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On the Picard group of the moduli scheme of stable curves in positive characteristic
JOURNAL OF PURE A N D APPLIE D ALGEBRA
ELSEVIER
Journal of Pure and Applied Algebra 124 (1998) 1 5
On the Picard group of the moduli scheme of stable curves in positive characteristic E. Ballico* Department of Mathematics, University of Trento, 38050 Povo (TN), Italy Communicated by C.A. Weibel; received 1 February 1995; revised 28 April 1996
Abstract
Here we prove that the moduli space of genus 9 stable curves in characteristic p > 0 has (up to torsion) the same Betti numbers for the 6tale topology as the characteristic 0 moduli space and if p > 84(9 - 1) the same Picard group (up to torsion). :{! 1998 Elsevier Science B.V.
1991 Math. Subj. Class.." 14HI0, 14C22
O. I n t r o d u c t i o n
Let M~- (Z) be the m o d u l i scheme (over the integers) of stable genus g curves. F o r every p r i m e p M 2 (p) will d e n o t e the m o d u l i scheme of stable genus g curves over an a l g e b r a i c a l l y closed field of characteristic p. It is k n o w n (see [9]) that Pic(M~- (C)) ® Q has r a n k [g/2] + 2 a n d has as basis a certain class ~. a n d the b o u n d a r y c o m p o n e n t s A i, 0 _< i _< [9/2] (a general element of A0 is an integral curve with a unique node, a general element of A~, 1 _< i _< [g/ /2], is the union of a s m o o t h curve of genus i a n d a s m o o t h curve of genus g - i intersecting at a point). This follows from a t o p o l o g i c a l t h e o r e m of Harer. T h e aim of this note is to check that this is true also in positive characteristic p if p is large with respect to g. T o prove this we will prove (with no restriction on p) t h a t for all j the 1-adic 6tale c o h o m o l o g y g r o u p s H~t(l'~l~ (p), Qi) a n d H~t(M[ (C), Q~) are i s o m o r p h i c , i.e. we will p r o v e the following results 0.1 a n d 0.2.
0.1. Fix an integer g >_ 5 and primes p, I with 1 ~ p. Then dim (H J r ( M ; (p), Qt)) = dim(HJt(M~ - (C), Qt)),
Theorem
*Fax: + 39-461881624; e-mail: [email protected]. 0022-4049/'98/$19.00 :~ 1998 Elsevier Science B.V. All rights reserved PlI S 0 0 2 2 - 4 0 4 9 ( 9 6 ) 0 0 1 1 8 - 3
2
E. Ballieo/Journal of Pure and Applied Algebra 124 (1998) l-5
Theorem 0.2. For all integers g >_ 5, for i = 1,2 and all primes p, l with I ¢ p H~t (M~ (p), Qt) is freely generated by the algebraic classes with the same name as in characteristic 0 (i.e. the ones described before for i = 1 and the ones described in [1] for i = 2). In [13] the existence of an intersection product was proved in the Chow group (tensor Q) of M ] ( p ) for the primes p > 84(g - 1) (see step (c) of the proofs of Theorems 0.1,0.2 and 0.3 in Section 1). Indeed M~ (p) is the moduli scheme associated to the smooth moduli stack of stable genus 9 curves. In particular we may consider the numerical equivalence for algebraic cycles of M~- (p). F r o m Theorems 0.1 and 0.2 we will obtain the following result. Theorem 0.3 Fix an integer g >_ 5 and a prime p > 84(g - 1). Then the Neron-Severi group NS(M~(p)) of M ] ( p ) is freely generared by the [g/2] + 2 classes 2 and Ai, 0 < i _< [g/2]. The proofs of Theorems 0.1, 0.2 and 0.3 will use a reduction to the characteristic 0 case (i.e. to the topological theorem of Harer) and a strong result proved in [10] (see below). The knowledge of the Picard group (up to torsion) of M o- (p) is obviously very interesting. However, we want to point out that in the range of (g, p) in which it is proved here the more interesting potential application (i.e. the study of the Kodaira dimension of M q- (p)) is true for a simpler reason. Indeed, first by the quoted result of Vistoli, up to torsion we may speak about intersection products of Weil divisors and curves of M~-(p). Then note that it is easy to check that the characteristic 0 proof of Harris - Mumford [4] of the fact the M~- (C) is of general type for large g works for p > 84(g - 1) just because no characteristic p ramification problem occurs in the computations and in the foundations (e.g. the theory of admissible covering with curves of genus g as total space). Recently (see [10], Th. 3.1.1]) it was proved that if we omit just one prime, say r, over Spec(Z)(r) there is a smooth covering M of M~j (Z~rl). We will apply the existence of M to check Theorems 0.1 and 0.2.
1. The proofs Unless otherwise clear from the context, in this paper we work over an algebraically closed field K of characteristic p. Since all the assertions in this paper are invariants under extension of algebraically closed fields we may freely assume in a proof that we are working over the algebraic closure of the minimal field. We divide the proof of Theorems 0.1, 0.2 and 0.3 into 7 steps and a proposition (see Proposition 1.1), only step (a) being necessary for Theorems 0.1 and 0.2. The remaining steps contain also material showing that if p > 84(g - 1) the order of the Q-divisor classes modulo Cartier divisors is bounded independently on p. Very recently in [2] Edidin
E. Ballico/Journal of Pure and AppliedAlgebra 124 (1998) 1-5
3
introduced a good intersection theory for these moduli spaces and this type of objects (see in particular [2, Section 3.2]). (a) Fix a prime r ¢ p. Let M be the smooth covering of M~- defined over Spec(Z)~r~ whose existence was proved in [10]. M i s a Galois covering for some group G by [10], first 4 lines of p. 12 in Section 3, after 3.1.2. Set M(p) := M(Spec(K)) and let M(0) be the corresponding characteristic 0 scheme. By smooth base change for 6tale cohomology [8, VIA. 1] the specialization maps from the groups H~,(M, Q~) to the corresponding groups of M(p) and M(0) are isomorphisms. Furthermore, the smooth specialization maps commute with the action of G and hence induce isomorphisms between the corresponding G-invariant parts. Since H~,(M(O),Q~)~= H~,(M~-(O),Qz) while H~t(M(p), Q~)C,contains H~t(M~ (p), Ql) we obtain Theorem 0.1. Theorem 0.2 follows from Theorem 0.1 and the computations of HZe,(Mr(O), Ql) made by Harer and of H4dM~ (0), Ql) made in [1] (using another result of Harer). (b) From now on we fix a prime p > 84(g - 1). Let M be the restriction of M o (2") to the localization of Z at all primes > 84(g - 1). Since p > 2g + 1 it is known (see [5, Ch. IV, Ex. 2.5] or [11]) that the classical upper bound 84(,q - 1) for the order of the automorphism group of a genus g smooth curve holds true even in characteristic p. In particular no prime > 84(9 - 1) divides ord(Aut(C)) for any smooth genus g curve C. (c) Here we will check that the upper bound 84(g - 1) for the primes dividing ord(Aut(C)) holds for every stable curve C of genus ,q. Fix a genus g stable curve C. Let Y be the normalization of C and mark the points of Y which go to the singular points of C. Call T this marked curve. Every a u t o m o r p h i s m of C induces a permutation of the set of marked points of Y. Fix one such automorphism, s. Since Y has at most 2g - 2 components (the upper bound being possible only if all connected components of Y are p1), and we are interested only in primes p > 2g - 2, we may assume that s sends each irreducible component of C into itself. It is easy to check that card(Sing(C)) _< g + (2g - 2). Hence T has at most 6g - 4 marked points. Hence for primes > 6g - 4 we reduce to the case in which s fixes every marked point of T. Hence s is contained in the product of the automorphism groups (as marked smooth curves) of the connected components of T. Fix one such component, Z, and call H its automorphism group. If p~(Z) >__2, we have ord(H) _< 84(p~(Z) - 1) _< 84(g - 1), as wanted. Assume pa(Z) = I. Then Z has some marked point and taking any of them (which by assumption are fixed by s) as origin to consider Z as an Abelian variety, we have ord(s) _< 3 (e.g. by Lefschetz trace formula [8, V. 2.5]). If Z~-P ~, then Z has at least 3 marked points because C is a stable curve. Hence s lZ is the identity (e.g. by Lefschetz trace formula [8, V.2.5]). (d) By formal deformation theory and algebraization theory it is known that M at a geometric point corresponding to a curve C e M r (p) is locally for the 6tale topology the quotient of a smooth scheme by Aut(C). Since p does not divide card(Aut(C)) for any C, all these groups are linearly reductive. Hence Mr(p) has only rational singularities, it is Q-orthocyclic (or equivalently Q-orienting) in the sense of [6, Section 2 and 3] or equivalently it is an Alexander scheme in the sense of [13]; we will say for
4
E. Ballico/Journal of Pure and Applied Algebra 124 (1998) 1-5
short that M~-(p) has quotient (prime to p) singularities. The same results of course are true for every prime r > 84(9 - 1). Recall that in [13] it was proved the existence of an intersection product in the Chow group tensor Q of M j (r) for every such prime r (or see [6], Section 3 and in particular from the last part of p. 337 on); furthermore, the same references give that the Chow group tensor Q is a commutative ring. Now consider the moduli scheme M~- first over Spec(Z), then over its generic point Spec(Q) and then M g (Q) over the algebraic closure of Q. By [3, Prop. 20,3] and [6, p. 338], the divisors )~ and As on M~(Q) specialize to the divisors with the same name on M j (r) and their numerical classes on the intersections of two (or more) of them are the same. (e) By [7, Corollary at p. 155], every pair (C, G) with C smooth genus g _> 2 curve and G ___Aut(C), card(G) prime to p, can be lifted to characteristic 0. This result implies that the same is true if C is a stable singular curve whose normalization has no elliptic or rational component. Since in our setting even Aut(D) with D elliptic curve has order prime to p, we lift to characteristic 0 also the stable curves with components with normalization of genus _< 1. (f) Recall that by [12] the isolated of codimension _> 3 quotient singularities by a finite linearly reductive (i.e. with order prime to the characteristic) are rigid. Motivated by this result and by step (c) we will analyze the integral subvarieties of codimension _< 2 formed by stable curves with nontrivial automorphism group. First note that it is classical (i.e. known in characteristic 0) that for g >_ 5 no such subvariety contains a smooth curve. By steps (c) and (e) this is true also in characteristic p > 8 4 ( g - I). The divisor A1 is the unique codimension 1 subvariety formed by curves with a nontrivial automorphism. Since M~(r) is normal, it is generically smooth along A1 and indeed when the elliptic tail has no extra automorphism. In codimension 2 there is the closure of the subset o r a l formed by the curves C = A w E with A smooth of genus 9, E elliptic and such that (taking as 0 the point Ac~E) the dimension 1 Abelian variety has more automorphisms that + Id. For primes > 84( 9 - 1 ) such curves have the same j-invariant (j = 0 and 1728) as in the characteristic 0 case. Then (again contained in A1) there is the closure of the set of curves A u R with R rational and with a node, the extra automorphism exchanging the branches of the nodes of R but keeping fixed the point Ac~R. The last codimension 2 subvariety contained in Sing(M] (r)) is the closure of the set of curves B ~ F with B smooth of genus g - 2 and F smooth of genus 2 and such that Bc~F is a Weierstrass point of F. (g) Note that every Weil divisor on M ~ ( p ) induces a G-invariant divisor on its Galois covering M(p). Hence Theorem 0.3 follows from the part i = 1 of Theorem 0.2, step (d) and the following result.
Proposition
1.1. Let X be an Alexander scheme with only quotient (prime to p) singularities. Then a divisor on X is numerically equivalent to 0 if and only if its class in Hat(X, Q1 ) vanishes.
E. Ballico /Journal of Pure and Applied Algebra 124 (1998) 1-5
5
Proof. In the case of s m o o t h surfaces, see for i n s t a n c e [8, Ch. V, R e m a r k 3.9 (a)]. F o r the g e n e r a l case fix a d i v i s o r D o f X w h o s e class in HZt(x, 0_+)v a n i s h e s a n d a n i n t e g r a l c u r v e Y c X. S i n c e X has q u o t i e n t ( p r i m e to p) s i n g u l a r i t i e s we r e d u c e to t h e case in w h i c h D is a C a r t i e r divisor. T o c h e c k D" Y t a k e a surface S w i t h o n l y i s o l a t e d singularities containing of S.
Y, a n d
reduce
to the a s s e r t i o n
on
a desingularization
[]
Acknowledgements T h e a u t h o r w a n t s to t h a n k h e r e the referee of a p r e v i o u s (essentially w r o n g ) p a p e r w i t h t h e s a m e title a n d c l a i m i n g s i m i l a r results. T h e a t t t h o r was p a r t i a l l y s u p p o r t e d by MURST
and GNSAGA
of C N R (Italy).
References [1] D. Edidin, The codimension-two homology of the moduli space of stable curves is algebraic, Duke Math. J. 67 (1993) 241-272. [2] D. Edidin, Equivariant intersection theory, Duke Algebraic Geometry preprint list 9603008. [3] W. Fulton, Intersection Theory (Springer, Berlin, 1984). [4] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982) 23--86, [5] R. Hartshorne, Algebraic Geometry (Springer, Berlin, 1977). [6] S. Kleiman (with the collaboration of A. Thorup), Intersection theory and enumerative geometry, a decade in review, in: Algebraic Geometry--- Bowdoin 1985, Proc. Symposia in Pure Math. 46, Part II (Amer. Math. Soc., Providence, RI, 1987) 321 370. [7] O.A. gaudal and K. Lested, Deformation of curves, I. Moduli for hyperelliptic curves, in: Algebraic Geometry, Proc. Tromso 1977, Lecture Notes in Mathematics, Vol. 687 (Springer, Berlin, 1977) 150 167. [8] J. Milne, Etale Cohomology (Princeton University Press, Princeton, N J, 1980). [9] D. Mumford, Stability of projective varieties, l'Ensegn. Math. 24 (1977) 39 110. [10] M. Pikaart and A.J. de Jong, Moduli of curves with non-abelian level structure, preprint no. 891, Dept. Math. University Utrecht, December 1994. [11] P. Roquette, Absch~itzung der Automorphismenanzahl yon Funktionenk6rpern bei Primzahlcharakteristik, Math. Z. 117 (1970) 157-163. [12] M. Schlessinger, Rigidity of quotient singularities, Invent. Math. 14 (1971) 17 26. [13] A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989) 613 670.
ELSEVIER
Journal of Pure and Applied Algebra 124 (1998) 1 5
On the Picard group of the moduli scheme of stable curves in positive characteristic E. Ballico* Department of Mathematics, University of Trento, 38050 Povo (TN), Italy Communicated by C.A. Weibel; received 1 February 1995; revised 28 April 1996
Abstract
Here we prove that the moduli space of genus 9 stable curves in characteristic p > 0 has (up to torsion) the same Betti numbers for the 6tale topology as the characteristic 0 moduli space and if p > 84(9 - 1) the same Picard group (up to torsion). :{! 1998 Elsevier Science B.V.
1991 Math. Subj. Class.." 14HI0, 14C22
O. I n t r o d u c t i o n
Let M~- (Z) be the m o d u l i scheme (over the integers) of stable genus g curves. F o r every p r i m e p M 2 (p) will d e n o t e the m o d u l i scheme of stable genus g curves over an a l g e b r a i c a l l y closed field of characteristic p. It is k n o w n (see [9]) that Pic(M~- (C)) ® Q has r a n k [g/2] + 2 a n d has as basis a certain class ~. a n d the b o u n d a r y c o m p o n e n t s A i, 0 _< i _< [9/2] (a general element of A0 is an integral curve with a unique node, a general element of A~, 1 _< i _< [g/ /2], is the union of a s m o o t h curve of genus i a n d a s m o o t h curve of genus g - i intersecting at a point). This follows from a t o p o l o g i c a l t h e o r e m of Harer. T h e aim of this note is to check that this is true also in positive characteristic p if p is large with respect to g. T o prove this we will prove (with no restriction on p) t h a t for all j the 1-adic 6tale c o h o m o l o g y g r o u p s H~t(l'~l~ (p), Qi) a n d H~t(M[ (C), Q~) are i s o m o r p h i c , i.e. we will p r o v e the following results 0.1 a n d 0.2.
0.1. Fix an integer g >_ 5 and primes p, I with 1 ~ p. Then dim (H J r ( M ; (p), Qt)) = dim(HJt(M~ - (C), Qt)),
Theorem
*Fax: + 39-461881624; e-mail: [email protected]. 0022-4049/'98/$19.00 :~ 1998 Elsevier Science B.V. All rights reserved PlI S 0 0 2 2 - 4 0 4 9 ( 9 6 ) 0 0 1 1 8 - 3
2
E. Ballieo/Journal of Pure and Applied Algebra 124 (1998) l-5
Theorem 0.2. For all integers g >_ 5, for i = 1,2 and all primes p, l with I ¢ p H~t (M~ (p), Qt) is freely generated by the algebraic classes with the same name as in characteristic 0 (i.e. the ones described before for i = 1 and the ones described in [1] for i = 2). In [13] the existence of an intersection product was proved in the Chow group (tensor Q) of M ] ( p ) for the primes p > 84(g - 1) (see step (c) of the proofs of Theorems 0.1,0.2 and 0.3 in Section 1). Indeed M~ (p) is the moduli scheme associated to the smooth moduli stack of stable genus 9 curves. In particular we may consider the numerical equivalence for algebraic cycles of M~- (p). F r o m Theorems 0.1 and 0.2 we will obtain the following result. Theorem 0.3 Fix an integer g >_ 5 and a prime p > 84(g - 1). Then the Neron-Severi group NS(M~(p)) of M ] ( p ) is freely generared by the [g/2] + 2 classes 2 and Ai, 0 < i _< [g/2]. The proofs of Theorems 0.1, 0.2 and 0.3 will use a reduction to the characteristic 0 case (i.e. to the topological theorem of Harer) and a strong result proved in [10] (see below). The knowledge of the Picard group (up to torsion) of M o- (p) is obviously very interesting. However, we want to point out that in the range of (g, p) in which it is proved here the more interesting potential application (i.e. the study of the Kodaira dimension of M q- (p)) is true for a simpler reason. Indeed, first by the quoted result of Vistoli, up to torsion we may speak about intersection products of Weil divisors and curves of M~-(p). Then note that it is easy to check that the characteristic 0 proof of Harris - Mumford [4] of the fact the M~- (C) is of general type for large g works for p > 84(g - 1) just because no characteristic p ramification problem occurs in the computations and in the foundations (e.g. the theory of admissible covering with curves of genus g as total space). Recently (see [10], Th. 3.1.1]) it was proved that if we omit just one prime, say r, over Spec(Z)(r) there is a smooth covering M of M~j (Z~rl). We will apply the existence of M to check Theorems 0.1 and 0.2.
1. The proofs Unless otherwise clear from the context, in this paper we work over an algebraically closed field K of characteristic p. Since all the assertions in this paper are invariants under extension of algebraically closed fields we may freely assume in a proof that we are working over the algebraic closure of the minimal field. We divide the proof of Theorems 0.1, 0.2 and 0.3 into 7 steps and a proposition (see Proposition 1.1), only step (a) being necessary for Theorems 0.1 and 0.2. The remaining steps contain also material showing that if p > 84(g - 1) the order of the Q-divisor classes modulo Cartier divisors is bounded independently on p. Very recently in [2] Edidin
E. Ballico/Journal of Pure and AppliedAlgebra 124 (1998) 1-5
3
introduced a good intersection theory for these moduli spaces and this type of objects (see in particular [2, Section 3.2]). (a) Fix a prime r ¢ p. Let M be the smooth covering of M~- defined over Spec(Z)~r~ whose existence was proved in [10]. M i s a Galois covering for some group G by [10], first 4 lines of p. 12 in Section 3, after 3.1.2. Set M(p) := M(Spec(K)) and let M(0) be the corresponding characteristic 0 scheme. By smooth base change for 6tale cohomology [8, VIA. 1] the specialization maps from the groups H~,(M, Q~) to the corresponding groups of M(p) and M(0) are isomorphisms. Furthermore, the smooth specialization maps commute with the action of G and hence induce isomorphisms between the corresponding G-invariant parts. Since H~,(M(O),Q~)~= H~,(M~-(O),Qz) while H~t(M(p), Q~)C,contains H~t(M~ (p), Ql) we obtain Theorem 0.1. Theorem 0.2 follows from Theorem 0.1 and the computations of HZe,(Mr(O), Ql) made by Harer and of H4dM~ (0), Ql) made in [1] (using another result of Harer). (b) From now on we fix a prime p > 84(g - 1). Let M be the restriction of M o (2") to the localization of Z at all primes > 84(g - 1). Since p > 2g + 1 it is known (see [5, Ch. IV, Ex. 2.5] or [11]) that the classical upper bound 84(,q - 1) for the order of the automorphism group of a genus g smooth curve holds true even in characteristic p. In particular no prime > 84(9 - 1) divides ord(Aut(C)) for any smooth genus g curve C. (c) Here we will check that the upper bound 84(g - 1) for the primes dividing ord(Aut(C)) holds for every stable curve C of genus ,q. Fix a genus g stable curve C. Let Y be the normalization of C and mark the points of Y which go to the singular points of C. Call T this marked curve. Every a u t o m o r p h i s m of C induces a permutation of the set of marked points of Y. Fix one such automorphism, s. Since Y has at most 2g - 2 components (the upper bound being possible only if all connected components of Y are p1), and we are interested only in primes p > 2g - 2, we may assume that s sends each irreducible component of C into itself. It is easy to check that card(Sing(C)) _< g + (2g - 2). Hence T has at most 6g - 4 marked points. Hence for primes > 6g - 4 we reduce to the case in which s fixes every marked point of T. Hence s is contained in the product of the automorphism groups (as marked smooth curves) of the connected components of T. Fix one such component, Z, and call H its automorphism group. If p~(Z) >__2, we have ord(H) _< 84(p~(Z) - 1) _< 84(g - 1), as wanted. Assume pa(Z) = I. Then Z has some marked point and taking any of them (which by assumption are fixed by s) as origin to consider Z as an Abelian variety, we have ord(s) _< 3 (e.g. by Lefschetz trace formula [8, V. 2.5]). If Z~-P ~, then Z has at least 3 marked points because C is a stable curve. Hence s lZ is the identity (e.g. by Lefschetz trace formula [8, V.2.5]). (d) By formal deformation theory and algebraization theory it is known that M at a geometric point corresponding to a curve C e M r (p) is locally for the 6tale topology the quotient of a smooth scheme by Aut(C). Since p does not divide card(Aut(C)) for any C, all these groups are linearly reductive. Hence Mr(p) has only rational singularities, it is Q-orthocyclic (or equivalently Q-orienting) in the sense of [6, Section 2 and 3] or equivalently it is an Alexander scheme in the sense of [13]; we will say for
4
E. Ballico/Journal of Pure and Applied Algebra 124 (1998) 1-5
short that M~-(p) has quotient (prime to p) singularities. The same results of course are true for every prime r > 84(9 - 1). Recall that in [13] it was proved the existence of an intersection product in the Chow group tensor Q of M j (r) for every such prime r (or see [6], Section 3 and in particular from the last part of p. 337 on); furthermore, the same references give that the Chow group tensor Q is a commutative ring. Now consider the moduli scheme M~- first over Spec(Z), then over its generic point Spec(Q) and then M g (Q) over the algebraic closure of Q. By [3, Prop. 20,3] and [6, p. 338], the divisors )~ and As on M~(Q) specialize to the divisors with the same name on M j (r) and their numerical classes on the intersections of two (or more) of them are the same. (e) By [7, Corollary at p. 155], every pair (C, G) with C smooth genus g _> 2 curve and G ___Aut(C), card(G) prime to p, can be lifted to characteristic 0. This result implies that the same is true if C is a stable singular curve whose normalization has no elliptic or rational component. Since in our setting even Aut(D) with D elliptic curve has order prime to p, we lift to characteristic 0 also the stable curves with components with normalization of genus _< 1. (f) Recall that by [12] the isolated of codimension _> 3 quotient singularities by a finite linearly reductive (i.e. with order prime to the characteristic) are rigid. Motivated by this result and by step (c) we will analyze the integral subvarieties of codimension _< 2 formed by stable curves with nontrivial automorphism group. First note that it is classical (i.e. known in characteristic 0) that for g >_ 5 no such subvariety contains a smooth curve. By steps (c) and (e) this is true also in characteristic p > 8 4 ( g - I). The divisor A1 is the unique codimension 1 subvariety formed by curves with a nontrivial automorphism. Since M~(r) is normal, it is generically smooth along A1 and indeed when the elliptic tail has no extra automorphism. In codimension 2 there is the closure of the subset o r a l formed by the curves C = A w E with A smooth of genus 9, E elliptic and such that (taking as 0 the point Ac~E) the dimension 1 Abelian variety has more automorphisms that + Id. For primes > 84( 9 - 1 ) such curves have the same j-invariant (j = 0 and 1728) as in the characteristic 0 case. Then (again contained in A1) there is the closure of the set of curves A u R with R rational and with a node, the extra automorphism exchanging the branches of the nodes of R but keeping fixed the point Ac~R. The last codimension 2 subvariety contained in Sing(M] (r)) is the closure of the set of curves B ~ F with B smooth of genus g - 2 and F smooth of genus 2 and such that Bc~F is a Weierstrass point of F. (g) Note that every Weil divisor on M ~ ( p ) induces a G-invariant divisor on its Galois covering M(p). Hence Theorem 0.3 follows from the part i = 1 of Theorem 0.2, step (d) and the following result.
Proposition
1.1. Let X be an Alexander scheme with only quotient (prime to p) singularities. Then a divisor on X is numerically equivalent to 0 if and only if its class in Hat(X, Q1 ) vanishes.
E. Ballico /Journal of Pure and Applied Algebra 124 (1998) 1-5
5
Proof. In the case of s m o o t h surfaces, see for i n s t a n c e [8, Ch. V, R e m a r k 3.9 (a)]. F o r the g e n e r a l case fix a d i v i s o r D o f X w h o s e class in HZt(x, 0_+)v a n i s h e s a n d a n i n t e g r a l c u r v e Y c X. S i n c e X has q u o t i e n t ( p r i m e to p) s i n g u l a r i t i e s we r e d u c e to t h e case in w h i c h D is a C a r t i e r divisor. T o c h e c k D" Y t a k e a surface S w i t h o n l y i s o l a t e d singularities containing of S.
Y, a n d
reduce
to the a s s e r t i o n
on
a desingularization
[]
Acknowledgements T h e a u t h o r w a n t s to t h a n k h e r e the referee of a p r e v i o u s (essentially w r o n g ) p a p e r w i t h t h e s a m e title a n d c l a i m i n g s i m i l a r results. T h e a t t t h o r was p a r t i a l l y s u p p o r t e d by MURST
and GNSAGA
of C N R (Italy).
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