ON THE FUNDAMENTAL GROUPS OF LOG CONFIGURATION SCHEMES

Math. J. Okayama Univ. 51 (2009), 1 26 ON THE FUNDAMENTAL GROUPS OF LOG CONFIGURATION SCHEMES Yuichiro HOSHI Abstract. In the present paper, we study the cuspidalization problem for the fundamental group of a curve by means of the geometry of the configuration scheme, which is a natural compactification of the usual configuration space of the curve. The goal of this paper is to show that the fundamental group of the configuration space is generated by the images from morphisms from a group extension of the fundamental groups of the configuration spaces of lower dimension, and that the fundamental group of the configuration space can be partially reconstructed from a collection of data concerning the fundamental groups of the configuration spaces of lower dimension. 1. Introduction In this paper, we study the cuspidalization problem for the fundamental groups of curves. This problem may be formulated as follows: Let X be a smooth, proper, geometrically connected curve of genus g 2 over a field K, and r a natural number. Then the r-th configuration space U r {}}{ of X is, by definition, the complement in the r-th product X K K X r {}}{ of the diagonal divisors {(x 1,,x r ) X K K X x i = x j }, where 1 i < j r; moreover, we have projections U (r+1) U obtained by forgetting the i-th factor, where 1 i r + 1. Roughly speaking, the cuspidalization problem (due to Mochizuki [cf. [11]]) refers the problem of reconstructing group-theoretically the output data consisting of the fundamental groups and morphisms π 1 (U ) π 1 (U (r+1) ) π 1 (U ) π 1 (U (r 1) ) π 1 (X) = π 1 (U (1) ) Mathematics Subject Classification. Primary 14H30; Secondary 14H10. Key words and phrases. cuspidalization, configuration scheme. 1

2 YUICHIRO HOSHI from induced by the above projections the input data consisting of the morphism π 1 (X) π 1 (SpecK) Gal(K sep /K), where K sep is a separable closure of K, induced by the structure morphism of X. The purpose of this paper is to give an approach to this problem from the viewpoint of geometry, i.e., we shall make use of the geometry of the configuration scheme associated to X X obtained by equipping a certain canonical compactification of U with def the structure defined by the divisor with normal crossings D = \ U. Let M g,r be the moduli stack of r-pointed stable curves of genus g over K whose r marked points are equipped with an ordering, M g,r M g,r the open substack of M g,r parametrizing smooth curves, and M g,r the stack obtained by equipping M g,r with the structure associated to the divisor with normal crossings M g,r \M g,r M g,r. Then the scheme X arises as the fiber product of the morphism of stacks SpecK M g,0 determined by the curve X and the morphism of stacks M g,r M g,0 obtained by forgetting the marked points. Note that for a set of prime numbers Σ, it follows from the purity theorem that if p Σ, where p is the characteristic of K, then the strict open immersion U X induces an isomorphism of the geometrically pro-σ fundamental group of U with the geometrically pro-σ fundamental group of X. The divisor at infinity D admits a decomposition D = I {1,2,,r};I 2 D I by considering the configurations of the r marked points. Our first result is as follows (cf. Theorem 4.1; Remark following Lemma 4.2): Theorem 1.1. Let r 3 be an integer, and i,j,k {1,2,,r} distinct elements of {1, 2,, r}. Then the following hold: (i) The images of the fundamental groups π 1 (D {i,j} ), π 1(D {j,k} ), and π 1 (D {i,j,k} ) in π 1(X ) topoically generate π 1(X ).

LOG CONFIGURATION SCHEMES 3 (ii) If K is of characteristic 0, then there exist exact sequences 1 Ẑ(1)(Ksep ) π 1 (D {i,j} ) π 1(X (r 1) ) 1 ; 1 Ẑ(1)(Ksep ) π 1 (D {i,j,k} ) π 1 (X (r 2) ) G K π 1 (P 1 K \ {0,1, }) 1. This result can be regarded as a arithmic anaue of [8], Remark 1.2. Note that even if K is of characteristic p > 0, by replacing fundamental group by geometrically pro-l fundamental group, where l is a prime number such that l p, one can obtain a similar result of (ii) in the statement of the above theorem (cf. Remark following Lemma 4.2). Our second result (cf. Theorem 5.2) asserts that one can reconstruct partially the fundamental groups of the configuration schemes of higher dimension from a collection of data concerning the fundamental groups of the configuration schemes of lower dimension by means of Theorem 1.1; more precisely, one can construct group-theoretically a profinite group π 1 (X (r+1) )G and projections q i : π 1 (X (r+1) )G π 1 (X ), where 1 i r + 1, in such a way that the morphism q i factors through the projection π 1 (X (r+1) ) π 1 (X ) induced by the projection X (r+1) X obtained by forgetting the i-th factor, and, moreover, the first arrow of the factorization π 1 (X (r+1) )G π 1 (X (r+1) ) π 1(X ) of q i is surjective from a collection of data concerning the fundamental groups of the configuration schemes X (k), where 0 k r. Here, we use the terminoy reconstruct as a sort of abbreviation for the somewhat lengthy but mathematically precise formulation given in the statement of Theorem 5.2. By this second result, if one can also reconstruct group-theoretically the kernel of the surjection π 1 (X (r+1) )G π 1 (X (r+1)) which appears as the first arrow in the above factorization of q i, then by taking the quotient by this kernel, one can reconstruct the desired profinite group π 1 (X (r+1) ) (cf. Proposition 5.1). However, unfortunately, no reconstruction of this kernel is performed in this paper. Moreover, it seems to the author that if such a reconstruction should prove to be possible, it is likely that the method of reconstruction of this kernel should depend on the arithmetic of K in an

4 YUICHIRO HOSHI essential way. On the other hand, in a subsequent paper [4], we obtain a solution of a pro-l version of the cuspidalization problem over a finite field by means of the results obtained in this paper. This paper is organized as follows: In Section 2, we define the configuration schemes of curves and consider the scheme-theoretic and scheme-theoretic properties of configuration schemes. In Section 3, we study the geometry of the divisors at infinity of configuration schemes. In Section 4, we study properties of the fundamental groups of configuration schemes and their divisors at infinity by means of the results obtained in Sections 2 and 3. In Section 5, we consider a partial reconstruction of the fundamental groups of higher dimensional configuration schemes as discussed above. Notations and Terminoies Groups: Let G be a profinite group, Σ a non-empty set of prime numbers, and n an integer. We shall say that n is a Σ-integer if the prime divisors of n are in Σ. We shall refer to the quotient lim G/H of G, where the projective limit is over all open normal subgroups H G such that the index [G : H] of H is a Σ-integer, as the maximal pro-σ quotient of G. We shall denote by G (Σ) the maximal pro-σ quotient of G. Log schemes: For a scheme X, we shall denote by X (respectively, M X ) the underlying scheme (respectively, the sheaf of monoids that defines the structure) of X. For a morphism f of schemes, we shall denote by f the underlying morphism of schemes. Let P be a property of schemes [for example, quasi-compact, connected, normal, regular ] (respectively, morphisms of schemes [for example, proper, finite, étale, smooth ]). Then we shall say that a scheme (respectively, a morphism of schemes) satisfies P if the underlying scheme (respectively, the underlying morphism of schemes) satisfies P. For fs schemes X, Y, and Z, we shall denote by X Y Z the fiber product of X and Z over Y in the category of fs schemes. In general, the underlying scheme of X Y Z is not naturally isomorphic to X Y Z. However, since strictness (note that a morphism f : X Y is called strict if the induced morphism f M Y M X on X is an isomorphism) is stable under base-change in the category of

LOG CONFIGURATION SCHEMES 5 arbitrary schemes, if X Y is strict, then the underlying scheme of X Y Z is naturally isomorphic to X Y Z. Note that since the natural morphism from the saturation of a fine scheme to the original fine scheme is finite, properness and finiteness are stable under fs basechange. If there exist both schemes and schemes in a commutative diagram, then we regard each scheme in the diagram as the scheme obtained by equipping the scheme with the trivial structure. We shall refer to the largest open subset (possibly empty) of the underlying scheme of an fs scheme on which the structure is trivial as the interior of the fs scheme. Let X and Y be schemes, and f : X Y a morphism of schemes. Then we shall refer to the quotient of M X by the image of the morphism f M Y M X induced by f as the relative characteristic sheaf of f. Moreover, we shall refer to the relative characteristic sheaf of the morphism X X induced by the natural inclusion O X M X as the characteristic sheaf of X. Fundamental groups: For a connected scheme X (respectively, scheme X ) equipped with a geometric point x X (respectively, geometric point x X ), we shall denote by π 1 (X,x) (respectively, π 1 (X, x )) the fundamental group of X (respectively, fundamental group of X ). Since one knows that the fundamental group is determined up to inner automorphisms independently of the choice of base-point, we shall often omit the base-point, i.e., we shall often denote by π 1 (X) (respectively, π 1 (X )) the fundamental group of X (respectively, fundamental group of X ). For a set Σ of prime numbers and a scheme X (respectively, scheme X ) which is of finite type and geometrically connected over a field K, we shall refer to the quotient of π 1 (X) (respectively, π 1 (X )) by the closed normal subgroup obtained as the kernel of the natural projection from π 1 (X K K sep ) (respectively, π 1 (X K K sep )), where K sep is a separable closure of K, to its maximal pro-σ quotient π 1 (X K K sep ) (Σ) (respectively, π 1 (X K K sep ) (Σ) ) as the geometrically pro-σ fundamental group of X (respectively, geometrically pro-σ fundamental group of X ). Thus, the geometrically pro-σ fundamental group π 1 (X) (Σ) of X (respectively, geometrically pro-σ fundamental group π 1 (X ) (Σ) of X ) fits into the following exact sequence: 1 π 1 (X K K sep ) (Σ) π 1 (X) (Σ) Gal(K sep /K) 1 (respectively,

6 YUICHIRO HOSHI 1 π 1 (X K K sep ) (Σ) π 1 (X ) (Σ) Gal(K sep /K) 1). 2. Log configuration schemes In this Section, we define the configuration scheme of a curve over a field and consider the geometry of such configuration schemes. Throughout this Section, we shall denote by X a smooth, proper, geometrically connected curve of genus g 2 over a field K whose (not necessarily positive) characteristic we denote by p. Let M g,r be the moduli stack of r-pointed stable curves of genus g over K whose r marked points are equipped with an ordering, and M g,r M g,r the open substack of M g,r parametrizing smooth curves (cf. [7]). Let us def def write M g = M g,0 and M g = M g,0. For 1 i r + 1, we shall denote by p M i : M g,r+1 M g,r the morphism of stacks obtained by forgetting the i-th marked point. Then by considering the morphism of stacks p M r+1 : M g,r+1 M g,r, we obtain a natural isomorphism of M g,r+1 with the universal r-pointed stable curve over M g,r (cf. [7], Corollary 2.6). Now the complement M g,r \ M g,r is a divisor with normal crossings in M g,r (cf. [7], Theorem 2.7). Let us denote by M g,r the fs stack obtained by equipping M g,r with the structure associated to the divisor with normal crossings M g,r \ M g,r. Then since a natural action of the symmetric group on r letters S r on M g,r given by permuting the marked points preserves the divisor M g,r \ M g,r, the action of S r on M g,r extends to an action on M g,r. First, we define the configuration scheme X as follows: Definition 1. Let r be a natural number. Then we shall define as the fiber product of the classifying morphism of stacks SpecK M g determined by the curve X SpecK and the morphism of stacks M g,r M g obtained by forgetting the marked points. Since M g,r M g is representable, is a scheme. We shall denote by X the fs scheme obtained by equipping with the structure induced by the structure of M g,r. We shall denote by U X the interior of X, and by D the reduced closed subscheme of obtained as the complement of U X in. Note that by definition, the scheme U X is naturally isomorphic to the usual r-th configuration space of X, and the action of S r on M g,r (respectively, M g,r) determines an action on (respectively, X ). For simplicity, we shall write U (respectively, D ) instead of U X (respectively, D X ) when there is no danger of confusion.

LOG CONFIGURATION SCHEMES 7 As is well-known, the pull-back of the divisor M g,r \ M g,r via p M r+1 : M g,r+1 M g,r is a subdivisor of the divisor M g,r+1 \ M g,r+1 (cf. [7], the proof of Theorem 2.7). Thus, there exists a unique morphism of stacks p M r+1 : M g,r+1 M g,r whose underlying morphism of stacks is p M r+1. Moreover, for an integer 1 i r, since p M i is obtained as the composite of the automorphism of M g,r+1 determined by the action of an element of S r and p M r+1, the morphism of stacks pm i also extends to a morphism of stacks M g,r+1 M g,r. We shall denote this morphism of stacks by p M i. Definition 2. Let r be a natural number. (i) Let 1 i r + 1 be an integer. Then p M i : M g,r+1 M g,r (respectively, p M i : M g,r+1 M g,r ) determines a morphism (respectively, X ). We shall denote this (r+1) X morphism by p X i (respectively, p i ). Note that by the definition of stable curves, p X i is proper, flat, geometrically connected, and geometrically reduced. For simplicity, we shall write p i (respectively, p i ) instead of p i (respectively, p i ) when there is no danger of confusion. (ii) Let I = {i 1,i 2,,i I }, where i 1 < i 2 < < i I, be a non-empty subset of {1, 2,, r}. Then we shall denote by pr I : X X (r I ) the morphism obtained as the compactification of the projection U X U X(r I ) given by mapping (x 1,,x r ) to (x i1,,x ii ), i.e., if the morphism U X U X(r I ) which induces a morphism U X (S) U X(r I ) (S) (x 1,,x r ) (x i1,,x ) ii for any scheme S over K, where x i is an S-valued point of X, is given as the compsite p UX(r I ) j r I p U X(r I +1) j r I +1 p UX(r 2) j r 2 p UX(r 1) j r 1, pr I where p UX(i) j is the projection U X(i+1) U X(i) obtained by forgetting the j-th factor, then we shall write def = p X (r I ) j r I p X (r I +1) j r I +1 p X (r 2) j r 2 p X (r 1) j r 1.

8 YUICHIRO HOSHI Note that it is immediate that this composite only depends on I, i.e., this composite is independent of the choice of the sequence (j r I,,j r 1 ). For simplicity, we shall write pr I instead of pr I when there is no danger of confusion. Next, let us consider the scheme-theoretic and scheme-theoretic properties of X in more detail. Proposition 2.1. Let r be a natural number, 1 i r + 1 an integer, and I a non-empty subset of {1,2,,r}. (i) The scheme is connected. (ii) The morphism p i is smooth. In particular, pr I smooth. (iii) The scheme X is regular. (iv) The scheme is regular, and the structure of X a divisor with normal crossings. is also is given by Proof. Assertion (i) follows from the fact that X (0) = SpecK is connected, together with the fact that the p i s are proper and geometrically connected. Next, we prove assertion (ii). The assertion for p r+1 fact that p M r+1 : M g,r+1 M g,r p i is a composite of an automorphism of X follows from the is smooth (cf. [5], Section 4). Since obtained by permuting of the marked points and p r+1, the morphism p i is also smooth. Assertion (iii) follows from assertion (ii), together with the regularity of the scheme X (0) = SpecK. Finally, we prove assertion (iv). Since the natural morphism X M g,r is strict, for any geometric point x, the stalk (M X /OX ) x of the characteristic sheaf of X at x is isomorphic to N n for some natural number n. Thus, assertion (iv) follows from assertion (iii). Proposition 2.2. Let r be a natural number, 1 i r + 1 an integer, and x X a strict geometric point, i.e., a strict morphism whose underlying morphism of schemes is a geometric point (cf. [3], Definition 1.1, (i)). Then the following sequence is exact: lim π 1 (X (r+1) X x λ ) via pr 1 π 1 (X (r+1) ) via p i π 1 (X ) 1.

LOG CONFIGURATION SCHEMES 9 Here, the projective limit is over all reduced covering points x λ x, i.e., a morphism obtained as the composite of a connected Kummer finite étale covering (x λ ) x and the strict morphism x λ (x λ ) whose def underlying morphism of schemes is the closed immersion x λ = (x λ ) red x λ defined by the ideal generated by the nilpotent elements of O x λ (cf. [3], Definition 1.1, (ii)). Proof. This follows immediately from Proposition 2.1; [3], Theorem 2.3. 3. Divisors at infinity of configuration schemes We continue with the notation of the preceding Section. In this Section, we consider the scheme-theoretic and scheme-theoretic properties of the divisors defining the structures of the configuration schemes. Definition 3. Let r 2 be an integer, and I a subset of {1,2,,r} of cardinality I # 2 equipped with the natural ordering. Then we shall denote by (C I X (r I # +1) K M 0,I # +1) the r-pointed stable curve of genus g obtained by applying the clutching morphism of stacks (cf. [7], Definition 3.8) β g,0,{1,2,,r}\i,i : M g,r I # +1 K M 0,I # +1 M g,r, where {1,2,,r} \ I is equipped with the natural ordering, to the (r I # + 1)-pointed stable curve of genus g X (r I # +2) K M 0,I # +1 X (r I # +1) K M 0,I # +1 obtained by base-changing p X(r I # +1) r I # +2 : X (r I # +2) X (r I # +1) and the (I # + 1)-pointed stable curve of genus 0 X (r I # +1) K M 0,I # +2 X (r I # +1) K M 0,I # +1 obtained by base-changing the universal curve M 0,I # +2 M 0,I # +1 over M 0,I # +1. Note that the clutching locus of X (r I # +2) K M 0,I # +1 X (r I # +1) K M 0,I # +1 (respectively, X (r I # +1) K M 0,I # +2 X (r I # +1) K M 0,I # +1) is the (r I # + 1)-st (respectively, (I # + 1)-st) section (cf. [7], Definition 3.8). Then it is immediate that the classifying morphism of stacks X (r I # +1) K M 0,I # +1 M g,r determined by the r-pointed stable curve of genus g (C I X (r I # +1) K M 0,I # +1)

10 YUICHIRO HOSHI factors through, and the resulting morphism X (r I # +1) K M 0,I # +1 is a closed immersion since it is a proper monomorphism. We shall denote by δ X I this closed immersion, by D X I the scheme-theoretic image of δ X I, by D I the scheme obtained by equipping D I with the structure induced by the structure of X, and by δ I : D I the strict closed immersion whose underlying morphism of schemes is X δ X I. For simplicity, we shall write D I (respectively, D I ; respectively, δ I ; respectively, δ I ) instead of D I (respectively, D I ; respectively, δ X I; respectively, δ I ) when there is no danger of confusion. Proposition 3.1. Let r 2; 1 i r + 1 be integers, and I a subset of {1,2,,r} of cardinality I # 2. (i) The scheme D I is irreducible. (ii) The divisor D of is J D J, where J ranges over the subsets of {1,2,,r} of cardinality 2. (iii) The closed subscheme of determined by the composite of the natural closed immersions defined in Definition 3 X (r I # +1) K M 0,I # +2 C I (respectively, X (r I # +2) K M 0,I # +1 C I ) is D (r+1)i {r+1} (respectively, D (r+1)i ). (iv) Let J be a subset of {1,2,,r} of cardinality 2. Then D I D J if and only if I J, J I, or I J =. (v) The inverse image of D I via p i is D (r+1)(i {r+1}) δ i D (r+1)i δ i, where δ i = ((1,2,,r + 1) (1,2,,i 1,i + 1,,r,r + 1,i)) S r+1, and I δ i = {δ i (k) k I}. Proof. First, we prove assertion (i). The smoothness of the morphism p (s)s+1 : X (s+1) X (s) and the morphism M 0,t+4 M 0,t+3 obtained by forgetting the (t+4)-th marked point, where s, t are natural numbers, imply the regularity of X (r I # +1) K M 0,I # +1 ; therefore, D I is normal (cf. [6], Theorem 4.1). Moreover, by a similar argument to the argument used in the proof of Proposition 2.1, (i), D I is connected. Thus, in light of the normality just observed, D is irreducible. Assertions (ii), (iii), (iv), and (v) follow from the construction of D I.

LOG CONFIGURATION SCHEMES 11 Proposition 3.2. Let r 2 be an integer, and I a subset of {1,2,,r} of cardinality I 2. Then the morphism D I D I = X (r I +1) K M 0,I +1 induced by the natural inclusion O D I M DI determines a morphism ν I : D I X (r I +1) K M 0,I +1 which is of type N, i.e., the underlying morphism of schemes is an isomorphism, and the relative characteristic sheaf is locally constant with stalk isomorphic to N (cf. [3], Definition 4.1). Moreover, let I [i] be the unique subset of {1,2,,r 1} such that for j {1,2,,r 1}, j I [i] if and only if { j I (if j < i) Then the following hold: j + 1 I (if j i). (i) If i I, then (I [i] ) = I 1, and the diagram X (r I +1) K M 0,I +1 ν I via p X (r I +1) K M 0,I ν X (r 1) I [i] (r 1)i D I δ I D (r 1)I [i] δ (r 1)I [i] X p (r 1)i X (r 1) commutes, where if I = {i 1,i 2,,i I }, i 1 < i 2 < < i I, and i = i j, then the left-hand vertical arrow is the morphism induced by the morphism M 0,I +1 M 0,I obtained by forgetting the j-th marked point. (ii) If i I, then (I [i] ) = I, and the diagram X (r I +1) K M 0,I +1 ν I via p X (r I ) K M 0,I +1 ν X (r 1) I [i] (r 1)i D I δ I D (r 1)I [i] δ (r 1)I [i] X p (r 1)i X (r 1) commutes, where if {1,2,,r} \ I = {i 1,i 2,,i r I }, i 1 < i 2 < < i r I, and i = i j, then the left-hand vertical arrow is the morphism induced by p (r I )j : X (r I +1) X (r I ).

12 YUICHIRO HOSHI Proof. By the definition of D I, the scheme obtained by forgetting the portion of the structure of D I defined by the divisor D I of is isomorphic to X (r I +1) K M 0,I +1 ; moreover, it follows from [3], Lemma 4.12, (i), together with the definition of the structure of D I, that the morphism D I D I induced by the natural inclusion OD I M DI factors through the morphism obtained by forgetting the portion of the structure of D I defined by the divisor D I of. Thus, we obtain a morphism D I X (r I +1) K M 0,I +1. On the other hand, by construction, it is a morphism of type N. The assertion that the diagrams in the statement of Proposition 3.2 commute follows from Proposition 3.1, (v), together with the definitions of D I and ν I. Definition 4. Let r 2 be an integer, and I a subset of {1,2,,r} of cardinality I 2. Then we shall denote by ν X I the underlying morphism of schemes of the morphism ν I, and by U I the open subscheme of D X I determined by the open immersion U X(r I +1) K M 0,I +1 X (r I +1) K M 0,I +1 ν 1 I D X I. For simplicity, we shall write ν I (respectively, ν I; respectively, U I ) instead of ν I (respectively, ν I; respectively, U X I) when there is no danger of confusion. 4. Fundamental groups of the configuration schemes We continue with the notation of the preceding Section. In this Section, we study fundamental facts concerning the fundamental groups of configuration schemes and their divisors at infinity. Let Σ be a non-empty set of prime numbers. We shall fix a separable closure K sep of K and denote by G K the absolute Galois group Gal(K sep /K) of K. Moreover, we shall denote by Λ the maximal pro-σ quotient of Ẑ(1)(Ksep ). Definition 5. Let r be a natural number. Then we shall denote by (respectively, M r+3,k ) the geometrically pro-σ fundamental group of X (respectively, the scheme M 0,r+3) and write Π X def = X (1), and

LOG CONFIGURATION SCHEMES 13 def P K = M 4,K. Moreover, if r 2, then for a subset I of {1,2,,r} of cardinality 2, we shall denote by I the geometrically pro-σ fundamental group of D I. For simplicity, we shall write Π (respectively, M r+3 ; respectively, P ; respectively, Π I) instead of Π (respectively, M r+3,k ; respectively, P K ; respectively, I ) when there is no danger of confusion. First, we show the following theorem. Theorem 4.1. Let X be a smooth, proper, geometrically connected curve of genus 2 over a field K, r 3 an integer, and i,j,k {1,2,,r} distinct elements of {1,2,,r}. Then, for any non-empty set of prime numbers Σ, the geometrically pro-σ fundamental group of X is topoically generated by the respective images of the geometrically pro-σ fundamental groups of D {i,j}, D {j,k}, and D {i,j,k}. Proof. It is immediate that it is enough to show the assertion in the case where Σ is the set of all prime numbers; thus, assume that Σ is the set of all prime numbers. Moreover, it is immediate that it is enough to show the assertion in the case where {i,j,k} = {1,2,3}; thus, assume that {i,j,k} = {1,2,3}. δ {2,3} X p (r 1)3 X (r 1) coincides ; in particular, this composite is of type N (cf. Proposition It follows that the composite D {2,3} with ν {2,3} via δ {2,3} via p (r 1)3 3.2). Therefore, the composite {2,3} is surjective (cf. [3], Lemma 4.5). Thus, it is enough to show that the respective images of {1,2} and Π {1,2,3} in Π (r 1) topoically generate the kernel of the morphism (r 1) induced by p (r 1)3. Let x X (r 1) be a strict geometric point of X (r 1) whose image lies in U (r 1){1,2}. Then it follows from Proposition 2.2 that the kernel of the morphism Π (r 1) induced by p (r 1)3 is included in the image of the natural morphism π 1 (X ) x, where X is the fs scheme x obtained as the fiber product of p (r 1)3 : X X (r 1) and x X (r 1). On the other hand, it follows from the definition of the open subscheme U (r 1){1,2} D (r 1){1,2}, together with Proposition 3.1, (v), that the schemes obtained by equipping the irreducible components of x with the structures induced by the structure of X are the schemes x

14 YUICHIRO HOSHI obtained as the respective fiber products of the diagrams x D {1,2} via p (r 1)3 X (r 1) ; x D {1,2,3} via p (r 1)3 X (r 1). Therefore, the assertion follows from the arithmic anaue of [13], Example 5.5 (cf. also [13], 5.5). Remark. Theorem 4.1 can be regarded as a arithmic anaue of [8], Remark 1.2. Definition 6. We shall say that a set of prime numbers Σ is innocuous in K if { the set of all prime numbers or {l} if p = 0 Σ = {l} if p 2, where l is a prime number which is invertible in K. In the rest of this paper, we assume that Σ is innocuous in K. Lemma 4.2. Let r,r be natural numbers, and I a subset of {1,2, r} of cardinality I 2. (i) The natural morphism U K M 0,r +3 X K M r +3 induces an isomorphism π 1 (U K M 0,r +3) (Σ) G K M, where r +3 π 1 ( ) (Σ) is the geometrically pro-σ fundamental group of ( ). In particular, the natural morphism U X (respectively, U I X (r I +1) K M 0,I +1 ) induces an isomorphism π 1 (U ) (Σ) Π (respectively, π 1 (U I ) (Σ) (r I +1) G K M I +1 ). (ii) Let 1 i r + 1 (respectively, 1 i r + 4) be an integer, and x U (respectively, x M 0,r+3 ) a geometric point of U (respectively, M 0,r+3 ). Then the sequence 1 π 1 (X (r+1) X x) (Σ) via pr 1 (r+1) Π 1 (respectively, 1 π 1 (M 0,r+4 M x) (Σ) via pr 1 M 0,r+3 r+4 M r+3 1)

LOG CONFIGURATION SCHEMES 15 is exact, where the third arrow is the morphism induced by the morphism p i (respectively, M 0,r+4 M 0,r+3 obtained by forgetting the i-th marked point). (iii) For a profinite group Γ (respectively, a scheme S), we shall denote by S(Γ) (respectively, Sét ) the classifying site of Γ, i.e., the site defined by considering the category of finite sets equipped with a continuous action of Γ (respectively, the étale site of S). Then we have natural morphisms of sites (U K M 0,r +3)ét S(π 1 (U K M 0,r +3) (Σ) ) S( G K M r +3 ). Let A be a finite G K M r +3 -module whose order is a Σ-integer, and n an integer. Then the morphisms H ń et (U K M 0,r +3, F A ) H n (π 1 (U K M 0,r +3) (Σ),A) H n ( G K M r +3,A) induced by the above morphisms of sites are isomorphisms, where F A is the locally constant sheaf on U K M 0,r +3 determined by A. Proof. Assertion (i) follows immediately from the purity theorem (cf. [9], Theorem 3.3, also [3], Remark 1.10), together with [3], Proposition 2.4, (ii). Next, we prove assertion (ii). To prove assertion (ii), by base-changing, we may assume that K is a separably closed field. Moreover, if Σ is the set of all prime numbers, then this follows from [8], Lemma 2.4, together with assertion (i). Thus, we may assume that Σ = {l} for a prime number l which is invertible in K. Then, to prove assertion (ii), it follows from [8], Lemma 3.1, (i), the assertion in the case where Σ is the set of all prime numbers, together with [1], Proposition 3, that it is enough to show that the representation π 1 (U ) Aut((π 1 (X (r+1) X x) (Σ) ) ab ) (respectively, π 1 (M 0,r+3) Aut((π 1 (M 0,r+4 M x) (Σ) ) ab )) 0,r+3 determined by the exact sequence appearing in the statement of assertion (ii) in the case where Σ is the set of all prime numbers factors through a pro-σ quotient of π 1 (U ) (respectively, π 1 (M 0,r+3)). On the other hand, it is immediate that this representation is trivial. This completes the proof of assertion (ii). Finally, we prove assertion (iii). The assertion that the second morphism in question is an isomorphism follows immediately from assertion (i); thus,

16 YUICHIRO HOSHI (pr 1,pr 3 ) we prove the assertion that the first morphism in question is an isomorphism. We prove this assertion by induction on r + r. If r + r = 0, then the assertion is well-known. If r 0, then by considering the partial compactification U (r 1) K X K M 0,r +3 U (r 1) K M 0,r +3 of (p (r 1)r,id) U K M 0,r +3 : U K M 0,r +3 U (r 1) K M 0,r +3, it follows from [2], Corollary 10.3, that the sheaf F def = R q ((p (r 1)r,id) U K M 0,r +3 ) F A is locally constant and constructible; moreover, it follows from [2], Theorem 7.3, that the (r 1) G K M -module F r +3 x is naturally isomorphic to H q (U, F A U ), where x U (r 1) K M 0,r +3 is a geometric point of U (r 1) K M 0,r +3, and U def = U U(r 1) x. On the other hand, the natural morphism H n (π,a) H ń et (U, F A U ) is an isomorphism, where π def = π 1 (U) (Σ). Indeed, one then verifies immediately that it is enough to verify that every étale cohomoy class of U with coefficients in F A U vanishes upon pull-back to some connected finite étale Σ-covering V U. Moreover, by passing to an appropriate V, we may assume that F A U is trivial. Then the vanishing assertion in question is immediate (respectively, a tautoy) for n = 0 (respectively, n = 1). Moreover, the vanishing assertion in question is immediate for n 3 by [2], Theorem 9.1. If U is affine, then since H ń et (U, F A U ) vanishes for n = 2 (cf. [2], Theorem 9.1), the assertion is immediate. If U is proper, then it is enough to take V U so that the degree of V U annihilates A (cf. e.g., the discussion at the bottom of [2], p. 136). Therefore, by considering the Hochschild-Serre spectral sequence (cf. [12], Theorem 2.1.5) associated to the exact sequence 1 π via (p G K Π Mr +3 (r 1)r,id) (r 1) G K Π Mr +3 1 obtained by assertion (ii) and the Leray spectral sequence associated to the morphism (p (r 1)r,id) U K M 0,r +3, it follows that it is enough to show the assertion in the case where the pair of natural numbers (r,r ) in the statement of assertion (iii) is (r 1,r ). Moreover, if r 0, then by a similar argument to the above argument, it is verified that it is enough to show the assertion in the case where the pair of natural numbers (r,r ) in the statement of assertion (iii) is (r,r 1). This completes the proof of assertion (iii). Remark. Let r 2 be an integer, and I a subset of {1,2,,r} of cardinality I 2. Then by Lemma 4.2, (i), (iii), it follows from Proposition

LOG CONFIGURATION SCHEMES 17 3.2, together with similar arguments to the arguments used in the proves of [10], Lemmas 4.3; 4.4, and [3], Proposition 4.22, that the morphism ν I determines an exact sequence 1 Λ I via ν I (r I +1) G K M I +1 1. Lemma 4.3. Let r 2; 1 i r be integers, I a subset of {1,2,,r} of cardinality I 2, and I [i] the subset of {1,2,,r 1} defined in the statement of Proposition 3.2. Then the following hold: (i) If i I, then the diagram via p (r 1)i I via ν I (r 1)I [i] via ν (r 1)I [i] (r I +1) G K M I +1 (r I +1) G K M I is cartesian, where if I = {i 1,i 2,,i I }, i 1 < i 2 < < i I, and i = i j, then the right-hand vertical arrow is the morphism induced by the morphism M 0,I +1 M 0,I obtained by forgetting the j-th marked point. (ii) If i I, then the diagram via p (r 1)i I via ν I (r 1)I [i] via ν (r 1)I [i] (r I +1) G K M I +1 (r I ) G K M I +1 is cartesian, where if {1,2,,r} \ I = {i 1,i 2,,i r I }, i 1 < i 2 < < i r I, and i = i j, then the right-hand vertical arrow is the morphism induced by p (r I )j : X (r I +1) X (r I ). Proof. This follows from Proposition 3.2; Remark following Lemma 4.2, together with the fact that the restriction of D I D to the generic (r 1)I [i] point of D I is strict.

18 YUICHIRO HOSHI Lemma 4.4. (i) Let r 2 be an integer, and I = {i,i + 1}, where i = 1, 2. Then the following diagram is cartesian: via ν I I via pr I (r 1) via pr (r 1){i} (2){1,2} via ν (2){1,2} Π X. (ii) Let r 3 be an integer. Then the following diagram is cartesian: via ν {1,2,3} {1,2,3} (r 2) G K P via pr {1,2} pr1 (r 2) via pr (r 2){1} Proof. This follows from Lemma 4.3 by induction on r. (2){1,2} via ν (2){1,2} Π X. Definition 7. (i) Let r 2 be an integer, and I = {i,i + 1}, where i = 1, 2. Then we shall write Π G I def = X (r 1) ΠX X (2) {1,2}, where the morphism implicit in the fiber product X (r 1) Π X (respectively, X (2) {1,2} Π X) is the morphism induced by pr (r 1){i} (respectively, ν X (2) {1,2}). On the other hand, by Lemma 4.4, (i), the morphism I Π X (r 1) induced by ν I I Π X (2) {1,2} induced by pr I Π G I I. We shall denote this isomorphism by α I. (ii) Let r 3 be an integer. Then we shall write Π G {1,2,3} and the morphism induce an isomorphism def = P K GK X (r 2) ΠX X (2) {1,2}, where the morphism implicit in the fiber product X (r 2) Π X (respectively, X (2) {1,2} Π X) is the morphism induced by pr (r 2){1}

LOG CONFIGURATION SCHEMES 19 (respectively, ν X (2) {1,2}). On the other hand, by Lemma 4.4, (ii), the morphism {1,2,3} Π X (r 2) GK P K and the morphism {1,2,3} Π X (2) {1,2} induce an isomorphism Π G {1,2,3} {1,2,3}. We shall denote this isomorphism by α {1,2,3}. induced by ν {1,2,3} induced by pr {1,2} 5. Partial reconstruction of the fundamental groups of higher dimensional configuration schemes We continue with the notation of the preceding Section; in particular, we assume that the set of prime numbers Σ is innocuous in K. In this Section, we show that the fundamental group of the configuration space can be partially reconstructed from a collection of data concerning the fundamental groups of the configuration spaces of lower dimension. Definition 8. Let r 3 be an integer. Then we shall denote by Π G the profinite group obtained as the free profinite product of We shall denote by {1,2} ; Π {2,3} ; Π {1,2,3}. f X : Π G the morphism determined by the morphisms D I δ I X where I = {1,2}, {2,3}, and {1,2,3}. Note that it follows from Theorem 4.1 that f X is surjective. Let I = {1,2}, {2,3}, or {1,2,3}. Then by the definition of Π G, we have a natural morphism I ΠG. We shall denote this morphism by δ G I. Next, we define the collection of data used in the partial reconstruction of the fundamental groups of higher dimensional configuration schemes performed in Theorem 5.2 below. Definition 9. Let r 2 be an integer. (i) We shall denote by D X (Σ), or D X(1) (Σ), the collection of data consisting of

20 YUICHIRO HOSHI (i-1) the profinite groups (i-2) the morphisms X (2), X (2) {1,2}, Π X, G K, P K ; via px (1) i X (2) Π X (i = 1, 2), and the morphisms induced by the respective structure morphisms (i-3) the morphism Π X G K ; P K G K ; via δ X (2) {1,2} X (2) {1,2} X (2). (ii) We shall denote by D X (Σ) the collection of data consisting of (ii-1) the profinite groups X (k) (1 k r + 1), X (2) {1,2}, G K, P K ; (ii-2) the morphisms via px (k 1) i X (k) X (k 1) (2 k r + 1, 1 i k), via ν X (2) {1,2} X (2) {1,2} Π X, and the morphisms induced by the respective structure morphisms (ii-3) the composites Π X G K ; P K G K ; Π G I α I I via δ I, where I = {1,2}, {2,3}, and {1,2,3}. (iii) We shall denote by D G (Σ) the collection of data consisting of (iii-1) the data obtained by replacing in (ii-1) by Π G ;

LOG CONFIGURATION SCHEMES 21 (iii-2) the data obtained by replacing r+1) in (ii-2) by Π G f X(r+1) via p i via p i r + 1); (iii-3) the data obtained by replacing (1 i (1 i (respectively, via δ I ) in (ii-3) by ΠG (respectively, via δ G I ). In the following, let Y be a smooth, proper, geometrically connected curve of genus 2 over a field L, and Σ Y a non-empty set of prime numbers which is innocuous in L. We shall fix a separable closure L sep of L and denote by G L the absolute Galois group Gal(L sep /L) of L. Definition 10. Let r 2 be an integer. (i) We shall refer to isomorphisms X (k) Π Y (k) (k = 1,2) ; X (2) {1,2} G K GL ; P K Π P L Y (2) {1,2} ; which are compatible with the morphisms given in the definitions of D X (Σ) and D Y (Σ Y ) as an isomorphism of D X (Σ) with D Y (Σ Y ). (ii) We shall refer to isomorphisms X (k) Π Y (k) (1 k r + 1) ; X (2) {1,2} G K GL ; P K Π P L Y (2) {1,2} ; which are compatible with the morphisms given in the definitions of D X (Σ) and D Y (Σ Y ) as an isomorphism of D X (Σ) with D Y (Σ Y ). (iii) We shall refer to isomorphisms Π G X (2) {1,2} Π G Y(r+1) ; X (k) Y (2) {1,2} ; G K Π Y (k) (1 k r) ; G L ; P K Π P L which are compatible with the morphisms given in the definitions of D G (Σ) and D G Y (Σ Y ) as an isomorphism of D G (Σ) with D G Y (Σ Y ). (iv) Let φ G : DG (Σ) D G Y (Σ Y ) be an isomorphism of D G (Σ) with D G Y (Σ Y ). Then by forgetting the isomorphism Π G Π G Y(r+1) in φ G, we obtain an isomorphism of D X (r 1) (Σ) with D Y(r 1) (Σ Y ).

22 YUICHIRO HOSHI We shall denote this isomorphism by F 1 (φ G ). Note that it is immediate that the correspondence is functorial. φ G F 1(φ G ) Remark. If there exists an isomorphism of D X (Σ) (respectively, D X (Σ); respectively, D G (Σ)) with D Y (Σ Y ) (respectively, D Y (Σ Y ); respectively, D G Y (Σ Y )), then Σ = Σ Y. Indeed, this follows from the fact that the abelianization of the kernel of Π X G K (respectively, Π Y G L ) is a non-trivial module which is free over Z (Σ) (respectively, Z (Σ Y ) ). Proposition 5.1. Let r 2 be an integer, Σ a set of prime numbers that is innocuous in K and L, and φ G : DG (Σ) D G Y (Σ) an isomorphism. Then if the isomorphism Π G Π G Y (r+1) in φ G induces an isomorphism of the kernel of the morphism f X(r+1) : Π G with the kernel of the morphism f Y(r+1) : Π G Y (r+1) Y (r+1), then there exists an isomorphism F(φ G ) : D (Σ) D Y (Σ). Moreover, the correspondence is functorial. φ G F(φG ) Proof. Since the morphism f X(r+1) : Π G (respectively, f Y(r+1) : Π G Y (r+1) Y (r+1) ) is surjective (cf. Theorem 4.1), by the assumption, we obtain an isomorphism φ : Π Y (r+1) induced by the isomorphism Π G Π G Y (r+1) in φ G. Therefore, by replacing the isomorphism Π G Π G Y(r+1) in φ G by φ, we obtain an isomorphism F(φG ) of the desired type. Theorem 5.2. Let r 2 be an integer, Σ a set of prime numbers which is innocuous in K and L, and φ (r 1) : D X(r 1) (Σ) D Y(r 1) (Σ) an isomorphism. Then there exists an isomorphism F +1 (φ (r 1) ) : D G (Σ) D G Y (Σ) such that Moreover, the correspondence is functorial. F 1 (F +1 (φ (r 1) )) = φ (r 1). φ (r 1) F +1 (φ (r 1) )

LOG CONFIGURATION SCHEMES 23 Proof. First, we define an isomorphism φ G {1,2} (respectively, φ G {2,3} ; respectively, φ G {1,2,3}) of Π G {1,2}, (respectively, ΠG {2,3} ; respectively, Π G {1,2,3} ) with ΠG Y (r+1) {1,2}, (respectively, ΠG Y (r+1) {2,3} ; respectively, Π G Y (r+1) {1,2,3}) as follows: (i) We define an isomorphism φ G {1,2} : Π G Π G as the {1,2} Y (r+1) {1,2} isomorphism induced by the isomorphisms Π Y, X (2) {1,2} Y (2) {1,2}, and Π X Π Y in φ (r 1). (ii) We define an isomorphism φ G {2,3} : Π G {2,3} Π G Y (r+1) {2,3} as the isomorphism induced by the isomorphisms Π X (2) {1,2} Y (2) {1,2}, and Π X Π Y in φ (r 1). P K Y, (iii) We define an isomorphism φ G {1,2,3} : Π G {1,2,3} Π G Y (r+1) {1,2,3} as the isomorphism induced by the isomorphisms Π P L, X (r 1) Π Y in φ (r 1). Π Y (r 1), X (2) {1,2} Y (2) {1,2}, G K G L, and Π X Then these isomorphisms φ G {1,2}, φ G {2,3}, and φ G {1,2,3} induce an isomorphism of the profinite group Π G obtained as the free profinite product of Π G {1,2} ; ΠG {2,3} ; ΠG {1,2,3} (cf. Definitions 7; 8) with the profinite group Π G Y (r+1) obtained as the free profinite product of Π G Y (r+1) {1,2} ; ΠG Y (r+1) {2,3} ; ΠG Y (r+1) {1,2,3}. Now we denote this isomorphism by φ G. On the other hand, for an integer 1 i r + 1, we define a projection q X i : Π G as follows: (i) If i = 1 or 2, then we define a morphism q {1,2} i : Π G {1,2} = ΠX X (2) {1,2} Π as the first projection. If i 3, then we define a morphism q {1,2} i : ΠG {1,2} Π as the composite Π G {1,2} via p X (r 1) i 2 id D X (2) {1,2} Π G {1,2}

24 YUICHIRO HOSHI α {1,2} via δ {1,2} {1,2}. (ii) We define a morphism q {2,3} 1 : ΠG {2,3} Π as the morphism obtained by replacing Π G {1,2} (respectively, p X (r 1) i 2 ) by Π G {2,3} (respectively, p X (r 1) 1 ) in the definition of q{1,2} i for i 3. If i = 2 or 3, then we define a morphism q {2,3} i : ΠG {2,3} = ΠX X (2) {1,2} Π as the first projection. If i 4, then we define a morphism q {2,3} i : ΠG {2,3} Π as the morphism obtained by replacing Π G {1,2} by ΠG {2,3} in the definition of q {1,2} i for i 3. (iii) If i = 1, 2, or 3, then we define a morphism q {1,2,3} i as the composite Π G {1,2,3} pr 2,3 Π G α {1,2} {1,2} If i 4, then we define a morphism q {1,2,3} i as the composite Π G {1,2,3} : Π G {1,2,3} via δ {1,2} {1,2}. : Π G {1,2,3} Π via id p P X K (r 2) i 3 id D X (2) {1,2} Π G {1,2,3} α {1,2,3} via δ {1,2,3} {1,2,3}. Moreover, by a similar procedure to the procedure that we applied to define the morphism q {1,2} i morphism q {1,2} (respectively, q{2,3} i Y i (respectively, q {2,3} Y i Then these morphisms q {1,2} i, q{2,3} i ; respectively, q{1,2,3} i ), we define a ; respectively, q{1,2,3} Y i )., and q{1,2,3} i (respectively, q {1,2} Y i, q{2,3} Y i, and q {1,2,3} Y i ) induce a morphism Π G (respectively, Π G Y (r+1) Y ). We denote this morphism by q X i (respectively, q Y i). Then by Lemma 4.4, together with constructions, for any 1 i r+1, the morphism q X i (respectively, q Y i) factors as the composite Π G f X(r+1) via p i (respectively, Π G f Y(r+1) Y (r+1) via p Y (r+1) Y i Y );

LOG CONFIGURATION SCHEMES 25 moreover, the diagram Π G q X i φ G Π G q Y i Y commutes, where the bottom horizontal arrow is the isomorphism in φ (r 1). Therefore, by equipping φ (r 1) with the isomorphism φ G, we obtain an isomorphism F +1 (φ (r 1) ) of DX G (Σ) with DY G (Σ) of the desired type. Acknowledgement I would like to thank Professor Shinichi Mochizuki for suggesting the topics, helpful discussions, warm encouragements, and valuable advices. I would also like to thank the referee for useful suggestions and comments. References [1] M. P. Anderson, Exactness properties of profinite completion functors, Topoy 13 (1974), 229-239. [2] E. Freitag and R. Kiehl, Étale cohomoy and the Weil conjecture, Springer-Verlag (1988). [3] Y. Hoshi, The exactness of the homotopy sequence, RIMS Preprint 1558 (2006). [4] Y. Hoshi, Absolute anabelian cuspidalizations of configuration spaces over finite fields, RIMS Preprint 1592 (2007). [5] F. Kato, Log smooth deformation and moduli of smooth curves, Internat. J. Math. 11 (2000), 215-232. [6] K. Kato, Toric singularities, Amer. J. Math. 116 (1994), 1073-1099. [7] F. F. Knudsen, The projectivity of the moduli space of stable curves II, Math. Scand. 52 (1983), 161-199. [8] M. Matsumoto, Galois representations on profinite braid groups on curves, J. Reine. Angew. Math. 474 (1996), 169-219. [9] S. Mochizuki, Extending families of curves over regular schemes, J. Reine. Angew. Math. 511 (1999), 43-71. [10] S. Mochizuki, Topics surrounding the anabelian geometry of hyperbolic curves, Galois Groups and Fundamental Groups, Math. Sci. Res. Inst. Publ. 41 (2003), 119-165. [11] S. Mochizuki, Absolute anabelian cuspidalizations of proper hyperbolic curves, J. Math. Kyoto Univ. 47 (2007), no. 3, 451-539. [12] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomoy of number fields, Grundlehren der Mathematischen Wissenschaften 323, Springer-Verlag (2000). [13] J. Stix, A general Seifert-Van Kampen theorem for algebraic fundamental groups, Publ. Res. Inst. Math. Sci. 42 (2006), no. 3, 763-786.

26 YUICHIRO HOSHI Yuichiro Hoshi Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 JAPAN e-mail address: yuichiro@kurims.kyoto-u.ac.jp (Received June 21, 2007) (Revised April 25, 2008)