + 1 − b) → TX(k − 1) → 0 since H0(TX(k − 1)) = 0, we get that H0(OQ(k + 1 − 2a, k + 1 − 2b)) = H0(OQ(k + 1 − a, k + 1 − b)) hence k + 1 − b ≤ −1, for otherwise k + 1 − a ≥ k + 1 − b ≥ 0, but then X is a base-component of |OQ(k + 1 − a, k + 1 − b)|, contradicting the fact that this linear system is base-point-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore b ≥ k + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) can be rewritten now as (a + b)(k − 1) = 3((a − 1)(b − 1) − 1) that is a = b(k + 2) 3b − k − 2 and it is readily seen that b ≥ k + 2 is equivalent to a ≤ k 2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore b ≥ 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But the maximum genus of a curve of type (a, b) with b ≥ 2a and degree d is attained when b = 2 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore g ≤ (1 3d − 1)(2 3d − 1) = 2 9d2 − d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This shows that the inequality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) cannot be strict, and therefore g ≤ 2 9d2−d+1, which is equivalent to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover, if equality holds in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1), then it holds in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) and therefore X is a curve of type (a, b) with b = 2 3d, hence b = 2a and 2a = b ≥ k + 2 ≥ 2a, so that k is even, a = k 2 + 1 and b = k + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Next consider the only remaining case, (d, g, k) = (6, 3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Again X′ is linearly normal, hence X = X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also we have equality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) and if X lies on a quadric, then it must be of type (2, 4) and we are done in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose therefore that X does not lie on a quadric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then it is easily seen that JX/P3(3) is 0-regular, hence globally generated, and we get that X is contained in a smooth cubic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore X is one of the curves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5 and TX(2) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Finally, to show that the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) is sharp for every even k ≥ 0, let X be a curve of type (k 2 + 1, k + 2) on a smooth quadric Q ⊂ P3, so that k = √8g+1−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' It remains to show that TX(k) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Set k = 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have TX(k − 1) = −KX + (k − 1)H = OQ(c, −1)|X and the exact sequence 0 → OQ(−1, −2c − 3) → OQ(c, −1) → OQ(c, −1)|X → 0 shows that Hi(OQ(c, −1)|X) = 0 for i ≥ 0, since Hi(OQ(c, −1)) = Hi(OQ(−1, −2c − 3)) = 0 for i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence TX(k) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Surfaces Throughout this section we will have that X ⊆ PN is a smooth irreducible surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We start by a characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX(k) is an Ulrich vector bundle if and only if (i) d = 4(g−1) 2k−1 (ii) HKX = (2k−3)d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) K2 X = 5χ(OX) + (k−1)(k−2)d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iv) H0(TX(k − 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (v) H2(TX(k − 2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that (i) and (ii) are equivalent, since HKX = 2(g − 1) − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now (ii) and (iii) are the conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) in [C1, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence the lemma follows by loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY Now we show the possible cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(k) is an Ulrich vector bundle, the following hold: (i) 0 ≤ k ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover, either (ii) k = 0 and (X, H) = (P2, OP2(2)), or (iii) k = 1 and X is a Del Pezzo surface of degree 5, or (iv) k = 2, q = 0 and X is a minimal surface of general type, or (v) k = 3, X is a minimal surface of general type with 2KX ≡ 3H, K2 X = 9d 4 , χ(OX) = d 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover X is a ball quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have that k ≥ 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now H1(TX) = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(iv), that is X is infinitesimally rigid and [BC, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3] implies that either X is a minimal surface of general type or X is a Del Pezzo surface of degree j ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In the latter case we have that HKX < 0 hence either k = 0 and we get (ii) by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9, or k = 1 and K2 X = 5 by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1(ii),(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This gives (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, if X is a minimal surface of general type then HKX > 0, hence k ≥ 2 by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Next, the Hodge index theorem H2K2 X ≤ (HKX)2 can be rewritten, using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1(ii),(iii) as χ(OX) ≤ (2k2 − 6k + 5)d 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Similarly, the Bogomolov-Miyaoka-Yau inequality K2 X ≤ 9χ(OX) can be rewritten as χ(OX) ≥ (k2 − 3k + 2)d 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Combining we get that (k2 − 3k + 2)d 8 ≤ (2k2 − 6k + 5)d 20 and this gives that k ≤ 3 and moreover that, if k = 3, then equality holds in both inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence, when k = 3 we have, as is well known, that X is a ball quotient and that H2KX ≡ (HKX)H, that is 2KX ≡ 3H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then K2 X = 9d 4 and χ(OX) = d 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus (i) and (v) are proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Alternatively (i) follows by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see (iv) observe that since k = 2 we have by the above that X is a minimal surface of general type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now if pg = 0 then q = 0 by [Be2, Lemma VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If pg ̸= 0 we have an inclusion H0(Ω1 X) ⊆ H0(Ω1 X(KX)) hence q = h0(Ω1 X) ≤ h0(Ω1 X(KX)) = h2(TX) = 0 since TX(2) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This proves (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ We now characterize the case k = 1 for surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then TX(1) is an Ulrich vector bundle if and only if X is a Del Pezzo surface of degree 5 and H = −2KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover in the latter case TX(1) is very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(1) is an Ulrich vector bundle, then Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2 implies that X is a Del Pezzo surface of degree 5 and H2 + 2HKX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let ε : X → P2 be the blow-up map, with exceptional divisors Ei over the points Pi ∈ P2, 1 ≤ i ≤ 4 and let L be a line in P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then we can write H ∼ aε∗L − 4 � i=1 biEi and, as H is very ample, we have, without loss of generality, b1 ≥ b2 ≥ b3 ≥ b4 ≥ 1, a ≥ b1 + b2 + 1 and H2 + 2HKX = 0 is a2 − 6a + 4 = 4 � i=1 (bi − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Setting ci = bi − 1, we get by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1 the following possibilities: (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' b1, b2, b3, b4) ∈ {(6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 1, 1, 1), (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 2, 2, 2), (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4, 2, 2, 1), (9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4, 4, 4, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 15 In the case (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 2, 2, 2) we have that H = −2KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We now exclude the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let H = 6ε∗L − 3E1 − E2 − E3 − E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We will prove that h2(TX(−1)) = h0(Ω1 X(H + KX)) ̸= 0, so that TX(1) cannot be an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To this end observe that, since ε∗Ω1 P2 ⊂ Ω1 X, we will be done in this case if we prove that H0(ε∗Ω1 P2(H + KX)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now H + KX = 3ε∗L − 2E1, hence H0(ε∗Ω1 P2(H + KX)) ∼= H0(IZ ⊗ Ω1 P2(3)) where Z ⊂ P2 is the 0-dimensional subscheme of length 2 supported on P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Finally h0(IZ ⊗ Ω1 P2(3)) ≥ h0(Ω1 P2(3)) − 6 = 2 > 0 and we are done in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consider now the exact sequences, for any 1 ≤ i ≤ 4, 0 → OEi(−Ei) → Ω1 X|Ei → Ω1 Ei → 0 that is 0 → OP1(1) → Ω1 X|Ei → OP1(−2) → 0 from which we get, for any 1 ≤ i ≤ 4, that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) h1(Ω1 X|Ei) = 1 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) H1(Ω1 X |Ei ⊗ OP1(2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In the two remaining cases we will prove that h1(TX(−1)) = h1(Ω1 X(H + KX)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let H = 7ε∗L − 4E1 − 2E2 − 2E3 − E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that H + KX − E4 + E1 = 4ε∗L − 2E1 − E2 − E3 − E4 is very ample by [DR, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6], hence H2(Ω1 X(H + KX − E4 + E1)) = 0 by Bott vanishing [T, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then the exact sequence 0 → Ω1 X(H + KX − E4) → Ω1 X(H + KX − E4 + E1) → Ω1 X|E1(H + KX − E4 + E1) → 0 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) imply that H2(Ω1 X(H + KX − E4)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now the exact sequence 0 → Ω1 X(H + KX − E4) → Ω1 X(H + KX) → Ω1 X|E4(H + KX) → 0 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) imply that h1(Ω1 X(H + KX)) ≥ h1(Ω1 X |E4(H + KX)) = h1(Ω1 X|E4) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let H = 9ε∗L − 4E1 − 4E2 − 4E3 − 3E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let C ∈ |ε∗L − E2 − E3| be the strict transform of a line through P2 and P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that H + KX − C + E1 = 5ε∗L − 2E1 − 2E2 − 2E3 − 2E4 is very ample by [DR, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6], hence H2(Ω1 X(H + KX − C + E1)) = 0 by Bott vanishing [T, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then the exact sequence 0 → Ω1 X(H + KX − C) → Ω1 X(H + KX − C + E1) → Ω1 X|E1(H + KX − C + E1) → 0 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) imply that H2(Ω1 X(H + KX − C)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now the exact sequence 0 → OC(−C) → Ω1 X|C → Ω1 C → 0 that is 0 → OP1(1) → Ω1 X|C → OP1(−2) → 0 gives that h1(Ω1 X |C) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Finally, from the exact sequence 0 → Ω1 X(H + KX − C) → Ω1 X(H + KX) → Ω1 X|C(H + KX) → 0 using that (H + KX)C = 0, we get that h1(Ω1 X(H + KX)) ≥ h1(Ω1 X|C(H + KX)) = h1(Ω1 X|C) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY This completes the proof under the assumption that TX(1) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose now that X is a Del Pezzo surface of degree 5 and H = −2KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Setting k = 1 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1, we have that d = 4(g − 1) and, in order to verify that TX(1) is an Ulrich vector bundle, we need to check that H0(TX) = H2(TX(−1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The first vanishing is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' As for the second, we first observe that for i < 2 we have hi(TX(−1)) = h2−i(Ω1 X(H + KX)) = h2−i(Ω1 X(−KX)) = 0 by Bott vanishing [T, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore h2(TX(−1)) = χ(TX(−1)) = d − 4(g − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Finally, as X does not contain lines in the embedding given by H = −2KX, we have that TX(1) is very ample by [LS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1] □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Properties of complete intersections We collect some properties inherited by the complete intersections Xi of X (as in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1), when TX(k) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then q(X) = q(Xi) for 2 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' By Kodaira vanishing we have that H1(OXi+1(−1)) = H2(OXi+1(−1)) = 0 as long as 2 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then the exact sequences 0 → OXi+1(−1) → OXi+1 → OXi → 0 imply that h1(OXi+1) = h1(OXi) for every 2 ≤ i ≤ n − 1, hence q(X) = q(Xi) for 2 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that k ≤ n − 2 and that TX(k) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then Hi(OXi) = 0 for all i such that max{1, k + 1} ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume that max{1, k + 1} ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since TX|Xi(k) is Ulrich, it follows by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(iv) that Hi(TX |Xi(k + m)) = 0 for all m ≥ −i, hence Hi(TX |Xi(−1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now the exact sequence 0 → TXi(−1) → TX|Xi(−1) → O⊕(n−i) Xi → 0 implies that Hi(OXi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that k ≤ n − 1 and that TX(k) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume that Hi(OX) = 0 for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then Hi(OXj) = 0 for all i ≥ 1 and for all j such that max{1, k + 1} ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume that i ≥ 1 and max{1, k + 1} ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We prove the lemma by induction on n − j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n − j = 0 then Xj = Xn = X and Hi(OX) = 0 for all i ≥ 1 just by our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Next suppose that n − j ≥ 1, so that max{1, k + 1} ≤ j ≤ n − 1, hence, in particular k ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consider the exact sequence 0 → OXj+1(−1) → OXj+1 → OXj → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If j = i, we have that Hj(OXj) = 0 by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, we have by induction that Hi(OXj+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now Hi+1(OXj+1(−1)) = 0 by Kodaira vanishing if i+1 < j+1 and by dimension reasons if i+1 > j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus Hi(OXj) = 0 if i ̸= j and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ We now collect some properties of the Xi’s that hold when TX(1) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that n ≥ 2 and that TX(1) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then: (i) H1(OXi) = 0 for 2 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) H2(OXi) = 0 for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) H1(OXi(1)) = 0 for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iv) H2(OXi(1)) = 0 for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (v) h0(OXi(1)) = d − g + i for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (vi) d ≥ n + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have Hi(OX) = 0 for i ≥ 1 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now (i) follows by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1 and (ii) follows by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see (iii) observe that, if i = 1 we have that X1 = C and TX(1)|C is an Ulrich vector bundle on C by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(ix), hence H1(TX |C) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then the exact sequence 0 → TC → TX|C → OC(1)⊕(n−1) → 0 shows that H1(OC(1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If i ≥ 2, since H1(OXi) = 0 by (i), the exact sequences (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) 0 → OXi → OXi(1) → OXi−1(1) → 0 imply by induction that H1(OXi(1)) = 0 and we get (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now (iv) is obvious for i = 1, while, for i ≥ 2, the exact sequences (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) and (ii) show by induction that H2(OXi(1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This proves (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that (v) follows for i = 1 by Riemann-Roch and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' For i ≥ 2, the exact sequences (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) and (i) show by induction that h0(OXi(1)) = 1 + h0(OXi−1(1)) = d − g + i, that is (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Finally, to see (vi), observe that g − 1 = n−1 n+2d by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i), hence g ≥ 2 and (v) gives that 3d n+2 = h0(OC(1)) ≥ 3, so that d ≥ n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover, if equality holds, we get that g = n and h0(OX(1)) = n + 2 by (v), hence X ⊂ PH0(H) = Pn+1 is a hypersurface of degree n + 2, so that KX = 0, contradicting Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence (vi) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' TX(k) Ulrich and special varieties in adjunction theory In this section we exclude some special varieties frequently arising in adjunction theory, under the hypothesis that TX(k) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The cases (X, H) = (Pn, OPn(1)), (Qn, OQn(1)) have been already treated in Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We start by recalling the following (see [BS, I]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let E be an effective divisor on (X, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The divisor E is called exceptional (i) of type 1 if (E, H|E) ∼= (Pn−1, OPn−1(1)) and NE/X ∼= OPn−1(−1), (ii) of type 2 if (E, H|E) ∼= (Pn−1, OPn−1(1)) and NE/X ∼= OPn−1(−2), (iii) of type 3 if (E, H|E) ∼= (Qn−1, OQn−1(1)) and NE/X ∼= OQn−1(−1), (iv) of type 4 if (E, H|E) is a linear Pn−2-bundle over a smooth curve B and (NE/X)|F ∼= OPn−2(−1), where F is a fiber of the structure morphism E → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Often these exceptional divisors will not be present under the condition that TX(k) is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see this we first prove Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let W be a variety of dimension s ≥ 1 and let OW (1) be a very ample line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then Ω1 W(1) is not globally generated if: (i) (W, OW (1)) ∼= (Ps, OPs(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) (W, OW (1)) is a (possibly singular) quadric hypersurface in Ps+1 and s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) (W, OW (1)) is a smooth Del Pezzo variety, s ≥ 2 and (W, OW (1)) ̸∈ {(P2, OP2(3)), (Q2, OQ2(2)), (P3, OP3(2))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (i) follows from det(Ω1 Ps(1)) = OPs(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see (ii), observe that the restricted Euler sequence 0 → Ω1 Ps+1|W (1) → H0(OW (1)) ⊗ OW → OW (1) → 0 implies that H0(Ω1 Ps+1|W(1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now the exact sequence 0 → OPs+1(−3) → OPs+1(−1) → OW (−1) → 0 implies that H1(OW (−1)) = 0 and the dual normal bundle sequence 0 → OW (−1) → Ω1 Ps+1|W(1) → Ω1 W(1) → 0 gives that H0(Ω1 W (1)) = 0, hence Ω1 W(1) is not globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Next, to see (iii), observe that, from the classification of Del Pezzo varieties [IP, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1] it follows, for the surface section W2, that (W2, OW2(1)) ̸∈ {(P2, OP2(3)), (Q2, OQ2(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence, as is well known, W2, and hence W, contains a line L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But now the surjection Ω1 W(1) → Ω1 L(1) = OP1(−1) gives that Ω1 W(1) is not globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Now 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume that TX(k) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have: (i) If k ≥ 1 and n ≥ 2, then (X, H) does not contain any exceptional divisor of type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) If k ≥ 2 and n ≥ 2, then (X, H) does not contain any exceptional divisor of type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) If k ≥ 2 and n ≥ 3, then (X, H) does not contain any exceptional divisors of types 3 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let E be an exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' It follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6 that Ω1 E(KX |E + (n + 1 − k)H|E) is globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now Ω1 E(KX |E + (n + 1 − k)H|E) ∼= \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Ω1 Pn−1(2 − k) if E is of type 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Ω1 Pn−1(3 − k) if E is of type 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Ω1 Qn−1(3 − k) if E is of type 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Further, when E is of type 4, let F be a fiber of the structure morphism of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Again it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6 that Ω1 F (KX |F + (n + 1 − k)H|F ) ∼= Ω1 Pn−2(3 − k) is globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently, we draw the conclusions from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ We now recall Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We say that (X, H) is a linear Pk-bundle over a smooth variety B if (X, H) ∼= (P(F), OP(F)(1)), where F is a very ample vector bundle on B of rank k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We say that (X, H) as above is a scroll (respectively a quadric fibration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' respectively a Del Pezzo fibration) over a normal variety Y of dimension m if there exists a surjective morphism with connected fibers φ : X → Y such that KX +(n−m+1)H = φ∗L (respectively KX +(n−m)H = φ∗L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' respectively KX + (n − m − 1)H = φ∗L), with L ample on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We now use the fibration to exclude several varieties as above, when TX(k) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume that TX(k) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let f : X → B be a fibration onto a normal variety B of dimension m ≥ 1, with general fiber F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then: (i) If m ≤ min{n − 1, k + 1}, then (F, H|F ) ̸= (Pn−m, OPn−m(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) If m ≤ min{n − 2, k}, then (F, H|F) ̸= (Qn−m, OQn−m(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) if m ≤ min{n − 2, k − 1}, then (F, H|F) is not a Del Pezzo variety, unless (F, H|F) ∈ {(P2, OP2(3)), (Q2, OQ2(2)), (P3, OP3(2))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have that Ω1 F(KF + (n + 1 − k)H|F) ∼= \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Ω1 Pn−m(m − k) if (F, H|F) = (Pn−m, OPn−m(1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Ω1 Qn−m(m − k + 1) if (F, H|F) = (Qn−m, OQn−m(1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Ω1 F(m − k + 2) if (F, H|F) is a Del Pezzo variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now Ω1 F(KF + (n + 1 − k)H|F ) is globally generated by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2 gives that, in each of the three cases, the inequality in m, n, k is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ We get a very useful consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(k) is an Ulrich vector bundle, then KX +(n−1)H is nef and H0(KX +(n−1)H) ̸= 0, unless (X, H, k) = (P2, OP2(2), 0) (the latter case actually occurs, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Recall that H0(KX + (n − 1)H) ̸= 0 if and only if KX + (n − 1)H is nef by [BS, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now if KX + (n − 1)H is not nef, it follows by [BS, Prop.’s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4] that (X, H) is either (Pn, OPn(1)), (Qn, OQn(1)), a linear Pn−1-bundle over a smooth curve or (P2, OP2(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The first three cases are excluded by Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5(i), while in the fourth case we have g = 0, hence k = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ We can now prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 19 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The assert is clear if either (X, H) = (P1, OP1(3)) or (P2, OP2(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Vice versa assume that TX is Ulrich for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n = 1, since TX = −KX is globally generated by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(vi), we have that X is either P1 or an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now the latter is excluded by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i), while in the former case TX = OP1(2) Ulrich implies that H = OP1(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now assume that n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6 gives that either (X, H) = (P2, OP2(2)), or KX + (n − 1)H is nef, leading, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii), to the contradiction 0 ≤ (KX + (n − 1)H)Hn−1 = − 2d n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ The following result will also be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that k ≥ 1 and that TX(k) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume that X ∼= P(F) is a projective bundle over a normal projective variety B of dimension 1 ≤ m ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then B is smooth and F is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In particular, if m = 1, then q(X) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let π : X ∼= P(F) → B be the structure morphism and let ξ be the tautological bundle of P(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' By twisting F with a sufficiently ample line bundle we can assume that ξ is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then [BS, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1] implies that B is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since H0(TX) = 0, the cohomology of the exact sequence 0 → TX/B → TX → π∗TB → 0 gives that H0(TX/B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now the cohomology of the exact sequence 0 → OX → π∗F∗ ⊗ ξ → TX/B → 0 implies that h0(F ⊗ F∗) = h0(π∗F∗ ⊗ ξ) = h0(OX) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now if m = 1 and q(X) = 0 we have that B ∼= P1, hence F cannot be simple since rk F = n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Next we prove three results for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' For the first one, in order to apply the results of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Fujita in [F1], we give the following definition, that coincides with the one in [F1] when B is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let f : X → B be a fibration over a curve, L an ample line bundle on X such that on the general fiber F we have that KF = −(n − 2)L|F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We say that f is minimal if there is a line bundle L on B such that KX + (n − 2)L = f ∗L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then we have Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that k ≥ 2 and that k = 2 if n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover assume that TX(k) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then: (i) (X, H) is not a Del Pezzo fibration over a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) If n = 4 and KX + 2H is ample, then (X, KX + 2H) is not a minimal (P3, OP3(2))-fibration over a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' For the sake of contradiction, let L be H in case (i) and KX + 2H in case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume that we have a fibration f : X → B over a smooth curve B such that (X, L) is a Del Pezzo fibration in case (i) (see Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) and (X, L) is a minimal (P3, OP3(2))-fibration in case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that f is minimal also in case (i) by Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let F be a general fiber of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In case (i) we have that F is a smooth variety of dimension n − 1 and KF = KX|F = −(n − 2)L|F, hence F is a Del Pezzo variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since L = H, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5 implies that (F, H|F) = (P3, OP3(2)), hence n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus (F, L|F ) is the same in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We now claim that every fiber of f is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Indeed, if not, let F0 be a reducible fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since F0 is connected, it must be singular, hence we can apply [F1, Table (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='20)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' It follows that we are in case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='17) of [F1, Table (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='20)], the degree of F0 is 8 and, if D is an irreducible component of F0, then (D, L|D) is a scroll over P1 and KX |D = −2L|D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Denoting a fiber of the structure morphism D → P1 by F ′ ∼= P2, we obtain L|F ′ = OP2(1) and KX|F ′ = −2L|F ′ = OP2(−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Set H|F ′ = OP2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In case (ii) we have that OP2(1) = L|F ′ = (KX + 2H)|F ′ = OP2(−2 + 2a) 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In case (i) we have that L = H and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But now Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6 gives that Ω1 P2(KX |F ′ + 3H|F ′) ∼= Ω1 P2(1) is globally generated, contradicting Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus every fiber of f is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now [F1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8)] implies that every fiber of f is P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since B is a smooth curve, it follows, as is well known, that X is a projective bundle over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, since n = 4, we have that k = 2 and q(X) = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(iii), contradicting Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that KX + 2H is ample and that TX(2) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then (X, KX + 2H) is not a quadric fibration over a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that (X, KX + 2H) is a quadric fibration π : X → B over a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since χ(OX) = 1 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(iii), it follows from [Lan, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (8)] that B ∼= P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Moreover, [Lan, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1] gives that if π∗(KX + 2H) ∼= 4� i=0 OP1(ai) and e = �4 i=0 ai, then there is b ∈ Z such that (KX + 2H)4 = 2e − b by [Lan, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (3)] and Ki X(KX + 2H)4−i = (−3)i2e + (−3)i−1(−4i + 2ie + (3 − 2i)b) for 1 ≤ i ≤ 4 by [Lan, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Solving these five equations we obtain KXH3 = 4e − 28b − 104 and d = H4 = 16b + 64 and therefore Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(iii) gives (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) 13d = 48(2 + e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since TX(2) is Ulrich we have that H4(TX(−2)) = 0 and the exact sequence 0 → TX/P1(−2) → TX(−2) → (π∗TP1)(−2) → 0 implies that H4((π∗TP1)(−2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence, by Serre duality 0 = h4((π∗TP1)(−2)) = h0((π∗OP1(−2))(KX+2H)) = h0(π∗(KX+2H)⊗OP1(−2)) = 4 � i=0 h0(OP1(ai−2)) and therefore ai ≤ 1 for 0 ≤ i ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But then e ≤ 5 and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) gives that 1 ≤ e + 2 ≤ 7 is divisible by 13, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose that KX + 2H is ample and that TX(2) is an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then (X, KX + 2H) is not a linear P2-bundle over a smooth surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume by contradiction that we have a P2-bundle structure π : X ∼= P(F) → B onto a smooth surface B, with KX + 2H = ξ, the tautological bundle, where F is a rank 3 vector bundle on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then H = aξ − π∗M for some a ∈ Z and M ∈ Pic(B), so that ξ = KX + 2H = (2a − 3)ξ + π∗(KB + c1(F) − 2M) giving a = 2 and 2M = KB + c1(F), thus (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) H ≡ 2ξ − 1 2π∗(KB + c1(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We will also use Grothendieck’s relation 3� j=0 (−1)jξ3−jπ∗cj(F) = 0, that is (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) ξ3 = ξ2π∗c1(F) − ξπ∗c2(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since ξ2f = 1 for every fiber f of π, we get from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) ξ3π∗c1(F) = c1(F)2, ξ3π∗KB = KBc1(F) and ξ4 = c1(F)2 − c2(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We first collect some invariants of X and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have: ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 21 (i) KXH3 = − 2 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) χ(OX) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) χ(OX(H)) = 2 + χ(OS) − d 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iv) h0(KX + 2H) = χ(OS) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (v) χ(OB) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (i) is obtained by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(iii) gives that Hi(OX) = 0 for i ≥ 1, hence Hi(OB) = 0 for i ≥ 1, giving (ii) and (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Next, to see (iii), consider the exact sequences 0 → OXi → OXi(H) → OXi−1(H) → 0 for i = 4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' They give χ(OX(H)) = 1+χ(OX3(H)) = 2+χ(OS(H)) and (iii) follows by Riemann-Roch since H2 |S = d and H|SKS = (KX + 2H)H3 = 4 3d by (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see (iv), observe that, since Rjπ∗(−ξ) = 0 for every j ≥ 0, we have that Hi(KX + H) = Hi(−ξ + π∗(KB + c1(F) − M)) = 0 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence the exact sequence 0 → KX + H → KX + 2H → KX3 + H|X3 → 0 implies that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5) h0(KX + 2H) = h0(KX3 + H|X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now we have q(S) = 0 by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1 and Hi(KX3) = 0 for i = 0, 1 by Serre duality and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence the exact sequence 0 → KX3 → KX3 + H|X3 → KS → 0 shows that χ(OS) − 1 = pg(S) = h0(KS) = h0(KX3 + H|X3) and we get (iv) by (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ We continue the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Next, we collect some relations among the invariants related to ξ, KB and the Chern classes of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The following identities hold: (i) d − 6c1(F)2 + 16c2(F) − 6K2 B + 4KBc1(F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) KBc1(F) − 8 + 2χ(OS) − c1(F)2 + 2c2(F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) K2 B − 1 + c1(F)2 − 3c2(F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iv) 3K2 B + c1(F)2 − 2c2(F) − 30 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (v) 4 − KBc1(F) + 9 4K2 B + 7 4c1(F)2 − 5c2(F) − χ(OS) + 1 6d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have by (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) that d = H4 = (2ξ − 1 2π∗(KB + c1(F)))4 = 16(c1(F)2 − c2(F)) + 6K2 B − 4KBc1(F) − 10c1(F)2 that is (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To see (ii) observe that, since π∗ξ = F and Rjπ∗ξ = 0 for j > 0 we have Hi(F) = Hi(ξ) = Hi(KX + 2H) = 0 for i > 0 by Kodaira vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, by Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='12(iv), (v) and Riemann-Roch we get χ(OS) − 1 = h0(KX + 2H) = h0(ξ) = h0(F) = χ(F) = 3 − 1 2KBc1(F) + 1 2c1(F)2 − c2(F) that is (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Next, consider the exact sequences (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6) 0 → TX/B → TX → π∗TB → 0 and 0 → OX → π∗F∗(ξ) → TX/B → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since TX(2H) is Ulrich, we have that χ(TX) = 0, hence, using Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='12(ii) we get (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7) χ(TB) = χ(π∗TB) = −χ(TX/B) = −χ(π∗F∗(ξ)) + 1 = −χ(F ⊗ F∗) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, by Riemann-Roch and Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='12(v), χ(TB) = 2K2 B − 10 and χ(F ⊗ F∗) = 9 + 2c1(F)2 − 6c2(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Replacing in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7) gives (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY Finally, to see (iv), we first compute c1(S2(F)) = 4c1(F) and c2(S2(F)) = 5c1(F)2 + 5c2(F), so that c1(S2(F)(−M)) = −3KB + c1(F), c2(S2(F)(−M)) = 15 4 K2 B − 5 2KBc1(F) − 5 4c1(F)2 + 5c2(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now Riemann-Roch gives (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8) χ(S2(F)(−M)) = 6 − KBc1(F) + 9 4K2 B + 7 4c1(F)2 − 5c2(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, χ(S2(F)(−M)) = χ(2ξ − π∗M) = χ(OX(H)) = 2 + χ(OS) − d 6 by Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='12(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Using (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8) we get (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ We now conclude the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Solving the five equations in Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='13 we get (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9) K2 B = − 7 48d + 7 and KBc1(F) = − 5 48d + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In particular d ≥ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, using Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='12(i), we get µ(TX) = −KXH3 4 = 1 6d and, using (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) µ(π∗TB) = c1(π∗TB)H3 2 = −π∗KB � 2ξ − 1 2π∗(KB + c1(F)) �3 2 = −8ξ3π∗KB − 6ξ2π∗(K2 B + KBc1(F)) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9) give µ(π∗TB) = −KBc1(F) + 3K2 B = −1 3d + 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since TX is semistable by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(v), we deduce by (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6) that 1 6d ≤ −1 3d + 12 that is d ≤ 24, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' TX(1) Ulrich in any dimension We study the case k = 1 in any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We start analyzing the properties of the curve section C and of the surface section S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(1) is an Ulrich vector bundle, then d ≥ 9 except, possibly, when d = 8, g = 5, n = 4 and h0(OC(1)) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) we know that g ≥ 2 and that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) (n − 1)d = (n + 2)(g − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4(v) we have d − g + 1 = h0(OC(1)) ≥ 3, hence g ≤ d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, if equality holds, then h0(OC(1)) = 3, so that d − 2 = g = �d−1 2 � , thus d = 3 and g = 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore 2 ≤ g ≤ d − 3, hence d ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But if d ≤ 8 the only possibility given by (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) is d = 8, g = 5, n = 4 and h0(OC(1)) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If TX(1) is an Ulrich vector bundle we have: (i) KSH|S = n−4 n+2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (ii) q(S) = pg(S) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iii) K2 S = − 3(n−2) 2(n+2)d − n−12 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (iv) S is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (i) follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii), while (ii) follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(iii), Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note now that the equation in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(vii) can be rewritten as 3(n − 2)d + 2(n + 2)K2 S + (n + 2)(n − 12) = 0 giving (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Finally assume that S is not ruled, so that κ(S) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6 gives that KS + H|S = (KX + (n − 1)H)|S is nef, hence KS(KS + H|S) ≥ 0, that is K2 S ≥ −KSH|S = − n−4 n+2d by (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then (iii) gives −3(n − 2) 2(n + 2)d − n − 12 2 = K2 S ≥ −n − 4 n + 2d so that (n + 2)(d + n − 12) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since n ≥ 3 it follows that d ≤ 9, and using Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1 we deduce that either d = 9, n = 3 or d = 8, g = 5, n = 4 and h0(OC(1)) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In the first case we get a contraction by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i), while in the second case d + n − 12 = 0, hence K2 S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' As C ⊂ P3 we deduce that S ⊂ P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But this contradicts the well-known formula for the invariants of a surface in P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore S is ruled, hence rational by (ii) and (iv) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ We are now ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n = 1 we know by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i) that TX(1) is not an Ulrich vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' If n = 2 this is Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose next that n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that H0(TX) = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(iii), hence X is neither Pn nor Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also q(X) = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have that (X, H) is not: (1) A projective bundle over a smooth curve by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (2) A Del Pezzo manifold by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i), since otherwise g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (3) A hyperquadric fibration over a smooth curve (in the sense of [I]), by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (4) A linear Pn−2-bundle over a smooth surface, by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also observe that X does not contain any exceptional divisor of type 1 by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence (X, H) is isomorphic to its reduction (X′, H′) (see [I, (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='11)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' It follows by [I, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7)] that KX +(n−2)H is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence S is minimal and rational by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(iv), a contradiction since a minimal rational surface does not have nef canonical bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus the case n ≥ 3 does not occur and the theorem is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' TX(2) Ulrich in any dimension We prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' It follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2(iii) that H0(TX) = 0, hence X is neither Pn nor Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Note that Hi(OX) = 0 for i ≥ 1 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(iii) and KX is not nef, since Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii) gives that KXHn−1 = n(3��n) n+2 d < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We divide the proof according to the value of τ(X, H) (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We will also use the notions of first and second reduction of (X, H), as defined in [BS, Defs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Case A: τ(X, H) ≥ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This case does not occur since Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6(iii) implies that τ(X, H) ≤ n − 2n n+1 < n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Case B: n − 2 ≤ τ(X, H) < n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then KX +(n−1)H is ample, hence the first reduction exists and is isomorphic to (X, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Therefore [BS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4] implies that τ(X, H) = n − 2 and then [BS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3] gives that (X, H) is one of the following: (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) a Mukai variety, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) a Del Pezzo fibration over a smooth curve, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) a quadric fibration over a normal surface, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) a scroll over a normal threefold, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5) (X, H) contains an exceptional divisor of type 2, 3, or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY Now, the case (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) is ruled out by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Case (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) is excluded for n = 4 by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9(i) and for n ≥ 5 by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also the cases (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) are ruled out by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5(ii) and (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Finally the case (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5) is excluded by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus also Case B does not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Case C: τ(X, H) < n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then the first and second reductions exist and are both isomorphic to (X, H), since KX + (n − 2)H is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We first claim that KX3 is not nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In fact, assume that KX3 is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the one hand, χ(OX3) = 1 by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On the other hand, 3c2(X3) − c1(X3)2 is pseff by [M, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1], hence 3c2(X3)KX3 ≥ K3 X3 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But then Riemann-Roch gives χ(OX3) = − 1 24c2(X3)KX3 ≤ 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence KX3 is not nef and [BS, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1] gives the following cases: (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) n = 5 and (X, KX + 3H) is a linear P4-bundle over a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) n = 4 and (X, KX + 2H) is a Del Pezzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) n = 4 and (X, KX + 2H) is a quadric fibration over a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) n = 4 and (X, KX + 2H) is a scroll over a normal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5) n = 4 and (X, H) contains an exceptional divisor of type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6) n = 4 and (X, KX + 2H) is a (P3, OP3(2))-fibration over a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In case (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) we have a contradiction by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In case (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) we have 4KX + 6H = 0, hence 4KXH3+6H4 = 0 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii) gives the contradiction d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Cases (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5) do not occur by Lemmas 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='10 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In Case (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6) we observe that the fibration is obtained in [F2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1)] by contracting an extremal ray, hence it is minimal (see Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8) and the image is a normal, hence smooth, curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus this case is excluded by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='9(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hence we are left with case (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have a surjective morphism π : X → B and denoting by F a general fiber, we have (F, (KX + 2H)|F ) ∼= (P2, OP2(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now all fibers of π are 2-dimensional by [BS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1], hence we get by [BS, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1] that B is a smooth surface and (X, KX + 2H) is a linear P2-bundle over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But this case is excluded by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This concludes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ References [A] 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London Mathematical Society Lecture Note Series, 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' iv+132 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 14 [BC] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Bauer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Catanese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On rigid compact complex surfaces and manifolds.' 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+page_content=' De Gruyter Exposi- tions in Mathematics, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Walter de Gruyter & Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=', Berlin, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5, 17, 18, 19, 23, 24 [C1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Casnati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Special Ulrich bundles on non-special surfaces with pg = q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 28 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 8, 1750061, 18 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 13 [C2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Casnati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Tangent, cotangent, normal and conormal bundles are almost never instanton bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Preprint 2022, in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1 [CH] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Casanellas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hartshorne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Stable Ulrich bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' With an appendix by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Geiss, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Schreyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 23 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 8, 1250083, 50 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 5 [CMRPL] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Costa, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Mir´o-Roig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Pons-Llopis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Ulrich bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' De Gruyter Studies in Mathematics, 77, De Gruyter 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1 [CP] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Campana, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' P˘aun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Foliations with positive slopes and birational stability of orbifold cotangent bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 129 (2019), 1-49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5 [DR] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Di Rocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' k-very ample line bundles on del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 179 (1996), 47-56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 15 [ES] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Eisenbud, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Schreyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Resultants and Chow forms via exterior syzygies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 16 (2003), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 537-579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 3 [F1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Fujita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On del Pezzo fibrations over curves.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Fulger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lehmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Morphisms and faces of pseudo-effective cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 112 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4, 651-676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5 [FL2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Fulger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lehmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Positive cones of dual cycle classes.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Logarithmic bounds on Fujita’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Preprint 2021, arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='11705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 8 [Ha1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Hartshorne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Algebraic geometry.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=', Nice, 1979), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 83-112, Progr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=', 7, Birkh¨auser, Boston, Mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=', 1980 12 [Ho] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' H¨oring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On a conjecture of Beltrametti and Sommese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Algebraic Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 21 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4, 721-751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 99 (1986), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 457-472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 17, 23 [IP] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Iskovskikh, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Prokhorov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Fano varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In: Algebraic geometry, V, 1-247, Encyclopaedia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 47, Springer-Verlag, Berlin, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 17 [K] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Kawamata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On effective non-vanishing and base-point-freeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Kodaira’s issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Asian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4 (2000), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 173-181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 7 [Lan] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 20 [Lop] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lopez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On varieties with Ulrich twisted normal bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Preprint 2022, arXiv:2205:06602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 9 [Laz] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lazarsfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Positivity in algebraic geometry, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Ergebnisse der Mathematik und ihrer Grenzgebiete, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Folge 48, Springer-Verlag, Berlin, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3 [LS] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Lopez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Sierra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' A geometrical view of Ulrich vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Preprint 2021, arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='05979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' To appear on Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' IMRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5, 16 [M] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Miyaoka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The Chern classes and Kodaira dimension of a minimal variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In: Algebraic geometry, Sendai, 1985, 449-476, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=', 10, North-Holland, Amsterdam, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 24 [MS] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Mori, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Sumihiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' On Hartshorne’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Kyoto Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 18 (1978), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 523-533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4 [T] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Totaro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Bott vanishing for algebraic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 373 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5, 3609-3626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 15, 16 [W] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Wahl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' A cohomological characterization of Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 72 (1983), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 315-322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Some numerical lemmas Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' c1, c2, c3, c4) ∈ Z5 be such that c1 ≥ c2 ≥ c3 ≥ c4 ≥ 0, a ≥ c1 + c2 + 3 and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) a2 − 6a + 4 = c2 1 + c2 2 + c2 3 + c2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' c1, c2, c3, c4) ∈ {(6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 0, 0, 0), (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 1, 1, 1), (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 1, 1, 0), (9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) a − 3 ≥ c1 + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Now, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) imply that (a − 3)2 − 5 = c2 1 + c2 2 + c2 3 + c2 4 ≥ (c1 + c2)2 − 5, that is (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) c2 3 + c2 4 ≥ 2c1c2 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But 2c2 3 ≥ c2 3 + c2 4 and 2c1c2 ≥ 2c2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently, we get (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) 5 ≥ 2(c2 − c3)(c2 + c3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, one of the following should happen: (α) c2 = c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (β) c2 = c3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' First assume that case (β) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) yields 2c3 ≤ 1 which gives c3 = 0, c2 = 2 and hence c4 = o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) gives (a + c1 − 3)(a − c1 − 3) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus we have one of the following possibilities a + c1 = 4, a − c1 = 9, or a + c1 = 5, a − c1 = 6, or a + c1 = 6, a − c1 = 5, or a + c1 = 9, a − c1 = 4 but none of them have integer solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Assume now that case (α) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Set c2 = c3 = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3), we obtain c2 + c2 4 ≥ 2c1c − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since c ≥ c4, we get (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='5) 5 ≥ 2c(c1 − c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' The above implies one of the following happens: 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' LOPEZ, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' RAYCHAUDHURY (α1) c2 = c3 = c4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (α2) c1 = c2 = c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (α3) c2 = c3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This case has two sub-cases, namely c1 = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (α4) c2 = c3 = 1, c1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose we are in case (α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) gives (a − 3)2 − 5 = c2 1, so that (a + c1 − 3)(a − c1 − 3) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case, either a + c1 = 4, a − c1 = 8, giving the contradiction c1 = −2, or a + c1 = 8, a − c1 = 4, giving a = 6, c1 = 2 and the solution (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose we are in case (α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) we conclude (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6) 5 ≥ (c − c4)(c + c4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' As before, we obtain the following cases: (α21) c = c1 = c2 = c3 = c4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (α22) c = c4 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' (α23) c = c4 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We first deal with (α21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case, from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1), we obtain (a + 2c − 3)(a − 2c − 3) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus, we have either a + 2c = 4, a − 2c = 8, giving the contradiction c = −1, or a + 2c = 8, a − 2c = 4, giving the solution (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We now deal with (α22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='6) we obtain c + c4 − 2 ≤ 5, hence c4 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus (c, c4) ∈ {(3, 2), (2, 1), (1, 0)} and using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) we see that it has no integer solutions except in the first case, giving the solution (9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We now deal with (α23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' As before, in this case we have c4 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This implies c = 2, c4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) does not have any integer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This concludes case (α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose we are in case (α3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We know that (c1, c2, c3, c4) ∈ {(1, 1, 1, 1), (2, 1, 1, 0), (3, 1, 1, 1), (3, 1, 1, 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) we see that we have no integer solutions except in the last case, giving (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 1, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Suppose we are in case (α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then (c1, c2, c3, c4) ∈ {(3, 2, 2, 2), (3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 2, 1), (3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 2, 0)} and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) has no integer solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' This concludes case (α) and the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Let z = (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' b1, b2, b3, b4, b5, b6) ∈ Z7 with b1 ≥ b2 ≥ b3 ��� b4 ≥ b5 ≥ b6 satisfying the following (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='7) a2 − 6 � i=1 b2 i = 10, 3a − 6 � i=1 bi = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then z ∈ {(4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 1, 1, 1, 1, 1, 1), (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 2, 2, 1, 1, 1), (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 2, 2, 2, 2, 1), (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 2, 2, 2), (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 3, 3, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We first use the Cauchy-Scwartz’s inequality (�6 i=1 bi)2 ≤ 6(�6 i=1 b2 i ) to obtain (a2−12a+32) ≤ 0 whence 4 ≤ a ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We further observe that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='8) 6 � i=1 (b2 i − bi) = a2 − 3a − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, (b2 i − bi) ≥ 0 for all i ≥ 1, and b1 > 0 as 3a − 6 > 0 for a ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Case 1: a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have �6 i=1(b2 i − bi) = 0 whence |bi| ≤ 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Since �6 i=1 bi = 6, we have bi = 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Case 2: a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have �6 i=1(b2 i − bi) = 6 whence |bi| ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, �6 i=1 b2 i = 15 and �6 i=1 bi = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case �6 i=2(b2 i − bi) = 0 whence |bi| ≤ 1 for all i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently �6 i=1 bi ≤ 8 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b1 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case we must have b1 = b2 = b3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently �6 i=4(b2 i − bi) = 0 whence |bi| ≤ 1 for all i ≥ 4 whence the only solution is z = (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 2, 2, 2, 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE 27 Case 3: a = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have �6 i=1(b2 i − bi) = 14 whence |bi| ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also �6 i=1 b2 i = 26 and �6 i=1 bi = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=2(b2 i − bi) = 2 whence |bi| ≤ 2 for i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently b2 = b3 = b4 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' But then �6 i=1 b2 i ≥ 28 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case �6 i=3(b2 i − bi) = 2 whence |bi| ≤ 2 for i ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently, b3 = b4 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus b5 + b6 = 2 and b2 5 + b2 6 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b2 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case we have the only solution z = (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 2, 2, 2, 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) b1 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case bi = 2 for all i whence �6 i=1 b2 i = 24 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Case 4: a = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have �6 i=1(b2 i − bi) = 24 whence |bi| ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, �6 i=1 b2 i = 39 and �6 i=1 bi = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=2(b2 i − bi) = 4 whence |bi| ≤ 2 for all i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently bi = 2 for all i, thus �6 i=1 b2 i = 45 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=3(b2 i − bi) = 0 whence |bi| ≤ 1 for all i ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently �6 i=1 bi ≤ 12 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=4(b2 i − bi) = 0 whence |bi| ≤ 1 for i ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently �6 i=1 bi ≤ 13 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b3 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then bi = 2 for all i ≥ 3 whence �6 i=1 b2 i = 41 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) b2 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=1 bi ≤ 14 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) b1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then b2 = b3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b4 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=5(b2 i − bi) = 0 whence |bi| ≤ 1 for i ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently �6 i=1 bi ≤ 14 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b4 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then b4 = b5 = b6 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We get only one solution z = (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 2, 2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) b1 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=1 bi ≤ 12 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Case 5: a = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' We have �6 i=1(b2 i − bi) = 36 whence |bi| ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Also, �6 i=1 b2 i = 54 and �6 i=1 bi = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b1 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=2(b2 i − bi) = 6 whence |bi| ≤ 3 for i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case �6 i=3(b2 i − bi) = 0 whence |bi| ≤ 1 for i ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently �6 i=1 bi ≤ 13 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b2 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case �6 i=1 bi ≤ 16 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=2(b2 i − bi) = 16 whence |bi| ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=3(b2 i − bi) = 4 whence |bi| ≤ 2 for i ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus �6 i=1 bi ≤ 17 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b2 = 3 which implies b3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=4(b2 i − bi) = 4 whence |bi| ≤ 2 for i ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently �6 i=1 bi ≤ 17 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) b2 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=1 bi ≤ 15 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) b1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1) b3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=4(b2 i − bi) = 0 whence |bi| ≤ 1 for i ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently �6 i=1 bi ≤ 15 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b3 = 3 which implies b4 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Thus �6 i=5(b2 i −bi) = 0 whence |bi| ≤ 1 for i ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently �6 i=1 bi ≤ 16 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) b3 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=1 bi ≤ 16 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='2) b2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then b3 = b4 = b5 = 3 and b6 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Consequently �6 i=1 b2 i = 56 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='3) b2 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Then �6 i=1 bi ≤ 14 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' Subcase 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='4) b1 ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' In this case we have the only solution z = (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' 3, 3, 3, 3, 3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' □ Angelo Felice Lopez, Dipartimento di Matematica e Fisica, Universit`a di Roma Tre, Largo San Leonardo Murialdo 1, 00146, Roma, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' e-mail lopez@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='uniroma3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='it Debaditya Raychaudhury, Department of Mathematics, University of Toronto, Bahen Centre, 40 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' George St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=', Room 6290, Toronto, ON M5S 2E4, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content=' email: debaditya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='raychaudhury@utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'} +page_content='ca' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfVgTd/content/2301.03104v1.pdf'}