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+arXiv:2301.04864v1  [math.AG]  12 Jan 2023
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY
+IN POSITIVE CHARACTERISTIC
+ZHI HU, YU YANG, AND RUNHONG ZONG
+ABSTRACT. In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable
+curves in positive characteristic. It shows that the topological structures of moduli spaces of curves can be understood from
+the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures relating the tame fundamental
+groups of curves over algebraically closed fields of characteristic p > 0 to the moduli spaces of curves. These conjectures
+are generalized versions of the weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields
+of characteristic p > 0 which was formulated by Tamagawa. Moreover, we prove that the conjectures hold for certain points
+lying in the moduli space of curves of genus 0.
+CONTENTS
+1.
+Introduction
+1
+1.1.
+The mystery of fundamental groups in positive characteristic
+1
+1.2.
+Topology structures of moduli spaces of curves and anabelian geometry
+2
+1.3.
+Main results
+3
+1.4.
+Some further developments
+4
+1.5.
+Structure of the present paper
+6
+1.6.
+Acknowledgements
+6
+2.
+Conjectures
+6
+2.1.
+The weak Hom-version conjecture
+6
+2.2.
+The pointed collection conjecture
+7
+3.
+Reconstructions of marked points
+8
+3.1.
+Anabelian reconstructions
+9
+3.2.
+The set of marked points
+9
+3.3.
+Reconstructions of inertia subgroups
+12
+3.4.
+Reconstructions of inertia subgroups via surjections
+14
+3.5.
+Reconstructions of additive structures via surjections
+21
+4.
+Main theorems
+23
+4.1.
+The first main theorem
+23
+4.2.
+The second main theorem
+29
+References
+31
+1. INTRODUCTION
+1.1. The mystery of fundamental groups in positive characteristic.
+sec111
+1.1.1.
+Let k be an algebraically closed field of characteristic p ≥ 0, and let (X, DX) be a smooth pointed stable curve
+of type (gX, nX) over k (i.e. 2gX + nX − 2 > 0, see
+K[K, Definition 1.1 (iv)]), where X denotes the underlying curve,
+DX denotes the (ordered) finite set of marked points, gX denotes the genus of X, and nX denotes the cardinality
+1
+
+2
+ZHI HU, YU YANG, AND RUNHONG ZONG
+#(DX) of DX. We put UX := X \ DX. By choosing a base point of UX, we have the tame fundamental group
+πt
+1(UX) of UX.
+If p = 0, it is well-known that πt
+1(UX) is isomorphic to the profinite completion of the topological fundamental
+group of a Riemann surface of type (gX, nX). Hence, almost no geometric information about UX can be carried out
+from πt
+1(UX). By contrast, if p > 0, the situation is quite different from that in characteristic 0. The tame fundamental
+group πt
+1(UX) is very mysterious and its structure is no longer known, in particular, there exist anabelian phenomena
+for curves over algebraically closed fields of characteristic p > 0.
+1.1.2.
+Firstly, let us explain some general background about anabelian geometry. In the 1980s, A. Grothendieck
+suggested a theory of arithmetic geometry called anabelian geometry (
+G[G]). The central question of the theory is as
+follows: Can we reconstruct the geometric information of a variety group-theoretically from various versions of its
+algebraic fundamental group? The original anabelian geometry suggested by Grothendieck focused on varieties over
+arithmetic fields, in particular, the fields finitely generated over Q. In the case of curves in characteristic 0, anabelian
+geometry has been deeply studied (e.g.
+N[N],
+T1
+[T1]) and, in particular, the most important case (i.e., the fields finitely
+generated over Q, or more general, sub-p-adic fields) has been established completely(
+M[M]). Note that the actions of
+the Galois groups of the base fields on the geometric fundamental groups play a crucial role for recovering geometric
+information of curves over arithmetic fields.
+Next, we return to the case where k is an algebraically closed field of characteristic p > 0. In
+T2
+[T2], A. Tamagawa
+discovered that there also exist anabelian phenomena for curves over algebraically closed fields of characteristic p.
+This came rather surprisingly since it means that, in positive characteristic, the geometry of curves can be determined
+by their geometric fundamental groups without Galois actions. Since the late 1990s, this kind of anabelian phenom-
+enon has been studied further by M. Raynaud (
+R2
+[R2]), F. Pop-M. Saïdi (
+PS
+[PS]), Tamagawa (
+T2
+[T2],
+T4
+[T4],
+T5
+[T5]), and the
+second author of the present paper (
+Y1
+[Y1],
+Y2
+[Y2],
+Y4
+[Y4]). More precisely, they focused on the so-called weak Isom-
+version of Grothendieck’s anabelian conjecture for curves over algebraically closed fields of characteristic p > 0 (or
+the “weak Isom-version conjecture” for short) formulated by Tamagawa (
+T3
+[T3, Conjecture 2.2]), which says that curves
+are isomorphic if and only if their tame (or étale) fundamental groups are isomorphic. At the present, this conjecture
+is still wide-open.
+1.2. Topology structures of moduli spaces of curves and anabelian geometry. In the present paper, we study a
+new kind of anabelian phenomenon concerning curves over algebraically closed fields of characteristic p > 0 which
+shows that the topological structures of moduli spaces of curves can be understood by their fundamental groups.
+1.2.1.
+Let Fp be the prime field of characteristic p > 0, and let Mord
+g,n,Z be the moduli stack over Z parameterizing
+smooth n-pointed stable curves of type (g, n) (in the sense of
+K[K]). We put Mord
+g,n,Fp := Mord
+g,n,Z ×Z Fp. Note that
+the set of marked points of an n-smooth pointed stable curve admits a natural action of the n-symmetric group Sn.
+Moreover, we denote by Mg,n,Fp := [Mord
+g,n,Fp/Sn] the quotient stack, and denote by Mg,n,Fp the coarse moduli space
+of Mg,n,Fp.
+Let q ∈ Mg,n,Fp be an arbitrary point, k(q) the residue field of q, kq an algebraically closed field containing k(q),
+and Vq := {q} the topological closure of {q} in Mg,n,Fp. Write (Xkq, DXkq ) for the smooth pointed stable curve
+of type (g, n) over kq determined by the natural morphism Speckq → Mg,n,Fp and put UXkq := Xkq \ DXkq . In
+particular, we put (Xkq, DXkq ) := (Xq, DXq) and UXq := Xq \ DXq if kq is an algebraic closure of k(q). Since
+the isomorphism class of the tame fundamental group πt
+1(UXkq ) depends only on q, we shall write πt
+1(q) for the tame
+fundamental group πt
+1(UXkq ).
+sec122
+1.2.2.
+We maintain the notation introduced above. The weak Isom-version conjecture of Tamagawa can be reformu-
+lated as follows:
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+3
+Weak Isom-version Conjecture . Let qi ∈ Mg,n,Fp, i ∈ {1, 2}, be an arbitrary point of Mg,n,Fp. The set of
+continuous isomorphisms of profinite groups
+Isompg(πt
+1(q1), πt
+1(q2))
+is non-empty if and only if Vq1 = Vq2 (namely, UXq1 ∼= UXq2 as schemes).
+The weak Isom-version conjecture means that moduli spaces of curves can be reconstructed “as sets” from the iso-
+morphism classes of the tame fundamental groups of curves. This conjecture has been only confirmed by Tamagawa
+(
+T4
+[T4, Theorem 0.2]) in the following case:
+Suppose that q1 is a closed point of M0,n,Fp. Then the weak Isom-version conjecture holds true.
+Next, we propose a new conjecture as follows, that is the weak Hom-version of the Grothendieck conjecture for
+curves over algebraically closed fields of characteristic p > 0 (or is called weak Hom-version conjecture for simplic-
+ity), as a generalization of the weak Isom-version conjecture.
+Weak Hom-version Conjecture . Let qi ∈ Mg,n,Fp, i ∈ {1, 2}, be an arbitrary point of Mg,n,Fp. The set of open
+continuous homomorphisms of profinite groups
+Homop
+pg(πt
+1(q1), πt
+1(q2))
+is non-empty if and only if Vq1 ⊇ Vq2.
+The weak Hom-version conjecture means that the sets of deformations of a smooth pointed stable curve can be re-
+constructed group-theoretically from the sets of open continuous homomorphisms of their tame fundamental groups.
+Therefore, it provides a new kind of anabelian phenomenon:
+The moduli spaces of curves in positive characteristic can be understood not only as sets but also “as
+topological spaces” from the sets of open continuous homomorphisms of tame fundamental groups
+of curves in positive characteristic.
+Roughly speaking, this means that a smooth pointed stable curve corresponding to a geometric point over q2 can be
+deformed to a smooth pointed stable curve corresponding to a geometric point over q1 if and only if the set of open
+continuous homomorphisms of tame fundamental groups Homop
+pg (πt
+1(q1), πt
+1(q2)) is not empty.
+1.3. Main results.
+1.3.1.
+The main result of the present paper is the following (see Theorem
+them-4
+4.6 (iv) for a more general statement):
+maintheorem
+Theorem 1.1. The Weak Hom-version Conjecture holds when q1 is a closed point of M0,n,Fp.
+Theorem
+maintheorem
+1.1 follows from the following “Hom-type" anabelian result (see Theorem
+them-3
+4.4 for a more precise statement)
+which is a generalization of Tamagawa’s result (i.e.
+T4
+[T4, Theorem 0.2]):
+them-0-1
+Theorem 1.2. Let qi ∈ Mg,n,Fp, i ∈ {1, 2}, be an arbitrary point of Mg,n,Fp. Suppose that q1 is a closed point of
+Mg,n,Fp. Then the set of open continuous homomorphisms
+Homop
+pg(πt
+1(q1), πt
+1(q2))
+is non-empty if and only if UXq1 ∼= UXq2 as schemes.
+Note that Theorem
+them-0-1
+1.2 is essentially different from
+T4
+[T4, Theorem 0.2]. The reason is the following: We a priori do
+not know whether or not
+Isompg(πt
+1(q1), πt
+1(q2))
+
+4
+ZHI HU, YU YANG, AND RUNHONG ZONG
+is non-empty even through Homop
+pg(πt
+1(q1), πt
+1(q2)) is non-empty. In fact, for arbitrary qi ∈ Mg,n,Fp, i ∈ {1, 2}, we
+have
+Isompg(πt
+1(q1), πt
+1(q2)) = ∅, Homop
+pg(πt
+1(q1), πt
+1(q2)) ̸= ∅
+in general (
+T5
+[T5, Theorem 0.3]).
+On the other hand, to verify Theorem
+them-0-1
+1.2, we need to establish various anabelian reconstructions from open contin-
+uous homomorphisms of tame fundamental groups which are much harder than the case of isomorphisms in general.
+We explain in more detail about this point in the following.
+1.3.2.
+Let us explain the main differences between the proofs of Tamagawa’s result (i.e.
+T4
+[T4, Theorem 0.2]) and
+our result (i.e. Theorem
+them-0-1
+1.2), and the new ingredient in our proof. First, we recall the key points of the proof of
+Tamagawa’s result. Roughly speaking, Tamagawa’s proof consists of two parts:
+(1) He proved that the sets of inertia subgroups of marked points and the field structures associated to inertia
+subgroups of marked points of smooth pointed stable curves can be reconstructed group-theoretically from
+tame fundamental groups. This is the most difficult part of Tamagawa’s proof.
+(2) By using the inertia subgroups and their associated field structures, if g = 0, he proved that the coordinates of
+marked points can be calculated group-theoretically.
+The group-theoretical reconstructions in Tamagawa’s proofs (1) and (2) are isomorphic version reconstructions.
+This means that the reconstructions should fix an isomorphism class of a tame fundamental group. To explain this,
+let us show an example. Let UXi, i ∈ {1, 2}, be a curve of type (gX, nX) over an algebraically closed field k of
+characteristic p > 0 introduced above, πt
+1(UXi) the tame fundamental group of UXi, φ : πt
+1(UX1) → πt
+1(UX2) an
+open continuous homomorphism, H2 ⊆ πt
+1(UX2) an open subgroup, and H1 := φ−1(H2). In Tamagawa’s proof,
+since φ is an isomorphism, we have H1 ≃ H2. Then the group-theoretical reconstruction for types implies that the
+type (gXH1 , nXH1 ) and the type (gXH2 , nXH2 ) of the curves corresponding to H1 and H2, respectively, are equal. This
+is a key point in the proof of Tamagawa’s group-theoretical reconstruction of the inertia subgroups of marked points.
+Unfortunately, his method cannot be applied to the present paper. The reason is that we need to treat the case where
+φ is an arbitrary open continuous homomorphism. Since H1 is not isomorphic to H2 in general (e.g. specialization
+homomorphism), we do not know whether or not (gXH1 , nXH1 ) = (gXH2 , nXH2 ). This is one of the main difficulties
+of “Hom-type” problems appeared in anabelian geometry. Similar difficulties for generalized Hasse-Witt invariants
+will appear if we try to reconstruct the field structure associated to inertia subgroups of marked points.
+To overcome the difficulties mentioned above, we have the following key observation:
+The inequalities of Avrp(Hi) (i.e., the p-averages of generalized Hasse-Witt invariants (see
+paverage
+3.4.3)) in-
+duced by φ play roles of the comparability of (outer) Galois representations in the theory of anabelian
+geometry of curves over algebraically closed fields of characteristic p > 0.
+In the present paper, our method for reconstructing inertia subgroups of marked points is completely different from
+Tamagawa’s reconstruction. We develop a new group-theoretical algorithm for reconstructing the inertia subgroups of
+marked points whose input datum is a profinite group which is isomorphic to πt
+1(UXi), i ∈ {1, 2}, and whose output
+data are inertia subgroups of marked points (Theorem
+them-2
+3.18). Moreover, we prove that the group-theoretical algorithm
+and the reconstructions for field structures are compatible with arbitrary surjection φ (Proposition
+pro-4
+3.19). By using
+Theorem
+them-2
+3.18 and Proposition
+pro-4
+3.19, we may prove that Tamagawa’s calculation of coordinates is compatible with our
+reconstructions. This implies Theorem
+them-0-1
+1.2.
+1.4. Some further developments.
+1.4.1. Moduli spaces of fundamental groups. Let us explain some further developments for the anabelian phenomenon
+concerning the weak Hom-verson conjecture. In
+Y6
+[Y6], the second author of the present paper introduced a topological
+space Πg,n (or more general, Πg,n) determined group-theoretically by the tame fundamental groups of smooth pointed
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+5
+stable curves (or more general, the geometric log étale fundamental groups of arbitrary pointed stable curves) of
+type (g, n) which is called the moduli spaces of fundamental groups of curves, whose underlying set is the sets of
+isomorphism classes of fundamental groups, and whose topology is determined by the sets of finite quotients of
+fundamental groups. Moreover, he posed the so-called homeomorphism conjecture, roughly speaking, which says
+that (by quotiening a certain equivalence relation induced by Frobenius actions) the moduli spaces of curves are
+homeomorphic to the moduli spaces of fundamental groups.
+In the present literatures, the term “anabelian” means that a geometric object can be determined by its fundamental
+group. Furthermore, the homeomorphism conjecture concerning moduli spaces of fundamental groups supplies a new
+point of view to understand anabelian phenomena as follows:
+The term “anabelian” means that not only a geometric object can be determined by its fundamental
+groups, but also a certain moduli space of geometric objects can be determined by the fundamental
+groups of geometric objects.
+Under this point of view, the homeomorphism conjecture is regarded as the analogue of a famous theorem in the theory
+of classic Teichmüller spaces which states that the Teichmüller spaces of complex hyperbolic curves are homeomorphic
+to the spaces of discrete and faithful representations of topological fundamental groups of underlying surfaces into the
+group PSL2(R).
+Now Theorem
+maintheorem
+1.1 implies that M0,4,Fp is homeomorphic to Π0,4 as topological spaces (note that Tamagawa’s
+result (i.e.
+T4
+[T4, Theorem 0.2]) only says that the natural map M0,4,Fp → Π0,4 is a bijection as sets). Moreover, based
+on
+Y1
+[Y1],
+Y3
+[Y3],
+Y4
+[Y4],
+Y5
+[Y5], and the main results of the present paper, the homeomorphism conjecture is confirmed
+for 1-dimensional moduli spaces of pointed stable curves in
+Y6
+[Y6] and
+Y7
+[Y7]. For the homeomorphism conjecture in
+the case of higher dimensional moduli spaces of curves, the weak Hom-version conjecture and the pointed collection
+conjecture (see Section
+pcc
+2.2 of the present paper) are also the main steps toward understanding it (see
+Y8
+[Y8, Section
+1.2.3]).
+1.4.2. The sets of finite quotients of tame fundamental groups. We maintain the notation introduced in
+sec111
+1.1.1. The
+techniques developed in §
+mpanabelian
+3 of the present paper have important applications for understanding the set of finite quotients
+πt
+A(UX) of the tame fundamental groups πt
+1(UX) of UX.
+Note that, if UX is affine, the set πét
+A(UX) of finite quotients of the étale fundamental groups πét
+1 (UX) of UX can
+be completely determined by its type (gX, nX) (i.e. Abhyankar’s conjecture proved by Raynaud and D. Harbater).
+However, the structure of πét
+1 (UX) cannot be carried out from πét
+A(UX) since πét
+1 (UX) is not topologically finitely
+generated when UX is affine.
+By contrast, the isomorphism class of πt
+1(UX) can be completely determined by πt
+A(UX) since πt
+1(UX) is topolog-
+ically finitely generated, and one cannot expect that there exists an explicit description for the entire set πt
+A(UX) since
+there exists anabelian phenomenon mentioned above (i.e. πt
+A(UX) depends on the isomorphism class of UX). On the
+other hand, for understanding more precisely the relationship between the structures of tame fundamental groups and
+the anabelian phenomena in positive characteristic world, it is naturally to ask the following interesting problem:
+How does the scheme structure of UX affect explicitly the set of finite quotients πt
+A(UX)?
+In
+Y9
+[Y9], by applying the techniques developed in §
+mpanabelian
+3 of the present paper and
+Y5
+[Y5, Theorem 1.2], we obtain the
+following result:
+Let q1 ∈ Mg1,n1,Fp and q2 ∈ M0,n2,Fp be arbitrary points and πt
+A(qi) the set of finite quotients of
+the tame fundamental group πt
+1(qi). Suppose that q2 is a closed point of M0,n2,Fp, and that πt
+1(q1) ̸∼=
+πt
+1(q2). Then we can construct explicitly a finite group Gq2 depending on q2 such that Gq2 ∈ πt
+A(q1)
+and Gq2 ̸∈ πt
+A(q2).
+
+6
+ZHI HU, YU YANG, AND RUNHONG ZONG
+1.5. Structure of the present paper. The present paper is organized as follows. In Section
+sec-1
+2, we formulate the
+the weak Hom-version conjecture and the pointed collection conjecture. In Section
+mpanabelian
+3, we give a group-theoretical
+algorithm for reconstructions of inertia subgroups associated to marked points, and prove that the group-theoretical
+algorithm is compatible with arbitrary open surjective homomorphisms of tame fundamental groups. In Section
+sec-5
+4, we
+prove our main results.
+1.6. Acknowledgements. The second author was supported by JSPS Grant-in-Aid for Young Scientists Grant Num-
+bers 16J08847 and 20K14283.
+2. CONJECTURES
+sec-1
+In this section, we formulate two new conjectures concerning anabelian geometry of curves over algebraically
+closed fields of characteristic p > 0.
+2.1. The weak Hom-version conjecture. In this subsection, we formulate the first conjecture of the present paper
+which we call “the weak Hom-version conjecture”.
+curves
+2.1.1.
+Let k be an algebraically closed field of characteristic p > 0, and let
+(X, DX)
+be a smooth pointed stable curve of type (gX, nX) over k, where X denotes the (smooth) underlying curve of genus
+gX and DX denotes the (ordered) finite set of marked points with cardinality nX := #(DX) satisfying
+K[K, Definition
+1.1 (iv)] (i.e. 2gX + nX − 2 > 0). Note that UX := X \ DX is a hyperbolic curve over k.
+Let (Y, DY ) and (X, DX) be smooth pointed stable curves over k, and let f : (Y, DY ) → (X, DX) be a morphism
+of smooth pointed stable curves over k. We shall say that f is étale (resp. tame, Galois étale, Galois tame) if f is étale
+over X (resp. f is étale over UX and is at most tamely ramified over DX, f is a Galois covering and is étale, f is a
+Galois covering and is tame).
+By choosing a base point of x ∈ UX, we have the tame fundamental group πt
+1(UX, x) of UX and the étale funda-
+mental group π1(X, x) of X. Since we only focus on the isomorphism classes of fundamental groups in the present
+paper, for simplicity of notation, we omit the base point and denote by πt
+1(UX) and π1(X) the tame fundamental
+group πt
+1(UX, x) of UX and the étale fundamental group π1(X, x) of X, respectively. Note that there is a natural
+continuous surjective homomorphism πt
+1(UX) ։ π1(X).
+moduli212
+2.1.2.
+Let Fp be an algebraic closure of Fp, and let Mord
+g,n,Fp be the moduli stack over Z parameterizing smooth
+pointed stable curves of type (g, n) in the sense of
+K[K, Definition 1.1]. The set of marked points of a smooth pointed
+stable curve admits a natural action of the n-symmetric group Sn, we put Mg,n,Z := [Mord
+g,n,Z/Sn] the quotient stack.
+Moreover, we denote by Mord
+g,n := Mg,n,Z ×Z Fp, Mg,n,Fp := Mg,n,Z ×Z Fp, and Mg,n := Mg,n,Z ×Z Fp, and
+denote by M ord
+g,n , Mg,n,Fp, and Mg,n the coarse moduli spaces of Mord
+g,n, Mg,n,Fp, and Mg,n, respectively.
+Let q ∈ M ord
+g,n be an arbitrary point and k(q) the residue field of q, and kq an algebraically closed field containing
+k(q). Write (Xkq, DXkq ) for the smooth pointed stable curve of type (g, n) over kq determined by the natural mor-
+phism Speckq → Speck(q) → M ord
+g,n and UXkq for Xkq \ DXkq . In particular, if kq is an algebraic closure of k(q), we
+shall write (Xq, DXq) for (Xkq, DXkq ).
+Since the isomorphism class of the tame fundamental group πt
+1(UXkq ) depends only on q (i.e., the isomorphism
+class does not depend on the choices of kq), we shall write πt
+1(q) and πt
+A(q) for πt
+1(UXkq ) and the set of finite quotients
+of πt
+1(UXkq ), respectively.
+FJ
+[FJ, Proposition 16.10.7] implies that for any points q1, q2 ∈ M ord
+g,n , πt
+1(q1) ∼= πt
+1(q2) as
+profinite groups if and only if πt
+A(q1) = πt
+A(q2) as sets.
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+7
+On the other hand, Let q ∈ M ord
+g,n and q′ ∈ Mg,n,Fp be arbitrary points. We denote by Vq ⊆ M ord
+g,n and Vq′ ⊆
+Mg,n,Fp the topological closures of q and q′ in M ord
+g,n and Mg,n,Fp, respectively.
+2.1.3.
+We have the following definition.
+Definition 2.1.
+def-2
+(1) Let c1, c2 ∈ M ord,cl
+g,n
+be closed points, where (−)cl denotes the set of closed points of (−). Then c1 ∼fe c2 if
+there exists m ∈ Z such that ν(c2) = ν(c(m)
+1
+), where c(m)
+1
+denotes the closed point corresponding to the curve
+obtained by mth Frobenius twist of the curve corresponding to c1. Here “fe" means “Frobenius equivalence".
+(2) Let q1, q2 ∈ M ord
+g,n be arbitrary points. We denote by Vq1 ⊇fe Vq2 if, for each closed point c2 ∈ V cl
+q2 , there
+exists a closed point c1 ∈ V cl
+q1 such that c1 ∼fe c2. Moreover, we denote by Vq1 =fe Vq2 if Vq1 ⊇fe Vq2 and
+Vq1 ⊆fe Vq2. Moreover, we also denote by q1 ∼fe q2 if Vq1 =fe Vq2.
+We have the following proposition.
+pro-5
+Proposition 2.2. Let ω : M ord
+g,n → Mg,n,Fp be the morphism induced by the natural morphism Mord
+g,n → Mg,n,Fp.
+Let i ∈ {1, 2}, and let qi ∈ M ord
+g,n and q′
+i := ω(qi) ∈ Mg,n,Fp. Then we have Vq1 ⊇fe Vq2 if and only if Vq′
+1 ⊇ Vq′
+2. In
+particular, we have Vq1 =fe Vq2 if and only if Vq′
+1 = Vq′
+2. Namely, we have Vq1 =fe Vq2 if and only if UXq1 ∼= UXq2
+as schemes.
+Proof. Suppose that qi, i ∈ {1, 2}, is a closed point of M ord
+g,n . If Vq1 ⊇fe Vq2, we see immeidately q1 ∼ q2. Thus, we
+obtain UXq1 ∼= UXq2 as schemes. This means q′
+1 = q′
+2. Conversely, if Vq′
+1 ⊇ Vq′
+2, then we have q′
+1 = q′
+2. Thus, we
+obtain q1 ∼ q2.
+Suppose that qi, i ∈ {1, 2}, is an aribtrary point of M ord
+g,n . If Vq1 ⊇fe Vq2, then the case of closed points implies
+V cl
+q′
+1 ⊇ V cl
+q′
+2 . Since Vq′
+1 and Vq′
+2 are irreducible, we obtain Vq′
+1 ⊇ Vq′
+2. Conversely, if Vq′
+1 ⊇ Vq′
+2, we note that Vqi is an
+irreducible component of (ω)−1(Vq′
+i). Then the case of closed points implies Vq1 ⊇fe Vq2.
+□
+2.1.4.
+Denote by Homop
+pg(−, −) the set of open continuous homomorphisms of profinite groups, and by
+Isompg(−, −) the set of isomorphisms of profinite groups. We have the following conjecture.
+Weak Hom-version Conjecture . Let qi ∈ Mg,n (resp. qi ∈ Mg,n,Fp), i ∈ {1, 2}, be an arbitrary point. Then we
+have
+Homop
+pg(πt
+1(q1), πt
+1(q2))
+is non-empty if and only if Vq1 ⊇fe Vq2 (resp. Vq1 ⊇ Vq2).
+The weak Hom-version conjecture means that the topological structures of the moduli spaces of smooth pointed stable
+curves can be understood by the tame fundamental groups of curves. In particular, the weak Hom-version conjecture
+implies the following conjecture which was essentially formulated by Tamagawa (
+T3
+[T3]).
+Weak Isom-version Conjecture . Let qi ∈ Mg,n (resp. qi ∈ Mg,n,Fp), i ∈ {1, 2}, be an arbitrary point. Then we
+have
+Isompg(πt
+1(q1), πt
+1(q2))
+is non-empty if and only if Vq1 =fe Vq2 (resp. Vq1 = Vq2).
+The weak Isom-version conjecture means that the set structures of the moduli spaces of smooth pointed stable curves
+can be understood by the tame fundamental groups of curves.
+pcc
+2.2. The pointed collection conjecture. In this subsection, we formulate the second conjecture of the present paper
+which we call “the pointed collection conjecture”. We maintain the notation introduced in
+moduli212
+2.1.2.
+
+8
+ZHI HU, YU YANG, AND RUNHONG ZONG
+2.2.1.
+Let q be an arbitrary point of M ord
+g,n and G ∈ πt
+A(q) an arbitrary finite group. We put
+UG := {q′ ∈ M ord
+g,n | G ∈ πt
+A(q′)} ⊆ M ord
+g,n .
+Then we have the following result.
+pro-6
+Proposition 2.3. Let q be an arbitrary point of M ord
+g,n and G ∈ πt
+A(q) an arbitrary finite group. Then the set UG
+contains an open neighborhood of q in M ord
+g,n .
+Proof. Proposition
+pro-6
+2.3 was proved by K. Stevenson when n = 0 and q is a closed point of Mg,0 (cf.
+Ste
+[Ste, Proposition
+4.2]). Moreover, by similar arguments to the arguments given in the proof of
+Ste
+[Ste, Proposition 4.2], Proposition
+pro-6
+2.3
+also holds for n ≥ 0.
+□
+def-3
+Definition 2.4. We denote by qgen the generic point of M ord
+g,n , and let
+C ⊆ πt
+A(qgen) =
+�
+q∈Mord,cl
+g,n
+πt
+A(q)
+be a subset of πt
+A(qgen). We shall say that C is a pointed collection if the following conditions are satisfied: (i)
+0 < #((�
+G∈C UG) ∩ M ord,cl
+g,n
+) < ∞; (ii) UG′ ∩ (�
+G∈C UG) ∩ M ord,cl
+g,n
+= ∅ for each G′ ∈ πt
+A(qgen) such that G′ ̸∈ C.
+On the other hand, for each closed point t ∈ M ord,cl
+g,n
+, we may define a set associated to t as follows:
+Ct := {G ∈ πt
+A(qgen) | t ∈ UG}.
+Note that, if t ∈ V cl
+q , then Ct ⊆ πt
+A(q). Moreover, we denote by
+Cq := {C is a pointed collection | C ⊆ πt
+A(q)}.
+2.2.2.
+At present, no published results are known concerning the weak Hom-version conjecture (or the weak Isom-
+version conjecture) for non-closed points. The main difficulty of proving the weak Hom-version conjecture (or the
+weak Isom-version conjecture) for non-closed points of M ord
+g,n is the following: For each q ∈ M ord
+g,n , we do not know
+how to reconstruct the tame fundamental groups of closed points of Vq group-theoretically from πt
+1(q).
+Once the tame fundamental groups of the closed points of Vq can be reconstructed group-theoretically from πt
+1(q),
+then the weak Hom-version conjecture for closed points of M ord
+g,n implies that the set of closed points of Vq can be
+reconstructed group-theoretically from πt
+1(q). Thus, the weak Hom-version conjecture for non-closed points of M ord
+g,n
+can be deduced from the weak Hom-version conjecture for closed points of M ord
+g,n .
+Let q ∈ M ord
+g,n . Since the isomorphism class of πt
+1(q) as a profinite group can be determined by the set πt
+A(q), the
+following conjecture tell us how to reconstruct group-theoretically the set of finite quotients of a closed point of Vq
+from πt
+A(q) (or πt
+1(q)).
+Pointed Collection Conjecture . For each t ∈ M ord,cl
+g,n
+, the set Ct associated to t is a pointed collection. Moreover,
+let q ∈ M ord
+g,n . Then the natural map
+colleq : V cl
+q
+→ Cq, [t] �→ Ct,
+is a bijection, where [t] denotes the image of t in V cl
+q
+:= V cl
+q / ∼fe.
+Write q′ ∈ Mg,n,Fp for the image ω(q). Then we have V cl
+q
+= V cl
+q′ . This means that the pointed collection conjecture
+holds if and only if the weak Hom-version conjecture holds.
+3. RECONSTRUCTIONS OF MARKED POINTS
+mpanabelian
+The main purposes of the present section are as follows: We will give a new mono-anabelian reconstruction of
+Ine(πt
+1(UX)), and prove that the mono-anabelian reconstruction (i.e., the group-theoretical algorithm) is compatible
+with any open continuous homomorphisms of tame fundamental groups of smooth pointed stable curves with a fixed
+type.
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+9
+3.1. Anabelian reconstructions. We maintain the notation introduced in
+curves
+2.1.1.
+3.1.1.
+Let us recall the definitions concerning “anabelian reconstructions".
+definition 1
+Definition 3.1. Let F be a geometric object and ΠF a profinite group associated to the object F. Suppose that we
+are given an invariant InvF depending on the isomorphism class of F (in a certain category), and that we are given
+an additional structure AddF (e.g. a family of subgroups, a family of quotient groups) on the profinite group ΠF
+depending functorially on F.
+We shall say that InvF can be mono-anabelian reconstructed from ΠF if there exists a group-theoretical algorithm
+whose input datum is ΠF, and whose output datum is InvF. We shall say that AddF can be mono-anabelian recon-
+structed from ΠF if there exists a group-theoretical algorithm whose input datum is ΠF, and whose output datum is
+AddF.
+Let Fi, i ∈ {1, 2}, be a geometric object and ΠFi a profinite group associated to Fi. Suppose that we are given
+an additional structure AddFi on the profinite group ΠFi depending functorially on Fi. We shall say that a map
+(or a morphism) AddF1 → AddF2 can be mono-anabelian reconstructed from an open continuous homomorphism
+ΠF1 → ΠF2 if there exists a group-theoretical algorithm whose input datum is ΠF1 → ΠF2, and whose output datum
+is AddF1 → AddF2.
+unicov313
+3.1.2.
+Let K be the function field of X, and let �K be the maximal Galois extension of K in a fixed separable closure
+of K, unramified over UX and at most tamely ramified over DX. Then we may identify πt
+1(UX) with Gal( �K/K).
+We define the universal tame covering of (X, DX) associated to πt
+1(UX) to be ( �
+X, D �
+X), where �
+X denotes the nor-
+malization of X in �K, and D �
+X denotes the inverse image of DX in �
+X. Then there is a natural action of πt
+1(UX) on
+( �
+X, D �
+X). For each �e ∈ D �
+X, we denote by I�e the inertia subgroup of πt
+1(UX) associated to �e (i.e., the stabilizer of �e
+in πt
+1(UX)). Then we have I�e ∼= �Z(1)p′, where �Z(1)p′ denotes the prime-to-p part of �Z(1). The following result was
+proved by Tamagawa (
+T4
+[T4, Lemma 5.1 and Theorem 5.2]).
+Proposition 3.2.
+proposition 1
+(1) The type (gX, nX) can be mono-anabelian reconstructed from πt
+1(UX).
+(2) Let �e and �e′ be two points of D �
+X distinct from each other. Then the intersection of I�e and I�e′ is trivial in
+πt
+1(UX). Moreover, the map
+D �
+X → Sub(πt
+1(UX)), �e �→ I�e,
+is an injection, where Sub(−) denotes the set of closed subgroups of (−).
+(3) Write Ine(πt
+1(UX)) for the set of inertia subgroups in πt
+1(UX), namely the image of the map D �
+X →
+Sub(πt
+1(UX)). Then Ine(πt
+1(UX)) can be mono-anabelian reconstructed from πt
+1(UX). In particular, the
+set of marked points DX and π1(X) can be mono-anabelian reconstructed from πt
+1(UX).
+sec-2
+3.2. The set of marked points. We maintain the notation introduced in
+curves
+2.1.1. Moreover, we suppose that gX ≥ 2
+and nX > 0.
+3.2.1.
+We will prove that the set of marked points can be regarded as a quotient set of a set of cohomological classes
+of a suitable covering of curves (i.e. Proposition
+pro-2
+3.3). The main idea is the following: By taking a suitable étale
+covering with a prime degree f : (Y, DY ) → (X, DX), for every marked point x ∈ DX, there exists a set of tame
+coverings with a prime degree which is totally ramified over the inverse image f −1(x). Then x can be regarded as the
+set of cohomological classes corresponding to such coverings.
+triple
+3.2.2.
+Let h : (W, DW ) → (X, DX) be a connected Galois tame covering over k. We put
+Ramh := {e ∈ DX | h is ramified over e}.
+
+10
+ZHI HU, YU YANG, AND RUNHONG ZONG
+Let (Y, DY ) be a smooth pointed stable curve over k. We shall say that
+(ℓ, d, f : (Y, DY ) → (X, DX))
+is an mp-triple associated to (X, DX) if the following conditions hold: (i) ℓ and d are prime numbers distinct from
+each other such that (ℓ, p) = (d, p) = 1 and ℓ ≡ 1 (mod d); then all dth roots of unity are contained in Fℓ; (ii) f is
+a Galois étale covering over k whose Galois group is isomorphic to µd, where µd ⊆ F×
+ℓ denotes the subgroup of dth
+roots of unity. Here, “mp” means “marked points”.
+Then we have a natural injection H1
+ét(Y, Fℓ) ֒→ H1
+ét(UY , Fℓ) induced by the natural surjection πt
+1(UY ) ։ π1(Y ).
+Note that every non-zero element of H1
+ét(UY , Fℓ) induces a connected Galois tame covering of (Y, DY ) of degree ℓ.
+We obtain an exact sequence
+0 → H1
+ét(Y, Fℓ) → H1
+ét(UY , Fℓ) → Div0
+DY (Y ) ⊗ Fℓ → 0
+with a natural action of µd.
+sec31aaa
+3.2.3.
+Let (Div0
+DY (Y ) ⊗ Fℓ)µd ⊆ Div0
+DY (Y ) ⊗ Fℓ be the subset of elements on which µd acts via the character
+µd ֒→ F×
+ℓ and M ∗
+Y ⊆ H1
+ét(UY , Fℓ) the subset of elements whose images are non-zero elements of (Div0
+DY (Y )⊗Fℓ)µd.
+For each α ∈ M ∗
+Y , write gα : (Yα, DYα) → (Y, DY ) for the tame covering induced by α. We define ǫ : M ∗
+Y → Z,
+where ǫ(α) := #DYα. Denote by
+MY := {α ∈ M ∗
+Y | #Ramgα = d} = {α ∈ M ∗
+Y | ǫ(α) = ℓ(dnX − d) + d}.
+Note that MY is non-empty.
+For each α ∈ MY , since the image of α is contained in (Div0
+DY (Y ) ⊗ Fℓ)µd, we obtain that the action of µd
+on Ramgα ⊆ DY is transitive. Thus, there exists a unique marked point eα ∈ DX such that f(y) = eα for each
+y ∈ Ramgα.
+For each e ∈ DX, we put
+MY,e := {α ∈ MY | gα is ramified over f −1(e)}.
+Then, for any marked points e, e′ ∈ DX distinct from each other, we have MY,e ∩ MY,e′ = ∅ and the disjoint union
+MY =
+�
+e∈DX
+MY,e.
+315
+3.2.4.
+Next, we define a pre-equivalence relation ∼ on MY as follows: Let α, β ∈ MY . Then α ∼ β if λα + µβ ∈
+MY for each λ, µ ∈ F×
+ℓ for which λα + µβ ∈ M ∗
+Y . Then we have the following proposition.
+pro-2
+Proposition 3.3. The pre-equivalence relation ∼ on MY is an equivalence relation. Moreover, the map
+ϑX : MY / ∼→ DX, [α] �→ eα,
+is a bijection, where [α] denotes the image of α in MY / ∼.
+Proof. Let β, γ ∈ MY . If Ramgβ = Ramgγ, then, for each λ, µ ∈ F×
+ℓ for which λβ + µγ ̸= 0, we have Ramgλβ+µγ =
+Ramgβ = Ramgγ. Thus we obtain that β ∼ γ. On the other hand, if β ∼ γ, we have Ramgβ = Ramgγ. Otherwise, we
+have #Ramgβ+γ = 2d. This means that β ∼ γ if and only if Ramgβ = Ramgγ. Then ∼ is an equivalence relation on
+MY .
+Let us prove that ϑX is a bijection. It is easy to see that ϑX is an injection. On the other hand, for each e ∈ DX,
+the structure of the maximal pro-ℓ tame fundamental groups implies that we may construct a connected tame Galois
+covering of h : (Z, DZ) → (Y, DY ) such that the element of H1
+ét(UY , Fℓ) induced by h is contained in MY . Then ϑX
+is a surjection. This completes the proof of Proposition
+pro-2
+3.3.
+□
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+11
+rem-2-1
+Remark 3.4. We claim that the set MY / ∼ does not depend on the choices of mp-triples associated to (X, DX). Let
+(ℓ∗, d∗, f ∗ : (Y ∗, DY ∗) → (X, DX))
+be an arbitrary mp-triple associated to (X, DX). Hence we obtain a resulting set MY ∗/ ∼ and a natural bijection ϑ∗
+X :
+MY ∗/ ∼→ DX. We will prove that there exists a natural bijection δ : MY ∗/ ∼
+≃
+−→ MY / ∼ such that ϑ∗
+X = ϑX ◦ δ.
+First, suppose that ℓ ̸= ℓ∗ and d ̸= d∗. Then we may construct a natural bijection δ : MY ∗/ ∼
+≃
+−→ MY / ∼ as
+follows. Let α ∈ MY and α∗ ∈ MY ∗. Write (Yα, DYα) → (Y, DY ) and (Yα∗, DYα∗) → (Y ∗, DY ∗) for the Galois
+tame coverings induced by α and α∗, respectively. We consider the following fiber product in the category of smooth
+pointed stable curves
+(Yα, DYα) ×(X,DX) (Yα∗, DYα∗)
+which is a smooth pointed stable curve over k. Thus, we obtain a connected tame covering (Yα, DYα) ×(X,DX)
+(Yα∗, DYα∗) → (X, DX) of degree dd∗ℓℓ∗. Then it is easy to check that ϑX([α]) = ϑ∗
+X([α∗]) if and only if the
+cardinality of the set of marked points of (Yα, DYα) ×(X,DX) (Yα∗, DYα∗) is equal to dd∗(ℓℓ∗(nX − 1) + 1). We put
+[α] := δ([α∗]) if ϑX([α]) = ϑ∗
+X([α∗]). Moreover, by the construction above, we obtain that ϑ∗
+X = ϑX ◦ δ. In the
+general case, we may choose an mp-triple
+(ℓ∗∗, d∗∗, f ∗∗ : (Y ∗∗, DY ∗∗) → (X, DX))
+associated to (X, DX) such that ℓ∗∗ ̸= ℓ, ℓ∗∗ ̸= ℓ∗, d∗∗ ̸= d, and d∗∗ ̸= d∗. Hence we obtain a resulting set MY ∗∗/ ∼
+and a natural bijection ϑ∗∗
+X : MY ∗∗/ ∼→ DX. Then the proof given above implies that there are natural bijections
+δ1 : MY ∗∗/ ∼
+≃
+−→ MY / ∼ and δ2 : MY ∗∗/ ∼
+≃
+−→ MY ∗/ ∼. Thus, we may put
+δ := δ1 ◦ δ−1
+2
+: MY ∗/ ∼
+≃
+−→ MY / ∼ .
+rem-2-2
+Remark 3.5. Let H ⊆ πt
+1(UX) be an arbitrary open normal subgroup and fH : (XH, DXH) → (X, DX) the Galois
+tame covering over k induced by the natural inclusion H ֒→ πt
+1(UX). Let
+(ℓ, d, f : (Y, DY ) → (X, DX))
+be an mp-triple associated to (X, DX) such that (#(πt
+1(UX)/H), ℓ) = (#(πt
+1(UX)/H), d) = 1. Then we obtain an
+mp-triple
+(ℓ, d, g : (Z, DZ) := (Y, DY ) ×(X,DX) (XH, DXH) → (XH, DXH))
+associated to (XH, DXH) induced by (ℓ, d, f : (Y, DY ) → (X, DX)), where (Y, DY ) ×(X,DX) (XH, DXH) denotes
+the fiber product in the category of smooth pointed stable curves. The mp-triple associated to (XH, DXH) induces a
+set MZ/ ∼ which can be identified with the set of marked points DXH of (XH, DXH) by applying Proposition
+pro-2
+3.3.
+Moreover, for each eX ∈ DX and each αY,eX ∈ MY,eX, αY,eX induces an element
+αZ =
+�
+eXH ∈f −1
+H (eX)
+αZ,eXH
+over (Z, DZ) via the natural morphism (Z, DZ) → (Y, DY ), where αZ,eXH ∈ MZ,eXH . On the other hand, for each
+e′
+XH ∈ DXH and each e′
+X ∈ DX, we have that fH(e′
+XH) = e′
+X if and only if there exists an element αY,e′
+X ∈ MY,e′
+X
+such that the following two conditions hold:
+• the element α′
+Z, induced by αY,e′
+X via the natural morphism (Z, DZ) → (Y, DY ), can be represented by a
+linear combination
+α′
+Z =
+�
+eXH ∈SXH
+α′
+Z,eXH ,
+where SXH is a subset of DXH, and αZ,eXH ∈ MZ,eXH ;
+• e′
+XH ∈ SXH.
+
+12
+ZHI HU, YU YANG, AND RUNHONG ZONG
+lem-1
+Lemma 3.6. Let (ℓ, d, f : (Y, DY ) → (X, DX)) be a triple associated to (X, DX) and gY the genus of Y . Then we
+have #(MY,e) = ℓ2gY +1 − ℓ2gY , e ∈ DX. Moreover, we have #(MY ) = nX(ℓ2gY +1 − ℓ2gY ).
+Proof. Let e ∈ DX. Write De ⊆ DY for the set f −1(e). Then MY,e can be naturally regarded as a subset of
+H1
+ét(Y \ De, Fℓ) via the natural open immersion Y \ De ֒→ Y. Write Le for the Fℓ-vector space generated by MY,e in
+H1
+ét(Y \ De, Fℓ). Then we have MY,e = Le \ H1
+ét(Y, Fℓ). Write He for the quotient Le/H1
+ét(Y, Fℓ). We have an exact
+sequence as follows:
+0 → H1
+ét(Y, Fℓ) → Le → He → 0.
+Since the action of µd on f −1(e) is transitive, we obtain dimFℓ(He) = 1. On the other hand, since dimFℓ(H1
+ét(Y, Fℓ)) =
+2gY , we obtain #(MY,e) = ℓ2gY +1 − ℓ2gY . Thus, we have #(MY ) = nX(ℓ2gY +1 − ℓ2gY ). This completes the proof
+of the lemma.
+□
+sec-3
+3.3. Reconstructions of inertia subgroups. We maintain the notation introduced in
+curves
+2.1.1.
+3.3.1.
+We will prove that the inertia subgroups of marked points can be mono-anabelian reconstructed from πt
+1(UX)
+(i.e. Proposition
+them-1
+3.10). The main idea is as follows: Let H ⊆ πt
+1(UX) be an arbitrary normal open subgroup
+and (XH, DXH) → (X, DX) the tame covering corresponding to H. Firstly, by using some numerical conditions
+induced by the Riemann-Hurwitz formula, the étale fundamental group π1(X) can be mono-anabelian reconstructed
+from πt
+1(UX). Then the results obtained in Section
+sec-2
+3.2 implies that DX can be mono-anabelian reconstructed from
+πt
+1(UX). Moreover, DXH can also be mono-anabelian reconstructed from H. Secondly, since the natural injection
+H ֒→ πt
+1(UX) induces a map of sets of cohomological classes obtained in Section
+sec-2
+3.2, we obtain that the natural map
+DXH → DX can be mono-anabelian reconstructed from H ֒→ πt
+1(UX). Thus, by taking a cofinal system of open
+normal subgroups of πt
+1(UX), we obtain a new mono-anabelian reconstruction of Ine(πt
+1(UX)).
+3.3.2.
+First, we have the following lemma.
+lem-2
+Lemma 3.7.
+(1) The prime number p (i.e., the characteristic of k) can be mono-anabelian reconstructed from πt
+1(UX).
+(2) The étale fundamental group π1(X) can be mono-anabelian reconstructed from πt
+1(UX).
+Proof. (1) Let P be the set of prime numbers, and let Q be an arbitrary open subgroup of πt
+1(UX) and rQ an integer
+such that
+#{l ∈ P | rQ = dimFl(Qab ⊗ Fl)} = ∞.
+Then we see immediately that the characteristic of k is the unique prime number p such that there exists an open
+subgroup T ⊆ πt
+1(UX) and rT ̸= dimFp(T ab ⊗ Fp).
+(2) Let H be an arbitrary open normal subgroup of πt
+1(UX). We denote by (XH, DXH) the smooth pointed stable
+curve of type (gXH, nXH) over k induced by H, and denote by fH : (XH, DXH) → (X, DX) the morphism of
+smooth pointed stable curves over k induced by the natural inclusion H ֒→ πt
+1(UX). We note that fH is étale if and
+only if gXH − 1 = #(πt
+1(UX)/H)(gX − 1). We put
+Et(πt
+1(UX)) := {H ⊆ πt
+1(UX) is an open normal subgroup : gXH − 1 = #(πt
+1(UX)/H)(gX − 1)}.
+Moreover, Proposition
+proposition 1
+3.2 (1) implies that gXH and gX can be mono-anabelian reconstructed from H and πt
+1(UX),
+respectively. Then the set Et(πt
+1(UX)) can be mono-anabelian reconstructed from πt
+1(UX). We obtain that
+π1(X) = πt
+1(UX)/
+�
+H∈Et(πt
+1(UX))
+H.
+This completes the proof of the lemma.
+□
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+13
+gpmptriple
+3.3.3.
+Suppose gX ≥ 2. Let us define a group-theoretical object corresponding to an mp-triple which was introduced
+in
+triple
+3.2.2. We shall say that (ℓ, d, y) is an mp-triple associated to πt
+1(UX) if the following two conditions hold:
+• ℓ and d are prime numbers distinct from each other such that (ℓ, p) = (d, p) = 1 and ℓ ≡ 1 (mod d); then all
+d-th roots of unity are contained in Fℓ;
+• y ∈ Hom(π1(X), µd) such that y ̸= 0, where µd ⊆ F×
+ℓ denotes the subgroup of d-th roots of unity.
+3.3.4.
+Moreover, by applying Lemma
+lem-2
+3.7, there is a triple (ℓ, d, y) associated to πt
+1(UX) which can be mono-
+anabelian reconstructed from πt
+1(UX). Let f : (Y, DY ) → (X, DX) be a Galois étale covering induced by y.
+Then we see immediately that (ℓ, d, f : (Y, DY ) → (X, DX)) is an mp-triple associated to (X, DX) defined in
+triple
+3.2.2. We denote by πt
+1(UY ) the kernel of the composition of the surjections πt
+1(UX) ։ π1(X)
+y։ µd. Since
+H1
+ét(Y, Fℓ) ∼= Hom(π1(Y ), Fℓ) and H1
+ét(UY , Fℓ) ∼= Hom(πt
+1(UY ), Fℓ), Lemma
+lem-2
+3.7 implies immediately that the fol-
+lowing exact sequence
+0 → H1
+ét(Y, Fℓ) → H1
+ét(UY , Fℓ) → Div0
+DY (Y ) ⊗ Fℓ → 0
+can be mono-anabelian reconstructed from πt
+1(UY ). Thus, Proposition
+proposition 1
+3.2 (1) implies that the set MY / ∼ defined in
+315
+3.2.4 can be mono-anabelian reconstructed from πt
+1(UY ). Note that, by Remark
+rem-2-1
+3.4, the set MY / ∼ does not depend
+on the choices of mp-triples. Then we put
+Dgp
+X := MY / ∼,
+where “gp" means “group-theoretical". By Proposition
+pro-2
+3.3, we may identify Dgp
+X with the set of marked points DX
+of (X, DX) via the bijection ϑX : Dgp
+X
+≃
+−→ DX defined in Proposition
+pro-2
+3.3.
+pro-3
+Proposition 3.8. Let H ⊆ πt
+1(UX) be an arbitrary open normal subgroup and
+fH : (XH, DXH) → (X, DX)
+the morphism of smooth pointed stable curves over k induced by the natural inclusion H ֒→ πt
+1(UX). Suppose
+gX ≥ 2. Then the sets Dgp
+X and Dgp
+XH can be mono-anabelian reconstructed from πt
+1(UX) and H, respectively.
+Moreover, the inclusion H ֒→ πt
+1(UX) induces a map γH,πt
+1(UX) : Dgp
+XH → Dgp
+X such that the following commutative
+diagram holds:
+Dgp
+XH
+ϑXH
+−−−−→ DXH
+γH,πt
+1(UX )�
+�γfH
+Dgp
+X
+ϑX
+−−−−→ DX,
+where γfH denotes the map of the sets of marked points induced by fH.
+Proof. We only need to prove the “moreover" part of Proposition
+pro-3
+3.8. We maintain the notation introduced in Remark
+rem-2-2
+3.5. Note that, for each eX ∈ DX and each eXH ∈ DXH, the sets MY,eX and MZ,eXH can be mono-anabelian
+reconstructed from πt
+1(UX) and H, respectively. Then the “moreover" part follows from Remark
+rem-2-2
+3.5.
+□
+rem-pro-3-1
+Remark 3.9. We maintain the notation introduced in Proposition
+pro-3
+3.8. Let π1(XH) be the étale fundamental group of
+XH. Then we have a natural surjection H ։ π1(XH). Note that π1(XH) admits an action of πt
+1(UX)/H induced by
+the outer action of πt
+1(UX)/H on H which is induced by the exact sequence
+1 → H → πt
+1(UX) → πt
+1(UX)/H → 1.
+Moreover, the action of πt
+1(UX)/H on π1(XH) induces an action of πt
+1(UX)/H on Dgp
+XH. On the other hand, it is
+easy to check that the action of πt
+1(UX)/H on Dgp
+XH coincides with the natural action of πt
+1(UX)/H on DXH when
+we identify Dgp
+X with DX.
+
+14
+ZHI HU, YU YANG, AND RUNHONG ZONG
+3.3.5.
+We have the following result.
+them-1
+Proposition 3.10. Write Ine(πt
+1(UX)) for the set of inertia subgroups in πt
+1(UX). Then Ine(πt
+1(UX)) can be mono-
+anabelian reconstructed from πt
+1(UX).
+Proof. Let CX := {Hi}i∈Z>0 be a set of open normal subgroups of πt
+1(UX) such that lim
+←−i πt
+1(UX)/Hi ∼= πt
+1(UX)
+(i.e., a cofinal system of open normal subgroups).
+Let �e ∈ D �
+X. For each i ∈ Z>0, we write (XHi, DXHi) for the smooth pointed stable curve of type (gXHi , nXHi )
+induced by Hi and eXHi ∈ DXHi for the image of �e. Then we obtain a sequence of marked points
+ICX
+�e
+: · · · �→ eXH2 �→ eXH1
+induced by CX. Note that the sequence ICX
+�e
+admits a natural action of πt
+1(UX). We may identify the inertia subgroup
+I�e associated to �e with the stabilizer of ICX
+�e
+.
+Moreover, since Proposition
+proposition 1
+3.2 (1) implies that (gXHi , nXHi ) can be mono-anabelian reconstructed from Hi, by
+choosing a suitable set of open normal subgroups CX, we may assume that gXH1 ≥ 2. If nXH1 = 0, Proposition
+them-1
+3.10
+is trivial. Then we may assume that nXH1 > 0.
+On the other hand, Proposition
+pro-3
+3.8 implies that, for each Hi, i ∈ Z>0, the set Dgp
+XHi can be mono-anabelian
+reconstructed from Hi. For each eXHi ∈ DXHi , we denote by
+egp
+XHi := ϑ−1
+XHi (eXHi ).
+Then the sequence of marked points ICX
+�e
+induces a sequence
+ICX
+�egp : · · · �→ egp
+XH2 �→ egp
+XH1 .
+By applying the “moreover” part of Proposition
+pro-3
+3.8, we see that ICX
+�egp can be mono-anabelian reconstructed from CX.
+Then Remark
+rem-pro-3-1
+3.9 implies that the stabilizer of ICX
+�egp is equal to the stabilizer of ICX
+�e
+. This completes the proof of the
+proposition.
+□
+sec-4
+3.4. Reconstructions of inertia subgroups via surjections. In this subsection, we will prove that the mono-
+anabelian reconstructions obtained in Proposition
+them-1
+3.10 are compatible with any open continuous homomorphisms
+(i.e. Theorem
+them-2
+3.18).
+sett331
+3.4.1. Settings. Let (Xi, DXi), i ∈ {1, 2}, be a smooth pointed stable curve of type (gX, nX) over an algebraically
+closed field ki of characteristic p > 0, UXi := Xi \ DXi, πt
+1(UXi) the tame fundamental group of UXi, and π1(Xi)
+the étale fundamental group of Xi. Then Lemma
+lem-2
+3.7 implies that π1(Xi) can be mono-anabelian reconstructed from
+πt
+1(UXi). Moreover, in this subsection, we suppose that nX > 0, and that φ : πt
+1(UX1) ։ πt
+1(UX2) is an arbitrary
+open continuous surjective homomorphism of profinite groups.
+Note that, since (Xi, DXi), i ∈ {1, 2}, is a smooth pointed stable curve of type (gX, nX), φ induces a natural
+surjection φp′ : πt
+1(UX1)p′ ։ πt
+1(UX2)p′, where (−)p′ denotes the maximal prime-to-p quotient of (−). Since
+πt
+1(UXi)p′, i ∈ {1, 2}, is topologically finitely generated, and πt
+1(UX1)p′ is isomorphic to πt
+1(UX2)p′ as abstract
+profinite groups, we obtain that φp′ : πt
+1(UX1)p′
+≃
+−→ πt
+1(UX2)p′ is an isomorphism (
+FJ
+[FJ, Proposition 16.10.6]).
+3.4.2.
+We explain the main idea in the proof of Theorem
+them-2
+3.18. Let H2 ⊆ πt
+1(UX2) be an arbitrary open normal
+subgroup and H1 := φ−1(H2) ⊆ πt
+1(UX1). We write (XHi, DXHi), i ∈ {1, 2}, for the smooth pointed smooth curve
+of type (gXHi , nXHi ) over ki induced by Hi. To prove the compatibility, we need to prove that, for any prime number
+ℓ ̸= p, the weight-monodromy filtration of Hab
+2
+⊗ Fℓ induces the weight-monodromy filtration of Hab
+1
+⊗ Fℓ via the
+natural surjection φ|H1 : H1 ։ H2. Note that the weight 1 part of Hab
+i
+⊗ Fℓ corresponds to π1(XHi)ab ⊗ Fℓ, and the
+weight 2 part of Hab
+i
+⊗ Fℓ corresponds to the image of the subgroup of Hi generated by the inertia subgroups of the
+marked points of DXHi. The key observation is as follows:
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+15
+The inequality of the limit of p-averages (see Proposition
+coro-p-average
+3.11 (1) below)
+Avrp(H1) ≥ Avrp(H2)
+of H1 and H2 induced by the surjection φ|H1 : H1 ։ H2 plays a role of the comparability of
+“Galois actions” in the theory of the anabelian geometry of curves over algebraically closed fields of
+characteristic p > 0.
+paverage
+3.4.3.
+Firstly, we have the following proposition.
+Proposition 3.11.
+coro-p-average
+(1) Let (X, DX) be a pointed stable curve of type (gX, nX) over an algebraically closed field k of characteristic
+p > 0, UX := X \ DX, and πt
+1(UX) the tame fundamental group of UX. Let r ∈ N be a natural number, and
+let Kpr−1 be the kernel of the natural surjection πt
+1(UX) ։ πt
+1(UX)ab ⊗ Z/(pr − 1)Z, where (−)ab denotes
+the abelianization of (−). Then we have
+Avrp(πt
+1(UX)) := lim
+r→∞
+dimFp(Kab
+pr−1 ⊗ Fp)
+#(πt
+1(UX)ab ⊗ Z/(pr − 1)Z) =
+�
+gX − 1,
+if nX ≤ 1,
+gX,
+if nX > 1.
+(2) We maintain the setting introduced in
+sett331
+3.4.1. Let H2 ⊆ πt
+1(UX2) be an open normal subgroup such that
+([πt
+1(UX2) : H2], p) = 1 and H1 := φ−1(H2). Write gHi, i ∈ {1, 2}, for the genus of the smooth pointed
+stable curve over ki corresponding to Hi ⊆ πt
+1(UXi). Then we have gH1 ≥ gH2.
+Proof. (1) is the Tamagawa’s result concerning the limit of p-averages of πt
+1(UX) (
+T4
+[T4, Theorem 0.5]). Let us prove
+(2). The surjection φ induces a surjection φp′ : πt
+1(UX1)p′ ։ πt
+1(UX2)p′, where (−)p′ denotes the maximal prime-
+to-p quotient of (−). Moreover, since πt
+1(UXi)p′, i ∈ {1, 2}, is topologically finitely generated, and πt
+1(UX1)p′
+is isomorphic to πt
+1(UX2)p′ as abstract profinite groups (since the types of (X1, DX1) and (X2, DX2) are equal to
+(gX, nX)), we obtain that φp′ is an isomorphism (cf.
+FJ
+[FJ, Proposition 16.10.6]).
+On the other hand, since [πt
+1(UX1) : H1] = [πt
+1(UX2) : H2] and ([πt
+1(UX2) : H2], p) = 1, we obtain that the
+natural homomorphism φp′
+H : Hp′
+1 ։ Hp′
+2 induced by φH := φ|H1 : H1 ։ H2 is also an isomorphism. This implies
+#(Hab
+1
+⊗ Z/(pr − 1)Z) = #(Hab
+2
+⊗ Z/(pr − 1)Z)
+for all r ∈ N. Let KHi,pr−1, i ∈ {1, 2}, be the kernel of the natural surjection Hi ։ Hab
+i
+⊗ Z/(pr − 1)Z. Then the
+surjection φH implies
+Avrp(H1) := lim
+r→∞
+dimFp(Kab
+H1,pr−1 ⊗ Fp)
+#(Hab
+1
+⊗ Z/(pr − 1)Z) ≥ Avrp(H2) := lim
+r→∞
+dimFp(Kab
+H2,pr−1 ⊗ Fp)
+#(Hab
+2 ⊗ Z/(pr − 1)Z).
+Thus, the corollary follows from (2).
+□
+3.4.4.
+We have the following lemmas.
+lem-3
+Lemma 3.12. Let ℓ be a prime number distinct from p. Then the isomorphism (φp′)−1 : πt
+1(UX2)p′
+≃
+−→ πt
+1(UX1)p′
+induces an isomorphism
+ψℓ
+X : H1
+ét(X1, Fℓ) ≃ Hom(π1(X1), Fℓ)
+≃
+−→ Hom(π1(X2), Fℓ) ≃ H1
+ét(X2, Fℓ).
+Proof. Let f1 : (Y1, DY1) → (X1, DX1) be an étale covering of degree ℓ over k1. Write f2 : (Y2, DY2) → (X2, DX2)
+for the connected Galois tame covering of degree ℓ over k2 induced by φp′. Then we will prove that f2 is also an étale
+covering over k2.
+Write gY1 and gY2 for the genus of Y1 and Y2, respectively. Since f1 is an étale covering of degree ℓ, the Riemann-
+Hurwitz formula implies gY1 = ℓ(gX1 − 1) + 1. On the other hand, the Riemann-Hurwitz formula implies gY2 =
+
+16
+ZHI HU, YU YANG, AND RUNHONG ZONG
+ℓ(gX2 − 1) + 1 + 1
+2(ℓ − 1)#(Ramf2). By applying Proposition
+coro-p-average
+3.11 (2), the surjection φ implies gY1 ≥ gY2. This
+means #(Ramf2) = 0. So f2 is an étale covering over k2. Then the morphism (φp′)−1 induces an injection
+ψℓ
+X : Hom(π1(X1), Fℓ) ֒→ Hom(π1(X2), Fℓ).
+Furthermore, since dimFℓ(Hom(π1(X1), Fℓ)) = dimFℓ(Hom(π1(X2), Fℓ)) = 2gX, we obtain that ψℓ
+X is a bijection.
+This completes the proof of the lemma.
+□
+lem-4
+Lemma 3.13. Suppose gX ≥ 2. Then the surjection φ : πt
+1(UX1) ։ πt
+1(UX2) induces a bijection
+ρφ : Dgp
+X1
+≃
+−→ Dgp
+X2,
+and the bijection ρφ can be mono-anabelian reconstructed from φ.
+Proof. Let (ℓ, d, y2) be an mp-triple associated to πt
+1(UX2) (see
+gpmptriple
+3.3.3). Then Lemma
+lem-3
+3.12 implies that φ induces an
+mp-triple (ℓ, d, y1) associated to πt
+1(UX1), where y1 := (ψd
+X)−1(y2) ∈ Hom(π1(X1), µd).
+Let fi : (Yi, DYi) → (Xi, DXi), i ∈ {1, 2}, be the étale covering of degree d over ki induced by yi. Then the
+mp-triple (ℓ, d, yi) associated to πt
+1(UXi) determines an mp-triple
+(ℓ, d, fi : (Yi, DYi) �� (Xi, DXi))
+associated to (Xi, DXi) over ki. Note that the types of (Y1, DY1) and (Y2, DY2) are equal.
+Write πt
+1(UYi), i ∈ {1, 2}, for the kernel of πt
+1(UXi) ։ π1(Xi)
+yi
+։ µd. By replacing (Xi, DXi) by (Yi, DYi),
+Lemma
+lem-3
+3.12 implies that (φ|p′
+πt
+1(UY1 ))−1 induces a commutative diagram as follows:
+0 −−−−→ H1
+ét(Y1, Fℓ) −−−−→ H1
+ét(UY1, Fℓ) −−−−→ Div0
+DY1 (Y1) ⊗ Fℓ −−−−→ 0
+ψℓ
+Y
+�
+ψt,ℓ
+Y
+�
+�
+0 −−−−→ H1
+ét(Y2, Fℓ) −−−−→ H1
+ét(UY2, Fℓ) −−−−→ Div0
+DY2 (Y2) ⊗ Fℓ −−−−→ 0,
+where all the vertical arrows are isomorphisms. We note that H1
+ét(Yi, Fℓ), H1
+ét(UYi, Fℓ), and Div0
+DYi (Yi) ⊗ Fℓ, i, ∈
+{1, 2}, are naturally isomorphic to Hom(π1(Yi), Fℓ), Hom(πt
+1(UYi), Fℓ), and Hom(πt
+1(UYi), Fℓ)/Hom(π1(Yi), Fℓ),
+respectively. Then Lemma
+lem-2
+3.7 implies that the commutative diagram above can be mono-anabelian reconstructed from
+φ|πt
+1(UY1 ) : πt
+1(UY1) ։ πt
+1(UY2).
+Write MYi ⊆ M ∗
+Yi for the subsets of H1
+ét(UYi, Fℓ) defined in
+sec31aaa
+3.2.3. Since the actions of µd on the exact sequences
+are compatible with the isomorphisms appearing in the commutative diagram above, we have ψt,ℓ
+Y (M ∗
+Y1) = M ∗
+Y2.
+Next, we prove ψt,ℓ
+Y (MY1) = MY2.
+Let α1 ∈ MY1 and gα1 : (Yα1, DYα1) → (Y1, DY1) the Galois tame covering of degree ℓ over k1 induced by α1.
+Write gα2 : (Yα2, DYα2) → (Y2, DY2) for the Galois tame covering of degree ℓ over k2 induced by α2 := ψt,ℓ
+Y (α1).
+Write gYα1 and gYα2 for the genus of Yα1 and Yα2, respectively. Then Proposition
+coro-p-average
+3.11 (2) and the Riemann-Hurwitz
+formula imply that gYα1 −gYα2 = 1
+2(d−#(Ramgα2 ))(ℓ−1) ≥ 0. This means d−#(Ramgα2 ) ≥ 0. Since α2 ∈ M ∗
+Y2,
+we have d | #(Ramgα2 ). Thus, either #(Ramgα2 ) = 0 or #(Ramgα2 ) = d holds.
+If #(Ramgα2 ) = 0, then gα2 is an étale covering over k2. Then Lemma
+lem-3
+3.12 implies that gα1 is an étale covering
+over k1. This provides a contradiction to the fact that α1 ∈ MY1. Then we have #(Ramgα2 ) = d. This means
+α2 ∈ MY2. Thus, we obtain ψt,ℓ
+Y (MY1) ⊆ MY2. On the other hand, Lemma
+lem-1
+3.6 implies #(MY1) = #(MY2). We
+have ψt,ℓ
+Y : MY1
+≃
+−→ MY2. Then Proposition
+pro-2
+3.3 implies that ψt,ℓ
+Y induces a bijection
+ρφ : Dgp
+X1
+≃
+−→ Dgp
+X2.
+Moreover, since MYi and M ∗
+Yi can be mono-anabelian reconstructed from πt
+1(UYi), the bijection ρφ can be mono-
+anabelian reconstructed from φ. This completes the proof of the lemma.
+□
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+17
+sec334
+3.4.5.
+Let H2 ⊆ πt
+1(UX2) be an arbitrary open normal subgroup and H1 := φ−1(H2). We write (XHi, DXHi ),
+i ∈ {1, 2}, for the smooth pointed stable curve of type (gXHi , nXHi ) over ki induced by Hi and fHi : (XHi, DXHi ) →
+(Xi, DXi) for the Galois tame coverings over ki induced by the inclusion Hi ֒→ πt
+1(UXi). Moreover, Proposition
+pro-3
+3.8
+implies that the inclusion Hi ֒→ πt
+1(UXi) induces a map γHi,πt
+1(UXi ) : Dgp
+XHi → Dgp
+Xi which fits into the following
+commutative diagram:
+Dgp
+XHi
+ϑXHi
+−−−−→ DXHi
+γHi,πt
+1(UXi )�
+�γfHi
+Dgp
+Xi
+ϑXi
+−−−−→ DXi,
+where γfHi denotes the map of the sets of marked points induced by fHi. We may identify πt
+1(UX1)/H1 with
+πt
+1(UX2)/H2 via the isomorphism πt
+1(UX1)/H1
+≃
+−→ πt
+1(UX2)/H2 induced by φ, and denote by G := πt
+1(UX1)/H1 ∼=
+πt
+1(UX2)/H2. Then we have the following lemma.
+lem-5
+Lemma 3.14. Suppose that gX ≥ 2, and that (gXH1 , nXH1 ) = (gXH2 , nXH2 ). Then the commutative diagram of
+profinite groups
+H1
+φ|H1
+−−−−→
+H2
+�
+�
+πt
+1(UX1)
+φ
+−−−−→ πt
+1(UX2)
+(3.1)
+induces a commutative diagram
+Dgp
+XH1
+ρφ|H1
+−−−−→ Dgp
+XH2
+γH1,πt
+1(UX1 )�
+�γH2,πt
+1(UX2 )
+Dgp
+X1
+ρφ
+−−−−→ Dgp
+X2.
+(3.2)
+Moreover, the commutative diagram (2) can be mono-anabelian reconstructed from (1).
+Proof. Proposition
+pro-3
+3.8 and Lemma
+lem-4
+3.13 imply the diagram
+Dgp
+XH1
+ρφ|H1
+−−−−→ Dgp
+XH2
+γH1,πt
+1(UX1 )�
+�γH2,πt
+1(UX2 )
+Dgp
+X1
+ρφ
+−−−−→ Dgp
+X2
+can be mono-anabelian reconstructed from the commutative diagram of profinite groups
+H1
+φ|H1
+−−−−→
+H2
+�
+�
+πt
+1(UX1)
+φ
+−−−−→ πt
+1(UX2).
+To verify Lemma
+lem-5
+3.14, it is sufficient to check that the diagram is commutative.
+Let egp
+XH1 ∈ Dgp
+XH1 , egp
+XH2 := ρφ|H1 (egp
+XH1 ) ∈ Dgp
+XH2 , egp
+X1 := γH1,πt
+1(UX1 )(egp
+XH1 ) ∈ Dgp
+X1, egp
+X2 := (γH2,πt
+1(UX2 ) ◦
+ρφ|H1)(egp
+XH1 ) ∈ Dgp
+X2, and egp,∗
+X1
+:= ρ−1
+φ (egp
+X2) ∈ Dgp
+X1. Let us prove
+egp
+X1 = egp,∗
+X1 .
+We put Sgp
+XH1 := γ−1
+H1,πt
+1(UX1)(egp,∗
+X1 ) and Sgp
+XH2 := γ−1
+H2,πt
+1(UX2 )(egp
+X2), respectively. Note that egp
+XH2 ∈ Sgp
+XH2 . To
+verify egp
+X1 = egp,∗
+X1 , it is sufficient to prove that egp
+XH1 ∈ Sgp
+XH1 . Moreover, for each i ∈ {1, 2}, we put
+eXi := ϑXi(egp
+Xi), eXHi := ϑXHi (egp
+Xi), e∗
+X1 := ϑX1(egp,∗
+X1 ), SXi := Sgp
+Xi, SXHi := Sgp
+XHi .
+
+18
+ZHI HU, YU YANG, AND RUNHONG ZONG
+Then to verify the lemma, we only need to prove that eXH1 ∈ ϑXH1 (SXH1 ).
+Let (ℓ, d, y2) be an mp-triple associated to πt
+1(UX2). Then Lemma
+lem-3
+3.12 implies that φ induces an mp-triple (ℓ, d, y1)
+associated to πt
+1(UX1), where y1 := (ψd
+X)−1(y2) ∈ Hom(π1(X1), µd). Let fi : (Yi, DYi) → (Xi, DXi), i ∈ {1, 2},
+be the tame covering of degree d over ki induced by yi. Then the mp-triple (ℓ, d, yi) associated to πt
+1(UXi) induces an
+mp-triple
+(ℓ, d, fi : (Yi, DYi) → (Xi, DXi))
+associated to (Xi, DXi) over ki. Note that since f1 and f2 are étale, the types of (Y1, DY1) and (Y2, DY2) are equal.
+On the other hand, we have an mp-triple
+(ℓ, d, g2 : (Z2, DZ2) := (Y2, DY2) ×(X2,DX2 ) (XH2, DXH2 ) → (XH2, DXH2 ))
+associated to (XH2, DXH2 ) induced by the natural inclusion H2 ֒→ πt
+1(UX2) and the mp-triple (ℓ, d, f2 : (Y2, DY2) →
+(X2, DX2)). By Lemma
+lem-3
+3.12 again, we obtain an mp-triple
+(ℓ, d, g1 : (Z1, DZ1) := (Y1, DY1) ×(X1,DX1 ) (XH1, DXH1 ) → (XH1, DXH1 ))
+associated to (XH1, DXH1 ) induced by φ|H1 and the triple (ℓ, d, g2 : (Z2, DZ2) → (XH2, DXH2 )).
+Let α2 ∈ MY2,eX2 . The final paragraph of the proof of Lemma
+lem-4
+3.13 implies that we have a bijection MY1 =
+�
+e∈DX1 MY1,e
+≃
+−→ MY2 = �
+e∈DX2 MY2,e induced by φ. Then α2 induces an element α1 ∈ MY1,e∗
+X1 . Write
+(Yα1, DYα1) and (Yα2, DYα2) for the smooth pointed stable curves over k1 and k2 induced by α1 and α2, respectively.
+Consider the connected Galois tame covering
+(Yα2, DYα2) ×(X2,DX2 ) (XH2, DXH2 ) → (Z2, DZ2)
+of degree ℓ over k2, and write β2 for an element of M ∗
+Z2 corresponding to this connected Galois tame covering. Then
+we have
+β2 =
+�
+c2∈SXH2
+tc2βc2,
+where tc2 ∈ (Z/ℓZ)× and βc2 ∈ MZ2,c2. On the other hand, the proof of Lemma
+lem-4
+3.13 implies that β2 induces an
+element
+β1 :=
+�
+c2∈SXH2 \{eXH2 }
+tc2βρ−1
+φ|H1
+(c2) + teXH2 βρ−1
+φ|H1
+(eXH2 )
+=
+�
+c2∈SXH2 \{eXH2 }
+tc2βρ−1
+φ|H1
+(c2) + teXH2 βeXH1 ∈ M ∗
+Z1.
+Then we have that the coefficient teXH2 of βeXH1 is not equal to 0. Thus, the composition
+(Yα1, DYα1) ×(X1,DX1 ) (XH1, DXH1 ) → (Z1, DZ1)
+g1
+→ (XH1, DXH1)
+is tamely ramified over eXH1 . This means that eXH1 is contained in SXH1 . This completes the proof of the lemma.
+□
+rem-lem-5-1
+Remark 3.15. Remark
+rem-pro-3-1
+3.9 implies that Dgp
+XHi , i ∈ {1, 2}, admits a natural action of G. Moreover, the commutative
+diagram
+Dgp
+XH1
+ρφ|H1
+−−−−→ Dgp
+XH2
+γH1,πt
+1(UX1 )�
+�γH2,πt
+1(UX2 )
+Dgp
+X1
+ρφ
+−−−−→ Dgp
+X2
+is compatible with the actions of G.
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+19
+3.4.6.
+Next, we prove that the condition (gXH1 , nXH1 ) = (gXH2 , nXH2 ) mentioned in Lemma
+lem-5
+3.14 can be omitted.
+Firstly, we treat the case of abelian groups.
+lem-6
+Lemma 3.16. We maintain the notation introduced in
+sec334
+3.4.5. Suppose that gX ≥ 2, and that G is an abelian group.
+Then we have (gXH1 , nXH1 ) = (gXH2 , nXH2).
+Proof. We write m for #G and put K2 := ker(πt
+1(UX2) ։ πt
+1(UX2)ab ⊗ Z/mZ). Then we see immediately that K2
+is contained in H2. Let K1 := φ−1(K2) ⊆ H1. Write (XKi, DXKi ) for the smooth pointed stable curves of type
+(gXKi , nXKi ) over ki induced by Ki and fKi : (XKi, DXKi ) → (Xi, DXi) for the tame covering over ki induced by
+the inclusion Ki ֒→ πt
+1(UXi). We identify πt
+1(UX1)/K1 with πt
+1(UX2)/K2 via the isomorphism induced by φ, and
+denote by A := πt
+1(UX1)/K1 ≃ πt
+1(UX2)/K2.
+Since each p-Galois tame covering is étale (i.e., Galois tame coverings whose Galois group is a p-group), we see
+immediately that (gXK1 , nXK1 ) = (gXK2 , nXK2 ). Then Lemma
+lem-5
+3.14 implies that the commutative diagram
+K1
+φ|K1
+−−−−→
+K2
+�
+�
+πt
+1(UX1)
+φ
+−−−−→ πt
+1(UX2)
+of profinite groups induces a commutative diagram
+Dgp
+XK1
+ρφ|K1
+−−−−→ Dgp
+XK2
+γK1,πt
+1(UX1 )�
+�γK2,πt
+1(UX2 )
+Dgp
+X1
+ρφ
+−−−−→ Dgp
+X2.
+Moreover, Remark
+rem-lem-5-1
+3.15 implies that the commutative diagram above admits a natural action of A. Then, for each
+egp
+XK1 ∈ Dgp
+XK1 , the inertia subgroup Iegp
+XK1 in A associated to egp
+XK1 (i.e., the stabilizer of egp
+XK1 under the action
+of A) is equal to the inertia subgroup Iegp
+XK2 in A associated to egp
+XK2 := ρφ|K1 (egp
+XK1 ) ∈ Dgp
+XK2 . On the other
+hand, write F for the kernel of the natural morphism A ։ G induced by the inclusion Ki ֒→ Hi, i ∈ {1, 2}.
+Since (XHi, DXHi ) ≃ (XKi, DXKi )/F, the set of ramification indices of the Galois tame covering (XKi, DXKi ) →
+(XHi, DXHi ) with Galois group F are equal to {#(F ∩ Iegp
+XKi )}egp
+XKi ∈Dgp
+XKi . Then by the Riemann-Hurwitz formula,
+we have (gXH1 , nXH1 ) = (gXH2 , nXH2 ). This completes the proof of the lemma.
+□
+Next, we treat the general case.
+lem-7
+Lemma 3.17. We maintain the notation introduced in
+sec334
+3.4.5. Suppose that gX ≥ 2 and nX ≥ 2. Then there exists an
+open normal subgroup P2 ⊆ πt
+1(UX2) which is contained in H2 such that the following holds:
+Write (XPi, DXPi ), i ∈ {1, 2}, for the smooth pointed stable curve of type (gXPi , nXPi ) over ki
+induced by Pi, where P1 = φ−1(P2). We have (gXP1 , nXP1 ) = (gXP2 , nXP2 ).
+Proof. First, suppose that G is a simple finite group. By applying Lemma
+lem-6
+3.16, we may assume that G is non-abelian.
+Moreover, we claim that we may assume that nX is a positive even number. Let us prove this claim. Suppose p ̸= 2.
+Let R2 ⊆ πt
+1(UX2) be an open subgroup such that #(πt
+1(UX2)/R2) = 2, and that R2 ⊇ ker(πt
+1(UX2) ։ π1(X2))
+(i.e., the cyclic Galois tame covering corresponding to R2 is étale). Let R1 := φ−1(R2) ⊆ πt
+1(UX1). Then we have
+that #(πt
+1(UX1)/R1) = 2, and that Lemma
+lem-3
+3.12 implies R1 ⊇ ker(πt
+1(UX1) ։ π1(X1)). By replacing Hi and
+πt
+1(UXi), i ∈ {1, 2}, by Hi ∩ Ri and Ri, respectively, we may assume that nX is a even positive number. Suppose
+that p = 2. Let ℓ be a prime number such that (ℓ, 2) = (ℓ, #G) = 1. By
+R1
+[R1, Théorème 4.3.1], there exists an open
+subgroup R∗
+2 ⊆ πt
+1(UX2) such that #(πt
+1(UX2)/R∗
+2) = ℓ, that R∗
+2 ⊇ ker(πt
+1(UX2) ։ π1(X2)), and that
+dimFp(R∗,ab
+2
+⊗ Fp) > 0.
+
+20
+ZHI HU, YU YANG, AND RUNHONG ZONG
+Let R∗
+1 := φ−1(R∗
+2) ⊆ πt
+1(UX1). Then we have that #(πt
+1(UX1)/R∗
+1) = ℓ, that dimFp(R∗,ab
+1
+⊗ Fp) > 0, and that
+Lemma
+lem-3
+3.12 implies R∗
+1 ⊇ ker(πt
+1(UX1) ։ π1(X1)). Thus, we may take an open subgroup R′
+2 ⊆ R∗
+2 such that
+πt
+1(UX2)/R′
+2 ≃ Z/2Z ⋊ Z/ℓZ,
+and that R′
+2 ⊇ ker(πt
+1(UX2) ։ π1(X2)).
+We put R′
+1 := φ−1(R′
+2).
+Then the construction of R′
+1 implies
+πt
+1(UX1)/R′
+1 ≃ Z/2Z ⋊ Z/ℓZ and R′
+1 ⊇ ker(πt
+1(UX1) ։ π1(X1)). By replacing Hi and πt
+1(UXi), i ∈ {1, 2},
+by Hi ∩ R′
+i and R′
+i, respectively, we may assume that nX is a even positive number. This completes the proof of the
+claim.
+Let #G := ptm′ such that (m′, p) = 1. Since nX is a positive even number, we may choose a Galois tame covering
+f2 : (Y2, DY2) → (X2, DX2)
+over k2 with Galois group Z/m′Z such that f2 is totally ramified over every marked point of DX2. Write (gY2, nY2)
+for the type of (Y2, DY2), Q2 ⊆ πt
+1(UX2) for the open normal subgroup induced by f2, Q1 := φ−1(Q2) ⊆ πt
+1(UX1),
+f1 : (Y1, DY1) → (X1, DX1)
+for the Galois tame covering over k1 with Galois group Z/m′Z induced by the natural inclusion Q1 ֒→ πt
+1(UX1), and
+(gY1, nY1) for the type of (Y1, DY1). Then Lemma
+lem-6
+3.16 implies that (gY1, nY1) = (gY2, nY2) and f1 is also totally
+ramified over every marked point of DX1.
+We consider the Galois tame covering
+(Zi, DZi) := (XHi, DXHi ) ×(Xi,DXi) (Yi, DYi) → (Xi, DXi), i ∈ {1, 2},
+over ki with Galois group G × Z/m′Z which is the composition of (Zi, DZi) → (Yi, DYi) and (Yi, DYi) →
+(Xi, DXi). Note that since G is a non-abelian simple finite group, (Zi, DZi) is connected. Moreover, by Abhyankar’s
+lemma, we obtain that (Zi, DZi) → (Yi, DYi) is an étale covering over ki. Since (gY1, nY1) = (gY2, nY2) and
+(Zi, DZi) → (Yi, DYi) is unramified, the Riemann-Hurwitz formula implies (gZ1, nZ1) = (gZ2, nZ2).
+Next, let us prove the lemma in the case where G is an arbitrary finite group. Let G1 ⊆ G2 ⊆ · · · ⊆ Gn := G
+be a sequence of subgroups of G such that Gi/Gi−1 is a simple group for all i ∈ {2, . . .n}. In order to verify the
+lemma, we see that it is sufficient to prove the lemma when n = 2. Let N2 be the kernel of the natural homomorphism
+πt
+1(UX2) ։ G ։ G1 and N1 := φ−1(N2). Then by replacing G by G1 and by applying the lemma for the simple
+group G1, we obtain an open normal subgroup M2 ⊆ πt
+1(UX2) which is contained in N2 such that (gXM1 , nXM1 ) =
+(gXM2 , nXM2 ), where M1 := φ−1(M2), and (gXMi , nXMi ), i ∈ {1, 2}, denotes the type of the smooth pointed stable
+curve corresponding to Mi.
+If Mi ⊆ Hi, i ∈ {1, 2}, then we may put Pi := Mi. If Hi, i ∈ {1, 2}, does not contain Mi, we put Oi := Mi ∩ Hi.
+Then we have Mi/Oi ≃ G/G1. Note that G/G1 is a simple group. Then the lemma follows from the lemma when
+we replace (Xi, DXi) and G by (XMi, DXMi ) and the simple group G/G1, respectively. This completes the proof of
+the lemma.
+□
+3.4.7.
+Now, we prove the main result of the present section.
+them-2
+Theorem 3.18. Let ( �
+Xi, D �
+Xi), i ∈ {1, 2}, be the universal tame covering of (Xi, DXi) defined in
+unicov313
+3.1.2.
+Let
+φ : πt
+1(UX1) ։ πt
+1(UX2) be an arbitrary open continuous surjective homomorphism. Then the group-theoretical
+algorithm of the mono-anabelian reconstruction concerning Ine(πt
+1(UXi)) obtained in Proposition
+them-1
+3.10 is compatible
+with the surjection φ : πt
+1(UX1) ։ πt
+1(UX2). Namely, the following holds: Let �e2 ∈ D �
+X2 and I�e2 ∈ Ine(πt
+1(UX2))
+the inertia subgroup associated to �e2. Then there exists an inertia subgroup I�e1 ∈ Ine(πt
+1(UX1)) associated to a point
+�e1 ∈ D �
+X1 such that
+φ(I�e1) = I�e2,
+and that the restriction homomorphism φ|I�e1 : I�e1 ։ I�e2 is an isomorphism.
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+21
+Proof. If nX = 0, then the theorem is trivial. We suppose nX > 0. Let m >> 0 be an integer such that (m, p) = 1.
+We put Ki := ker(πt
+1(UXi) ։ πt
+1(UXi)ab ⊗ Z/mZ), i ∈ {1, 2}. Write (XKi, DKi) for the smooth pointed stable
+curve of type (gXKi , nXKi ) over ki induced by Ki. Moreover, the condition m >> 0 implies gXK1 = gXK2 ≥
+2, nXK1 = nXK2 ≥ 2.
+By applying Lemma
+lem-7
+3.17, we may choose a set of open subgroups CX2 := {H2,j}j∈Z>0 of πt
+1(UX2) such that the
+following three conditions hold:
+• H2,1 = K2;
+• lim
+←−j πt
+1(UX2)/H2,j ≃ πt
+1(UX2) (i.e. CX2 is a cofinal system);
+• write {H1,j := φ−1(H2,j)}j∈Z>0 for the set of open subgroups of πt
+1(UX1) induced by φ, and for each
+j ∈ Z>0, write (XHi,j, DXHi,j ), i ∈ {1, 2}, for the smooth pointed stable curve of type (gXHi,j , nXHi,j ) over
+ki induced by Hi,j, then we have (gXH1,j , nXH1,j ) = (gXH2,j , nXH2,j ).
+For each j ∈ Z>0, we write eXH2,j ∈ DXH2,j for the image of �e2. Then we obtain a sequence of marked points
+I
+CX2
+�e2
+: · · · �→ eH2,2 �→ eH2,1.
+Proposition
+pro-3
+3.8 implies that, for each H2,j, j ∈ Z>0, the set Dgp
+XH2,j can be mono-anabelian reconstructed from H2,j.
+For each eXH2,j ∈ DXH2,j , we denote by
+egp
+XH2,j := ϑ−1
+XH2,j (eXH2,j ).
+Then the sequence of marked points ICX
+�e2
+induces a sequence
+ICX
+�egp
+2
+: · · · �→ egp
+XH2,2 �→ egp
+XH2,1 .
+Then Remark
+rem-pro-3-1
+3.9 implies that the inertia subgroup associated to �e2 is equal to the stabilizer of ICX
+�egp
+2 .
+By Lemma
+lem-5
+3.14 and Lemma
+lem-7
+3.17, I
+CX2
+�egp
+2
+induces a sequence as follows:
+· · · �→ egp
+XH1,2 := ρ−1
+φ|H1,2 (egp
+XH2,2 ) ∈ Dgp
+XH1,2 �→ egp
+XH1,1 := ρ−1
+φ|H1,1(egp
+XH2,1 ) ∈ Dgp
+XH1,1
+with an action of I�e2. Then Proposition
+them-1
+3.10 implies that we have a sequence
+· · · �→ eXH1,2 := ϑXH1,2 (egp
+XH1,2 ) ∈ DXH1,2 �→ eXH1,1 := ϑXH1,1 (egp
+XH1,1 ) ∈ DXH1,1
+with an action of I�e2
+Let Kker(φ) be the subfield of �K induced by the closed subgroup ker(φ) of πt
+1(UX1), �
+X1,ker(φ) the normalization of
+X1 in Kker(φ), and D �
+X1,ker(φ) the inverse image of DX1 in �
+X1,ker(φ). Then the sequence
+· · · �→ eXH1,2 �→ eXH1,1 .
+determines a point �e1,ker(φ) ∈ D �
+X1,ker(φ). We choose a point of �e1 ∈ D �
+X1 such that the image of �e1 in D �
+X1,ker(φ)
+is �e1,ker(φ). Then we have φ(I�e1) = I�e2. Moreover, since I�e1 and I�e2 are isomorphic to �Z(1)p′, the restriction
+homomorphism φ|I�e1 is an isomorphism. This completes the proof of the theorem.
+□
+sec-new6
+3.5. Reconstructions of additive structures via surjections. We maintain the settings introduced in
+sett331
+3.4.1.
+3.5.1.
+Let �e2 be an arbitrary point of D �
+X2. By applying Theorem
+them-2
+3.18, there exists a point �e1 ∈ D �
+X1 such that
+φ|I�e1 : I�e1
+≃
+−→ I�e2 is an isomorphism. Write Fp,i, i ∈ {1, 2}, for the algebraic closure of Fp in ki. We put
+F�ei := (I�ei ⊗Z (Q/Z)p′
+i ) ⊔ {∗�ei}, i ∈ {1, 2},
+where {∗�ei} is an one-point set, and (Q/Z)p′
+i denotes the prime-to-p part of Q/Z which can be canonically identified
+with �
+(p,m)=1 µm(ki). Moreover, let a�ei be a generator of I�ei. Then we have a natural bijection
+I�ei ⊗Z (Q/Z)p′
+i
+≃
+−→ Z ⊗Z (Q/Z)p′
+i , a�ei ⊗ 1 �→ 1 ⊗ 1.
+
+22
+ZHI HU, YU YANG, AND RUNHONG ZONG
+Thus, we obtain the following bijections
+I�ei ⊗Z (Q/Z)p′
+i
+≃
+−→ Z ⊗Z (Q/Z)p′
+i
+≃
+−→
+�
+(p,m)=1
+µm(ki)
+≃
+−→ F
+×
+p,i.
+This means that F�ei can be identified with Fp,i as sets, and hence admits a structure of field whose multiplicative group
+is I�ei ⊗Z (Q/Z)p′
+i , and whose zero element is ∗�ei.
+3.5.2.
+We will prove that φ|I�e1 : I�e1
+≃
+−→ I�e2 induces an isomorphism F�e1
+≃
+−→ F�e2 as fields (i.e. Proposition
+pro-4
+3.19).
+The main idea is as follows: First, we reduce the problem to the case where nX = 3 by applying Theorem
+them-2
+3.18.
+Second, the field structure of F�ei (i.e., the set of isomorphisms of F�ei and Fp,i as fields) can be translated to certain
+problem concerning generalized Hasse-Witt invariants (e.g. γχi(Mχi) in the proof of Proposition
+pro-4
+3.19). Then by
+applying Theorem
+them-2
+3.18 again, we obtained the result by comparing γχ1(Mχ1) with γχ2(Mχ2).
+3.5.3.
+We have the following proposition.
+pro-4
+Proposition 3.19. The field structure of F�ei, i ∈ {1, 2}, can be mono-anabelian reconstructed from πt
+1(UXi). More-
+over, the isomorphism φ|I�e1 : I�e1
+≃
+−→ I�e2 induces an isomorphism
+θφ,�e1,�e2 : F�e1
+≃
+−→ F�e2
+as fields.
+Proof. First, we claim that we may assume nX = 3. If gX = 0, then nX ≥ 3. Suppose that gX ≥ 1. Theorem
+them-2
+3.18 implies that φ : πt
+1(UX1) ։ πt
+1(UX2) induces an open continuous surjection φét : π1(X1) ։ π1(X2). Let
+H′
+2 ⊆ π1(X2) be an open normal subgroup such that #(π1(X2)/H′
+2) ≥ 3 and H′
+1 := (φét)−1(H′
+2). Write Hi ⊆
+πt
+1(UXi), i ∈ {1, 2}, for the inverse image of H′
+i of the natural surjection πt
+1(UXi) ։ π1(Xi), and (XHi, DXHi )
+for the smooth pointed stable curve of type (gXHi , nXHi ) over ki induced by Hi. Note that gXH1 = gXH2 ≥ 2 and
+nXH1 = nXH2 ≥ 3. By replacing (Xi, DXi) by (XHi, DXHi), we may assume gX ≥ 2 and nX ≥ 3. The surjection
+φ induces a bijection
+DX1
+ϑ−1
+X1
+−−−→ Dgp
+X1
+ρφ
+−→ Dgp
+X2
+ϑX2
+−−−→ DX2.
+Let D′
+X1 := {e1,1, e1,2, e1,3} ⊆ DX1 and D′
+X2 := {e2,1 := ϑX2 ◦ρφ◦ϑ−1
+X1(e1,1), e2,2 := ϑX2 ◦ρφ◦ϑ−1
+X1(e1,2), e2,3 :=
+ϑX2 ◦ ρφ ◦ ϑ−1
+X1(e1,3)} ⊆ DX2. Then (Xi, D′
+Xi), i ∈ {1, 2}, is a smooth pointed stable curve of type (gX, 3) over ki.
+Write Ii, i ∈ {1, 2}, for the closed subgroup of πt
+1(UXi) generated by the inertia subgroups associated to the elements
+of D �
+Xi whose images in DXi are contained in DXi \ D′
+Xi. Then we have an isomorphism
+πt
+1(Xi \ D′
+Xi) ∼= πt
+1(UXi)/Ii, i ∈ {1, 2}.
+Moreover, Theorem
+them-2
+3.18 implies that φ induces an open continuous surjective homomorphism
+φ′ : πt
+1(X1 \ D′
+X1) ։ πt
+1(X2 \ D′
+X2).
+Thus, by replacing (Xi, DXi), πt
+1(UXi), and φ by (Xi, D′
+Xi), πt
+1(Xi \ D′
+Xi), and φ′, respectively, we may assume
+nX = 3.
+Let r ∈ N. We denote by Fpr,�ei, i ∈ {1, 2}, the unique subfield of F�ei whose cardinality is equal to pr. On the
+other hand, we fix any finite field Fpr of cardinality pr and an algebraic closure Fp of Fp. By Proposition
+them-1
+3.10, we
+have that F×
+pr,�ei = I�ei/(pr − 1) can be mono-anabelian reconstructed from πt
+1(UXi). Then reconstructing the field
+structure of Fpr,�ei is equivalent to reconstructing Homfields(Fpr,�ei, Fpr) as a subset of Homgroup(F×
+pr,�ei, F×
+pr). Note
+that, in order to reconstruct the field structure of F�ei, it is sufficient to reconstruct the subset Homfields(Fpr,�ei, Fpr) for
+r in a cofinal subset of N with respect to division.
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+23
+Let χi ∈ Homgroups(πt
+1(UXi)ab ⊗ Z/(pr − 1)Z, F×
+pr). Write Hχi for the kernel of πt
+1(UXi) ։ πt
+1(UXi)ab ⊗
+Z/(pr − 1)Z
+χi
+→ F×
+pr, Mχi for Hab
+χi ⊗ Fp, and (XHχi , DXHχi ) for the smooth pointed stable curve over ki induced by
+Hχi. We define
+Mχi[χi] := {a ∈ Mχi ⊗Fp Fp | σ(a) = χi(σ)a for all σ ∈ πt
+1(UXi)ab ⊗ Z/(pr − 1)Z}
+and γχi(Mχi) := dimFp(Mχi[χi]) (i.e. a generalized Hasse-Witt invariant (see
+Y5
+[Y5, Section 2.2])). Then
+T4
+[T4, Remark
+3.7] implies γχi(Mχi) ≤ gX + 1. Moreover, we define two maps
+Resi,r : Homgroups(πt
+1(UXi)ab ⊗ Z/(pr − 1)Z, F×
+pr) → Homgroups(F×
+pr,�ei, F×
+pr),
+Γi,r : Homgroups(πt
+1(UXi)ab ⊗ Z/(pr − 1)Z, F×
+pr) → Z≥0, χi �→ γχi(Mχi),
+where the map Resi,r is the restriction with respect to the natural inclusion F×
+pr,�ei ֒→ πt
+1(UXi)ab ⊗ Z/(pr − 1)Z.
+Let m0 be the product of all prime numbers ≤ p − 2 if p ̸= 2, 3 and m0 = 1 if p = 2, 3. Let r0 be the order of p in
+the multiplicative group (Z/m0Z)×. Then
+T4
+[T4, Claim 5.4] implies the following result:
+there exists a constant C(gX) which depends only on gX such that, for each r > logp(C(gX) + 1)
+divisible by r0, we have
+Homfields(Fpr,�ei, Fpr) = Homsurj
+groups(F×
+pr,�ei, F×
+pr) \ Resi,r(Γ−1
+i,r ({gX + 1})), i ∈ {1, 2},
+where Homsurj
+groups(−, −) denotes the set of surjections whose elements are contained in
+Homgroups(−, −).
+Let κ2 ∈ Homgroups(πt
+1(UX2)ab ⊗ Z/(pr − 1)Z, F×
+pr). Then φ induces a character
+κ1 ∈ Homgroups(πt
+1(UX1)ab ⊗ Z/(pr − 1)Z, F×
+pr).
+Moreover, the surjection φ|Hκ1 induces a surjection Mκ1[κ1] ։ Mκ2[κ2]. Suppose that κ2 ∈ Γ−1
+2,r({gX + 1}). The
+surjection Mκ1[κ1] ։ Mκ2[κ2] implies γκ1(Mκ1) = gX + 1. This means κ1 ∈ Γ−1
+1,r({gX + 1}). On the other hand,
+the isomorphism φ|I�e1 : I�e1
+≃
+−→ I�e2 induces an injection
+Res2,r(Γ−1
+2,r({gX + 1})) ֒→ Res1,r(Γ−1
+1,r({gX + 1})).
+Since #(Homfields(Fpr,�e1, Fpr))
+=
+#(Homfields(Fpr,�e2, Fpr)),
+we obtain that φ|I�e1
+induces a bijection
+Homfields(Fpr,�e2, Fpr)
+≃
+−→ Homfields(Fpr,�e1, Fpr). Thus, φ|I�e1 induces a bijection
+Homfields(F�e2, Fp)
+≃
+−→ Homfields(F�e1, Fp).
+If we choose Fp = F�e2, then the image of idF�e2 via the bijection above induces an isomorphism θφ,�e1,�e2 : F�e1
+≃
+−→ F�e2
+as fields. This completes the proof of the proposition.
+□
+4. MAIN THEOREMS
+sec-5
+4.1. The first main theorem. In this subsection, we apply the results obtained in the previous sections to prove that
+the curves of type (0, n) over Fp can be reconstructed group-theoretically from open continuous homomorphism (i.e.
+Theorem
+them-3
+4.4).
+
+24
+ZHI HU, YU YANG, AND RUNHONG ZONG
+4.1.1. Settings. We fix some notation. Let (Xi, DXi), i ∈ {1, 2}, be a smooth pointed stable curve of type (gX, nX)
+over an algebraically closed field ki of characteristic p > 0, UXi := Xi \ DXi, πt
+1(UXi) the tame fundamental
+group of UXi, π1(Xi) the étale fundamental group of Xi, and ( �
+Xi, D �
+Xi) the universal tame covering of (Xi, DXi)
+associated to πt
+1(UXi) (
+unicov313
+3.1.2). Let km
+i , i ∈ {1, 2}, be the minimal algebraically closed subfield of ki over which UXi
+can be defined. Thus, by considering the function field of Xi, we obtain a smooth pointed stable curve (Xm
+i , DXm
+i )
+(i.e., a minimal model of (Xi, DXi) (cf.
+T3
+[T3, Definition 1.30 and Lemma 1.31])) such that UXi ∼= UXm
+i ×km
+i ki as
+ki-schemes, where UXm
+i := Xm
+i \ DXm
+i . Note that πt
+1(UXm
+i ) is naturally isomorphic to πt
+1(UXi). We shall denote by
+Fp,i the algebraic closure of Fp in ki. Moreover, we put
+d(Xi,DXi ) :=
+�
+0,
+if km
+i ∼= Fp,i,
+1,
+if km
+i ̸∼= Fp,i.
+4.1.2.
+Firstly, we have the following lemma.
+lemsurj
+Lemma 4.1. Let φ : πt
+1(UX1) → πt
+1(UX2) be an arbitrary open continuous homomorphism. Then φ is a surjection.
+Proof. We denote by Πφ the image of φ which is an open subgroup of πt
+1(UX2). Let (Xφ, DXφ) be the smooth
+pointed stable curve of type (gXφ, nXφ) over k2 induced by Πφ and fφ : (Xφ, DXφ) → (X2, DX2) the tame covering
+of smooth pointed stable curves over k2 induced by the inclusion Πφ ֒→ πt
+1(UX2). Since fφ is a tame covering,
+we have that nXφ ≥ nX. On the other hand, if gX = 0, we have gφ ≥ 0. If gX > 0, the Riemann-Hurwitz
+formula implies gXφ ≥ [πt
+1(UX2) : Πφ](gX − 1) + 1 ≥ gX. Then we have gφ ≥ gX and nXφ ≥ nX. Note that
+πt
+1(UX1) ։ Πφ ֒→ πt
+1(UX2) implies
+2gX + nX − 1 ≥ 2gXφ + nXφ − 1 ≥ 2gX + nX − 1.
+Then we obtain that 2gX + nX − 1 = 2gXφ + nXφ − 1. Moreover, Proposition
+coro-p-average
+3.11 (ii) and the natural surjection
+πt
+1(UX1) ։ Πφ induced by φ imply that gX ≥ gXφ. Then we obtain that gX = gXφ. Thus, we have (gX, nX) =
+(gXφ, nXφ). This means that the tame covering fφ : (Xφ, DXφ) → (X2, DX2) is totally ramified over every marked
+point of DX2.
+Let us prove [πt
+1(UX2) : Πφ] = 1. Suppose [πt
+1(UX2) : Πφ] ̸= 1. Since fφ is totally ramified, the Riemann-
+Hurwitz formula implies gXφ > gX if nX ̸= 0 and gX ̸= 0. This is a contradiction. If nX = 0, the Riemann-Hurwitz
+formula implies gX = 1 if gX ̸= 0. This contradicts the assumption that (Xi, DXi) is a pointed stable curve. Then
+we obtain gX = gXφ = 0. Moreover, by applying the Riemann-Hurwitz formula again, since nX = nXφ, we
+obtain nX = nXφ = 2. This contradicts the assumption that (Xi, DXi) is pointed stable curve. Then we have
+[πt
+1(UX2) : Πφ] = 1. This means that φ is a surjection.
+□
+4.1.3. Further settings. In the remainder of this subsection, we suppose (gX, nX) = (0, n). We fix two marked
+points e1,∞, e1,0 ∈ DX1 distinct from each other. Moreover, we choose any field k′
+1 ∼= k1, and choose any isomor-
+phism ϕ1 : X1
+≃
+−→ P1
+k′
+1 as schemes such that ϕ1(e1,∞) = ∞ and ϕ1(e1,0) = 0. Then the set of k1-rational points
+X1(k1) \ {e1,∞} is equipped with a structure of Fp-module via the bijection ϕ1. Note that since any k′
+1-isomorphism
+of P1
+k′
+1 fixing ∞ and 0 is a scalar multiplication, the Fp-module structure of X1(k1) \ {e1,∞} does not depend on the
+choices of k′
+1 and ϕ1 but depends only on the choices of e1,∞ and e1,0. Then we shall say that X1(k1) \ {e1,∞} is
+equipped with a structure of Fp-module with respect to e1,∞ and e1,0.
+By applying Theorem
+them-2
+3.18, in the next lemma, we will prove that Tamagawa’s group-theoretical criterion (i.e.,
+T2
+[T2,
+Lemma 3.3]) for linear conditions is compatible with arbitrary open continuous surjective homomorphism.
+lem-8
+Lemma 4.2. Let φ : πt
+1(UX1) ։ πt
+1(UX2) be an open continuous surjective homomorphism. By Lemma
+lem-4
+3.13, φ
+induces a bijection ρφ : Dgp
+X1
+≃
+−→ Dgp
+X2. We may identify Dgp
+Xi, i ∈ {1, 2}, with DXi via the bijection ϑXi : Dgp
+Xi
+≃
+−→
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+25
+DXi. Write e2,∞ and e2,0 for ρφ(e1,∞) and ρφ(e1,0), respectively. Let
+�
+e1∈DX1 \{e1,∞,e1,0}
+be1e1 = e1,0
+be a linear condition with respect to e1,∞ and e1,0 on (X1, DX1), where be1 ∈ Fp for each e1 ∈ DX1 \ {e1,∞, e1,0}.
+Then the linear condition
+�
+e1∈DX1 \{e1,∞,e1,0}
+be1ρφ(e1) = ρφ(e1,0) = e2,0
+with respect to e2,∞ and e2,0 on (X2, DX2) also holds.
+Proof. Let �e2,∞ ∈ D �
+X2 be a point over e2,∞. The set F�e2,∞ := (I�e2,∞ ⊗Z (Q/Z)p′
+2 ) ⊔ {∗�e2,∞} admits a structure
+of field, and Proposition
+pro-4
+3.19 implies that the field structure can be mono-anabelian reconstructed from πt
+1(UX2).
+Theorem
+them-2
+3.18 implies that there exists a point �e1,∞ ∈ D �
+X1 over e1,∞ such that φ(I�e1,∞) = �e2,∞. By Proposition
+pro-4
+3.19 again, the set F�e1,∞ := (I�e1,∞ ⊗Z (Q/Z)p′
+1 ) ⊔ {∗�e1,∞} admits a structure of field which can be mono-anabelian
+reconstructed from πt
+1(UX1), and φ induces an isomorphism θφ,�e1,∞,�e2,∞ : F�e1,∞
+≃
+−→ F�e2,∞ as fields.
+For each e1 ∈ DX1, we take b′
+e1 ∈ Z≥0 such that b′
+e1 ≡ be1 (mod p) and
+�
+e1∈DX1 \{e1,∞,e1,0}
+b′
+e1 ≥ 2.
+Let r ≥ 1 such that pr − 2 ≥ �
+e1∈DX1 \{e1,∞,e1,0} b′
+e1. For each �e1 ∈ D �
+X1 over e1, write I�e1,ab for the image of the
+natural morphism I�e1 ֒→ πt
+1(UX1) ։ πt
+1(UX1)ab. Moreover, since the image of I�e1,ab does not depend on the choices
+of �e1, we may write Ie1 for I�e1,ab. The structure of maximal prime-to-p quotient of πt
+1(UX1) implies that πt
+1(UX1)ab
+is generated by {Ie1}e1∈DX1 , and that there exists a generator ae1, e1 ∈ DX1, of Ie1 such that �
+e1∈DX1 ae1 = 1. We
+define
+Ie1,∞ → Z/(pr − 1)Z, ae1,∞ �→ 1,
+Ie1,0 → Z/(pr − 1)Z, ae1,0 �→ (
+�
+e1∈DX1 \{e1,∞,e1,0}
+b′
+e1) − 1,
+and
+Ie1 → Z/(pr − 1)Z, ae1 �→ −b′
+e1, e1 ∈ DX1 \ {e1,∞, e1,0}.
+Then the homomorphisms of inertia subgroups defined above induces a surjection δ1 : πt
+1(UX1) ։ πt
+1(UX1)ab ։
+Z/(pr − 1)Z. Note that ker(δ1) does not depend on the choices of the generators {ae1}e1∈DX1 .
+Let I�e2 := φ(I�e1), �e1 ∈ D �
+X1, and Ie2, e2 ∈ DX2 be the image of the natural homomorphism I�e2 ֒→ πt
+1(UX2) ։
+πt
+1(UX2)ab. Since (p, pr − 1) = 1, by Theorem
+them-2
+3.18, δ1 and the isomorphism φp′ : πt
+1(UX1)p′
+≃
+−→ πt
+1(UX2)p′ imply
+the following homomorphisms of inertia subgroups:
+Ie2,∞ → Z/(pr − 1)Z, ae2,∞ �→ 1,
+Ie2,0 → Z/(pr − 1)Z, ae2,0 �→ (
+�
+e1∈DX1 \{e1,∞,e1,0}
+b′
+e1) − 1,
+and
+Ie2 → Z/(pr − 1)Z, ae2 �→ −b′
+e1, e2 ∈ DX2 \ {e2,∞, e2,0},
+where ae2, e2 ∈ DX2, denotes the element induced by ae1, e1 ∈ DX1, via φ. Then the homomorphisms of inertia
+subgroups defined above induces a sujection δ2 : πt
+1(UX2) ։ πt
+1(UX2)ab ։ Z/(pr − 1)Z.
+We put Hδi := ker(δi), Mδi := Hab
+δi ⊗ Fp, i ∈ {1, 2}. Write (XHδi , DXHδi ) for the smooth pointed stable curve
+over ki induced by Hδi, where Hδ1 = φ−1(Hδ2). The Fp-vector space Mδi admits a natural action of I�ei,∞ via
+conjugation which coincides with the action via the following character
+χI�ei,∞,r : I�ei,∞ ֒→ πt
+1(UXi)
+δi։ Z/(pr − 1)Z = I�ei,∞/(pr − 1) ֒→ F×
+�ei,∞, i ∈ {1, 2}.
+
+26
+ZHI HU, YU YANG, AND RUNHONG ZONG
+We put Mδi[χI�ei,∞ ,r]
+:=
+{a
+∈
+Mδi ⊗Fp F�ei,∞ | σ(a)
+=
+χI�ei,∞,r(σ)a for all σ
+∈
+I�ei,∞} (in fact,
+dimF�ei,∞ (Mδi[χI�ei,∞ ,r]) is the first generalized Hasse-Witt invariant associated to the tame covering of UXi corre-
+sponding to Hδi ⊆ πt
+1(UXi) (see
+Y5
+[Y5, Section 2.2])). Since the action of I�ei,∞ on Mδi is semi-simple, we obtain a
+surjection Mδ1[χI�e1,∞,r] ։ Mδ2[χI�e2,∞,r] induced by φ|Hδ1 and θφ,�e1,∞,�e2,∞. On the other hand, the third and the
+final paragraphs of the proof of
+T2
+[T2, Lemma 3.3] imply that the linear condition
+�
+e1∈DX1 \{e1,∞,e1,0}
+be1e1 = e1,0
+with respect to e1,∞ and e1,0 on (X1, DX1) holds if and only if Mδ1[χI�e1,∞ ,r] = 0. Thus, we obtain Mδ2[χI�e2,∞ ,r] =
+0. Then the third and the final paragraphs of the proof of
+T2
+[T2, Lemma 3.3] imply that the linear condition
+�
+e1∈DX1\{e1,∞,e1,0}
+be1ρφ(e1) = e2,0
+with respect to e2,∞ and e2,0 on (X2, DX2) holds. This completes the proof of the lemma.
+□
+rem-lem-8-1
+Remark 4.3. Note that, if X1 = P1
+k, then the linear condition is as follows:
+�
+e1∈DX1 \{∞,0}
+be1e1 = 0
+with respect to ∞ and 0.
+4.1.4.
+Now, we prove the first main theorem of the present paper.
+them-3
+Theorem 4.4. We maintain the notation and settings introduced above. Then we have the following claims.
+(1) d(Xi,DXi ), i ∈ {1, 2}, can be mono-anabelian reconstructed from πt
+1(UXi).
+(2) Suppose km
+1 ∼= Fp,1. Then the set of open continuous homomorphisms
+Homop
+pg(πt
+1(UX1), πt
+1(UX2))
+is non-empty if and only if UXm
+1 ∼= UXm
+2 as schemes. In particular, if this is the case, we have km
+2 ∼= Fp,2 and
+Homop
+pg(πt
+1(UX1), πt
+1(UX2)) = Isompg(πt
+1(UX1), πt
+1(UX2)).
+Proof. Firstly, let us prove (2). The “if" part of (2) is trivial. We treat the “only if" part of (2). Suppose that
+Homop
+pg(πt
+1(UX1), πt
+1(UX2)) is a non-empty set, and let φ ∈ Homop
+pg(πt
+1(UX1), πt
+1(UX2)). Then Lemma
+lemsurj
+4.1 implies
+that φ is a surjection.
+We identify Dgp
+Xi, i ∈ {1, 2}, with DXi via the bijection ϑXi : Dgp
+Xi
+≃
+−→ DXi. Since φ is a surjection, Lemma
+lem-4
+3.13
+implies that φ induces a bijection ρφ : DX1
+≃
+−→ DX2. We put e2,0 := ρφ(e1,0) and e2,∞ := ρφ(e1,∞). Let �e2,0 ∈ D �
+X2
+be a point over e2,0. Theorem
+them-2
+3.18 implies that there exists a point �e1,0 ∈ D �
+X1 over e1,0 such that φ(I�e1,0) = I�e2,0.
+Then F�ei,0 := (I�ei,0 ⊗Z (Q/Z)p′
+i )⊔{∗�ei,0}, i ∈ {1, 2}, admits a structure of field. Moreover, Proposition
+pro-4
+3.19 implies
+that the field structure can be mono-anabelian reconstructed from πt
+1(UXi), and that φ induces a field isomorphism
+θφ,�e1,0,�e2,0 : F�e1,0
+≃
+−→ F�e2,0.
+Proposition
+proposition 1
+3.2 (1) implies that n can be mono-anabelian reconstructed from πt
+1(UXi), i ∈ {1, 2}. If n = 3,
+(ii) is trivial, so we may assume n ≥ 4. Moreover, since km
+1
+∼= Fp,1, without loss of generality, we may assume
+k1 = Fp,1 = F�e1,0, X1 = P1
+Fp,1, and
+DX1 = {e1,∞ = ∞, e1,0 = 0, e1,1 = 1, e1,2, . . . , e1,n−2}.
+Here, e1,2, . . . , e1,n−2 ∈ Fp,1 \ {e1,0, e1,1} are distinct from each other.
+Step 1: In this step, we will construct a linear condition on a certain tame covering of (X1, DX1).
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+27
+We see that there exists a natural number r prime to p such that Fp(ζr) contains rth roots of e1,2, . . . , e1,n−2, where
+ζr denotes a fixed primitive rth root of unity in Fp,1. Let s := [Fp(ζr) : Fp]. For each e1,u ∈ {e1,2, . . . , e1,n−2}, we
+fix an rth root e1/r
+1,u in Fp,1. Then we have
+e1/r
+1,u =
+s−1
+�
+v=0
+b1,uvζv
+r , u ∈ {2, . . ., n − 2},
+where b1,uv ∈ Fp for each u ∈ {2, . . . , n − 2} and each v ∈ {0, . . . , s − 1}.
+Let X1 \ {e1,∞} = SpecFp,1[x1], fH1 : (XH1, DXH1) → (X1, DX1) the Galois tame covering over Fp,1 with
+Galois group Z/rZ determined by the equation yr
+1 = x1, and H1 the open normal subgroup of πt
+1(UX1) induced by
+the tame covering fH1. Then fH1 is totally ramified over {e1,∞ = ∞, e1,0 = 0} and is étale over DX1 \ {∞, 0}. Note
+that XH1 = P1
+Fp,1, and the points of DXH1 over {e1,∞, e1,0} are {eH1,∞ := ∞, eH1,0 := 0}. We put
+eH1,u := e1/r
+1,u ∈ DXH1 , u ∈ {2, . . ., n − 2}, ev
+H1,1 := ζv
+r ∈ DXH1 , v ∈ {0, . . . , s − 1}.
+Thus, we obtain a linear condition
+eH1,u =
+s−1
+�
+v=0
+b1,uvev
+H1,1
+with respect to eH1,∞ and eH1,0 on (XH1, DXH1 ) for each u ∈ {2, . . . , n − 2}.
+Step 2: In this step, we will prove that the linear condition on a certain tame covering of (X1, DX1) constructed in
+Step 1 induces a linear condition on a certain tame covering of (X2, DX2) via the surjection φ.
+Write H2 for φ(H1). Since (r, p) = 1, we have the following commutative diagram of profinite groups:
+H1
+φ|H1
+−−−−→
+H2
+�
+�
+πt
+1(UX1)
+φ
+−−−−→ πt
+1(UX2)
+�
+�
+Z/rZ
+Z/rZ.
+We denote by fH2 : (XH2, DXH2) → (X2, DX2) the Galois tame covering over Fp,2 with Galois group Z/rZ induced
+by H2. Note that Lemma
+lem-6
+3.16 implies that (XH1, DXH1) and (XH2, DXH2 ) are equal types. Moreover, Lemma
+lem-5
+3.14
+implies that the following commutative diagram can be mono-anabelian reconstructed from the commutative diagram
+of profinite groups above:
+DXH1
+ρφ|H1
+−−−−→ DXH2
+�
+�
+DX1
+ρφ
+−−−−→ DX2.
+We put
+e2,∞ := ρφ(e1,∞), e2,u := ρφ(e1,u), u ∈ {0, . . ., n − 2},
+eH2,∞ := ρφ|H1(eH1,∞), eH2,0 := ρφ|H1(eH1,0), eH2,u := ρφ|H1 (eH1,u), u ∈ {2, . . ., n − 2},
+and
+ev
+H2,1 := ρφ|H1(ev
+H1,1), v ∈ {0, . . . , s − 1}.
+Remark
+rem-lem-5-1
+3.15 implies that fH2 is totally ramified over {e2,∞, e2,0} and is étale over X2 \ {e2,∞, e2,0}. Then we
+may assume that X2 = P1
+k2, and that e2,∞ = ∞, e2,0 = 0, e2,1 = 1. We regard e2,u, u ∈ {2, . . . , n − 2}, as an
+element of k2 \ {e2,0, e2,1}. Moreover, we have eH2,∞ = ∞ and eH2,0 = 0.
+
+28
+ZHI HU, YU YANG, AND RUNHONG ZONG
+We put ξr := θφ,�e1,0,�e2,0(ζr) which is an rth root of unity in F�e2,0. Since ζr(ev
+H1,1) = ev+1
+H1,1, we obtain ξr(ev
+H2,1) =
+ev+1
+H2,1, v ∈ {0, . . ., s − 2}. By applying Lemma
+lem-8
+4.2 for φ|H1 : H1 ։ H2, the following linear condition
+eH2,u =
+s−1
+�
+v=0
+b1,uvξv
+r(e0
+H2,1)
+with respect to eH2,∞ and eH2,0 on (XH2, DXH2 ) holds for each u ∈ {2, . . . , n − 2}. Since (eH2,u)r = e2,u,
+u ∈ {2, . . . , n − 2}, we obtain
+e2,u = (
+s−1
+�
+v=0
+b1,uvξv
+r (e0
+H2,1))r.
+Moreover, if we put e0
+H2,1 = 1, then we obtain that
+e2,u = (
+s−1
+�
+v=0
+b1,uvξv
+r )r
+for each u ∈ {2, . . . , n − 2}. Since θφ,�e1,0,�e2,0(ζr) = ξr, we have
+UX1 = UXm
+1 = P1
+Fp,1 \ {e1,∞ = ∞, e1,0 = 0, e1,1 = 1, e1,2, . . . , e1,n−2}
+≃
+−→ P1
+F�e2,0 \ {e2,∞ = ∞, e2,0 = 0, e2,1 = 1, θφ,�e1,0,�e2,0(e1,2), . . . , θφ,�e1,0,�e2,0(e1,n−2)}
+∼= P1
+Fp,2 \ {e2,∞ = ∞, e2,0 = 0, e2,1 = 1, e2,2, . . . , e2,n−2}
+and
+P1
+Fp,2 \ {e2,∞ = ∞, e2,0 = 0, e2,1 = 1, e2,2, . . . , e2,n−2} ×Fp,2 k2 ∼= UX2.
+This means UXm
+1 ∼= UXm
+2 as schemes. In particular, we have km
+2 ∼= Fp,2.
+Finally, we prove that Homop
+pg(πt
+1(UX1), πt
+1(UX2)) = Isompg(πt
+1(UX1), πt
+1(UX2)). The “⊇" part is trivial. We only
+need to prove the “⊆" part. We may assume Homop
+pg(πt
+1(UX1), πt
+1(UX2)) ̸= ∅. Let φ′ ∈ Homop
+pg(πt
+1(UX1), πt
+1(UX2)).
+Then πt
+1(UX1) is isomorphic to πt
+1(UX2) as abstract profinite groups. By Lemma
+lemsurj
+4.1, φ′ is a surjection. Then
+FJ
+[FJ,
+Proposition 16.10.6] implies that φ′ is an isomorphism. Thus, we obtain φ′ ∈ Isompro-gps(πt
+1(UX1), πt
+1(UX2)). This
+completes the proof of (2).
+Next, let us prove (1). Without loss of generality, we only treat the case where i = 1. Moreover, let (X, DX) :=
+(X1, DX1),
+DX = {e∞ = ∞, e0 = 0, e1 = 1, e2, . . . , en−2},
+k := k1, and Fp := F�e0. Let (r, Q) be a pair such that the following two conditions hold:
+• (r, p) = 1;
+• Q is an open normal subgroup of πt
+1(UX) such that πt
+1(UX)/Q ∼= Z/rZ, and that the Galois tame covering
+fQ : (XQ, DXQ) → (X, DX) over k induced by Q is totally ramified over {e∞, e0} and is étale over
+DX \ {e∞, e0}.
+By applying Theorem
+them-2
+3.18, we see immediately that the set of pairs defined above can be mono-anabelian recon-
+structed from πt
+1(UX).
+We fix a primitive r-th root of unity ζr in Fp and put sr := [Fp(ζr) : Fp]. Moreover, we put
+eQ,∞ := ∞, eQ,0 := 0, ev
+Q,1 := ζv
+r ∈ DXQ, v ∈ {0, . . . sr − 1},
+and let eQ,u ∈ DXQ, u ∈ {2, . . ., n}, such that fQ(eQ,u) = eu. Denote by
+LQ,u := {eQ,u −
+sr−1
+�
+v=0
+buvev
+Q,1 | buv ∈ Fp} ∩ {0}, u ∈ {2, . . . , n − 2}.
+By applying arguments similar to the arguments given in the proof of (2) above, we have that d(X,DX) = 0 if and
+only if there exists a pair (r, Q) defined above such that LQ,u ̸= ∅ for each u ∈ {2, . . . , n − 2}. Then the third and
+
+TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC
+29
+the final paragraphs of the proof of
+T2
+[T2, Lemma 3.3] implies that LQ,u, u ∈ {2, . . . , n − 2}, can be mono-anabelian
+reconstructed from Q. Thus, d(X,DX) can be mono-anabelian reconstructed from πt
+1(UX). This completes the proof
+of the theorem.
+□
+Remark 4.5. Note that Theorem
+them-3
+4.4 also holds if we replace πt
+1(UXi), i ∈ {1, 2}, by its maximal pro-solvable
+quotient πt
+1(UXi)sol. Then we obtain the following solvable version of Theorem
+them-3
+4.4 which is slightly stronger than
+the original theorem:
+We maintain the notation introduced above. Then d(Xi,DXi), i ∈ {1, 2}, can be mono-anabelian
+reconstructed from πt
+1(UXi)sol. Moreover, suppose that km
+1 ∼= Fp,1. Then the set of open continuous
+homomorphisms
+Homop
+pg(πt
+1(UX1)sol, πt
+1(UX2)sol)
+is non-empty if and only if UXm
+1
+∼= UXm
+2 as schemes. In particular, if this is the case, we have
+km
+2 ∼= Fp,2 and
+Homop
+pg(πt
+1(UX1)sol, πt
+1(UX2)sol) = Isompg(πt
+1(UX1)sol, πt
+1(UX2)sol).
+sec-6
+4.2. The second main theorem. In this subsection, by using Theorem
+them-3
+4.4, we prove a result concerning pointed
+collection conjecture and the weak Hom-version conjecture (i.e. Theorem
+them-4
+4.6). We maintain the notation introduced
+in
+moduli212
+2.1.2.
+def-4
+4.2.1.
+Let q ∈ M ord
+0,n be an arbitrary point, k(q) an algebraic closure of k(q), and
+UXq ≃ P1
+k(q) \ {a1 = 1, a2 = 0, a3 = ∞, a4, . . . , an}
+as k(q)-schemes. We shall say that q is a coordinated point if either q = qgen or the following three conditions are
+satisfied:
+• dim(Vq) = dim(M ord
+0,n ) − 1;
+• there exists i ∈ {4, . . ., n} such that ai ∈ Fp;
+• let ωi
+n,n−1 : M ord
+0,n → M ord
+0,n−1 be the morphism induced by the morphism Mord
+0,n → Mord
+0,n−1 obtained by
+forgetting the ith marked point; then ωi
+n,n−1(q) is the generic point of M ord
+0,n−1.
+Let t be a closed point of M ord
+0,n . Then there exists a set of coordinated points Pt := {qt,4, . . . , qt,n} such that
+{t} =
+�
+qt,j∈Pt
+Vqt,j .
+4.2.2.
+Now, we prove the second main theorem of the present paper.
+Theorem 4.6.
+them-4
+(1) For each closed point t ∈ M ord,cl
+0,n
+, the set Ct associated to t is a pointed collection (Definition
+def-3
+2.4). Moreover,
+for each pointed collection C ∈ Cqgen, there exists a closed point s ∈ M ord,cl
+0,n
+such that C = Cs.
+(2) Let q ∈ M ord
+0,n be an arbitrary point. Then the the natural map colleq : V cl
+q
+→ Cq, [t] �→ Ct, is an injection.
+(3) Let q ∈ M ord
+0,n be an arbitrary point. Suppose that there exists a set of coordinated points Pq such that
+Vq =
+�
+u∈Pq
+Vu.
+Then the pointed collection conjecture holds for q. In particular, the pointed collection conjecture holds for
+each closed point of M ord
+0,n .
+
+30
+ZHI HU, YU YANG, AND RUNHONG ZONG
+(4) Let qi ∈ M ord
+0,n , i ∈ {1, 2}, be an arbitrary point. Suppose that there exists a set of coordinated points Pq1
+such that
+Vq1 =
+�
+u∈Pq1
+Vu.
+Then the weak Hom-version conjecture holds. In particular, the weak Hom-version conjecture holds when q1
+is a closed point.
+Proof. Let us prove (1). We put Ft := {t′ ∈ M ord,cl
+0,n
+| t ∼fe t′}. Let t′′ be an arbitrary point of �
+G∈πt
+A(t) UG.
+Then, for each G
+∈ πt
+A(t), Homsurj
+pg (πt
+1(t′′), G) is non-empty, where Homsurj
+pg (−, −) denotes the subset of
+Homopen
+pg
+(−, −) whose elements are surjections. Since πt
+1(t′′) is topologically finitely generated, we obtain that the
+set Homsurj
+pg (πt
+1(t′′), G) is finite. Then the set of open continuous homomorphisms
+lim
+←−
+G∈πt
+A(t)
+Homsurj
+pg (πt
+1(t′′), G) = Homsurj
+pg (πt
+1(t′′), πt
+1(t))
+is non-empty. Thus, Theorem
+them-3
+4.4 implies t′′ ∈ Ft. This means
+(
+�
+G∈πt
+A(t)
+UG) ∩ M ord,cl
+g,n
+= Ft.
+Since UXt can be defined over a finite field, Ft is a finite set. Then Ct is a pointed collection.
+Let C ∈ Cqgen be a pointed collection and s a closed point of �
+G∈C UG. By replacing t by s, and by applying
+arguments similar to the arguments given in the proof above, we obtain C = Cs.
+(2) follows immediately from Theorem
+them-3
+4.4. Let us prove (3). If n = 4, then M ord
+0,4 is a one dimensional scheme.
+For each q ∈ M ord
+0,4 , the pointed collection conjecture follows immediately from Theorem
+them-3
+4.4. Then we may assume
+n ≥ 5. To verify (iii), (ii) implies that we only need to prove that colleq is a surjection. Suppose that q is a closed point
+of M ord
+0,n , then (iii) follows immediately from Theorem
+them-3
+4.4.
+Suppose that q is a non-closed point. This means dim(Vq) ≥ 1. If q = qgen, (3) follows from (1) and (2). Let us
+treat the case where q ̸= qgen. First, suppose that q is a coordinated point, and that
+UXq ≃ P1
+k(q) \ {1, 0, ∞, a4, . . . , an}.
+Without loss of generality, we may assume an ∈ Fp.
+For each pointed collection C ⊆ Cq, by applying (1), there exists a closed point t1 ∈ M ord,cl
+g,n
+such that Ct1 = C.
+Then we have an open continuous surjective homomorphism πt
+1(q) ։ πt
+1(t1). Let ω\n
+n,4 : M ord
+0,n → M ord
+0,4 be the
+morphism induced by the morphism Mord
+0,n → Mord
+0,4 obtained by forgetting the marked points except the first, the
+second, the third, and the nth marked points. We put t′′
+1 := ω\n
+n,4(t1) and q′′ := ω\n
+n,4(q). Note that t′′
+1 and q′′ are closed
+points of M0,4. Then Theorem
+them-2
+3.18 implies that the surjection πt
+1(q) ։ πt
+1(t1) induces an open continuous surjective
+homomorphism πt
+1(q′′) ։ πt
+1(t′′
+1). Thus, by Theorem
+them-3
+4.4, we obtain that q′′ ∼fe t′′
+1. Then without loss of generality,
+we may assume
+UXt1 ≃ P1
+Fp \ {1, 0, ∞, b4, . . . , bn−1, an}
+over Fp, where bi ∈ Fp for each i ∈ {4, . . . , n − 1}.
+On the other hand, let ωn
+n,n−1 : M ord
+0,n → M ord
+0,n−1 be the morphism induced by the morphism Mord
+0,n → Mord
+0,n−1
+obtained by forgetting the n-th marked point. We put t′
+1 := ωn
+n,n−1(t1) and q′ := ωn
+n,n−1(q), respectively. Since q is
+a coordinated point, q′ is the generic point of M ord
+0,n−1. Then we obtain t′
+1 ∈ V cl
+q′ . Moreover, we see Vq = ω−1
+n,n−1(q′).
+Thus, t1 = ω−1
+n,n−1(t′
+1) is a closed point of Vq. Then the pointed collection conjecture holds for q when q is a
+coordinated point.
+
+31
+Next, we prove the general case. If Vq = �
+u∈Pq Vu, then V cl
+q = �
+u∈Pq V cl
+u and �
+u∈Pq Cu = Cq. Moreover, since
+we have a bijection colleu : V cl
+u
+≃
+−→ Cu for each u ∈ Pq, we have that
+colleq : V cl
+q
+=
+�
+u∈Pq
+V cl
+u →
+�
+u∈Pq
+Cu = Cq
+is a bijection. This completes the proof of (3).
+Let us prove (4). We only need to prove the “only if" part of the weak Hom-version conjecture. Suppose that Vq2 is
+not essentially contained in Vq1. This implies that there exists a closed point t2 ∈ V cl
+q2 such that Ft2 ∩ Vq1 = ∅, where
+Ft2 := {t′
+2 ∈ M ord,cl
+0,n
+| t2 ∼fe t′
+2}. By (3), we have Ct2 ̸∈ Cq1. Thus, by Lemma
+lemsurj
+4.1, we obtain that
+Homop
+pg(πt
+1(q1), πt
+1(t2)) = ∅.
+This provides a contradiction to the assumption that Homop
+pg (πt
+1(q1), πt
+1(q2)) is non-empty. This completes the proof
+of (4).
+□
+Remark 4.7. Let q ∈ Mg,n be an arbitrary point. Stevenson posed a question as follows (see
+Ste
+[Ste, Question 4.3] for
+the case of n = 0): Does �
+G∈πt
+A(q) UG contain any closed points of Mg,n? By
+T5
+[T5, Theorem 0.3], �
+G∈πt
+A(q) UG
+contains a closed point of Mg,n if and only if q is a closed point of Mg,n. Furthermore, when g = 0 and q is a closed
+point, the proof of Theorem
+them-4
+4.6 (1) implies that
+(
+�
+G∈πt
+A(q)
+UG) ∩ M cl
+0,n = Fq,
+where Fq := {q′ ∈ M cl
+0,n | q ∼fe q′}.
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+SCHOOL OF MATHEMATICS, NANJING UNIVERSITY OF SCIENCE AND TECHNOLOGY, NANJING 210094, CHINA
+Email address: halfask@mail.ustc.edu.cn
+RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES, KYOTO UNIVERSITY, KYOTO 606-8502, JAPAN
+Email address: yuyang@kurims.kyoto-u.ac.jp
+DEPARTMENT OF MATHEMATICS, NANJING UNIVERSITY, NANJING 210093, CHINA
+Email address: rzong@nju.edu.cn
+