lanxicy.com

第一范文网 文档专家

第一范文网 文档专家

arXiv:math/9908085v2 [math.AG] 23 Aug 1999

THE PICARD GROUP OF THE MODULI OF HIGHER SPIN CURVES

TYLER J. JARVIS

1/r

Abstract. This article treats the Picard gr

oup of the moduli (stack) Sg

1/r Sg .

Generalized spin curves, or r of r -spin curves and its compacti?cation spin curves are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because they are the subject of a remarkable conjecture of E. Witten, and because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. We generalize results of Cornalba, describing and giving relations between 1/r 1/r many of the elements of the Picard group of the stacks Sg and Sg . These relations are important in the proof of the genus-zero case of Witten’s conjecture given in [13]. We use these relations to show that when 2 or 3 divides 1/r r , then Pic Sg has non-zero torsion. And ?nally, we work out some speci?c examples.

1. Introduction In this article we study the Picard group of the moduli (or rather the stack) Sg of higher spin curves, or r-spin curves, over the Deligne-Mumford stack of stable curves Mg . Smooth r-spin curves consist of a smooth algebraic curve X , a line bundle (invertible coherent sheaf) L, and an isomorphism from the rth tensor power L?r to the cotangent bundle ωX . The compacti?cation of the stack of smooth spin curves uses stable spin curves. These consist of a stable curve X , and for any divisor d of r, a rank-one, torsion-free sheaf Ed which is almost a dth root of the canonical (relative dualizing) sheaf. This is made precise in De?nition 2.2.5. The stack of spin curves provides a ?nite cover of the stack of stable curves. Although this stack has many similarities to the stack of stable maps, including the existence of classes analogous to Gromov-Witten classes, and an associated cohomological ?eld theory [13], it is not the stack of stable maps into any variety [13, §5.1]. These moduli spaces are especially interesting because of a conjecture of E. Witten relating the intersection theory on the moduli space of r-spin curves and KdV (Gelfand-Dikii) hierarchies of order r [20, 21]. This conjecture is a generalization of an earlier conjecture of his, which was proved by Kontsevich (see [14, 15] and [16]). As in the case of Gromov-Witten theory, one can construct a cohomology class c1/r

Date : February 8, 2008. 1991 Mathematics Subject Classi?cation. Primary 14H10,32G15, 81T40, Secondary 14N, 14M. Key words and phrases. Algebraic curves, moduli, higher spin curves. This material is based on work supported in part by the National Science Foundation under Grant No. DMS-9501617 and by the National Security Agency under Grant No. MDA904-99-10039.

1 1/r

2

TYLER J. JARVIS

and a potential function from the intersection numbers of c1/r and the tautological ψ classes associated to the universal curve. Witten conjectures that the potential of the theory corresponds to the tau-function of the order-r Gelfand-Dikii (KdVr ) hierarchy. In genus zero, the Witten conjecture is true [13], and the relations of Theorem 3.3.4 play a role in the proof. 1/r 1/r Construction of the stack Sg , and its compacti?cation Sg , was done in [3] for r = 2 and in [11] for all r ≥ 2. This article focuses on giving a description of the Picard group of Sg

1/r

and Sg .

1/r

1.1. Overview and Outline of the Paper. In Section 2.1 and 2.2 we give definitions of smooth and stable spin curves and some examples. The examples of Section 2.2.3 are typical of the spin curves that arise over the boundary divisors of Pic Mg . In Section 2.3 we recall the basic properties of the moduli spaces from [11]. In Section 3 we treat the Picard group of these spaces. The Picard group that we will work with in this paper is the Picard group of the 1/r stack Pic Sg (also called Pic f un ), as de?ned in [17] or [10]. The exact de?nition of the Picard group is given in Section 3.1, and the de?nitions of the boundary 1/r divisors and the tautological elements of Pic Sg are given in Section 3.2. We show in Section 3.2.4 that the boundary divisors and the Hodge class are independent in 1/r Pic Sg . This turns out to be a useful step toward showing that torsion exists in Pic Sg . In Section 3.3 we compute the main relations between elements of the Picard group, and we show some of the consequences of those relations. The general results are given in Theorems 3.3.4 and 3.3.4.bis, and some special cases of those results are given in Corollary 3.3.6. In the special case that r = 2, these results reduce to those of [3] and [4]. 1/r In section 3.4 we discuss the existence of torsion in Pic Sg when 2 or 3 divides 1/r r. In particular, Proposition 3.4.1 shows that when 2 divides r, Pic Sg has 41/r torsion, and when 3 divides r, Pic Sg has 3-torsion. This is in stark contrast to the case of Pic Mg , and Pic Mg , which are known to be free for g ≥ 3. [2]. Finally, in Section 4 we work out examples for genus 1 and general r, and for r = 2 and general genus. 1.2. Previous Work. The Picard group of the moduli space of curves is now fairly well understood, thanks primarily to the work of Arbarello-Cornalba [2] and Harer [8]. Most of the progress toward understanding the Picard group of the moduli of 2-spin curves is also due to Cornalba [3, 4] and Harer [9]. Section 3.3 on relations between classes in the Picard group is strongly motivated by Cornalba’s work in [3] and [4]. In [11] and [12] several di?erent compacti?cations of the moduli of r-spin curves are constructed. The best-behaved of these compacti?cations, and the one we will use here, is the stack of what are called stable r-spin curves or just r-spin curves in [11]. In the case of prime r, these are the same as the pure spin curves of [12]. 1.3. Conventions and Notation. By a curve we mean a reduced, complete, connected, one-dimensional scheme over an algebraically closed ?eld k . A semi-stable curve of genus g is a curve with only ordinary double points such that H 1 (X, OX ) has dimension g . And an n-pointed stable curve is a semi-stable curve X together

1/r

PICARD GROUP OF SPIN CURVE MODULI

3

with an ordered n-tuple of non-singular points (p1 , . . . , pn ), such that at least three marked points or double points of X lie on every smooth irreducible component of genus 0, and at least one marked point or double point of X lies on every smooth component of genus one. A family of stable (or semi-stable) curves is a ?at, proper morphism X → T whose geometric ?bres Xt are (semi) stable curves. Irreducible components of a semi-stable curve which have genus 0 (i.e., are birational to P1 ) but which meet the curve in only two points will be called exceptional curves. By line bundle we mean an invertible (locally free of rank one) coherent sheaf. By canonical sheaf we will mean the relative dualizing sheaf of a family of curves f : X → T , and this sheaf will be denoted ωX /T or ωf . Note that for a semistable curve, the canonical sheaf is a line bundle. When T is the spectrum of an algebraically closed ?eld, we will often write X for X , and ωX for ωX/T .

2. Spin Curves 2.1. Smooth Spin Curves. For any g ≥ 0, ?x a positive integer r with the property that r divides 2g ? 2. We de?ne a smooth r-spin curve to be a triple (X, L, b) of a smooth curve X of genus g , a line bundle L, and an isomorphism b, of ? the rth tensor power of L to the canonical bundle ωX of X ; that is, b : L?r - ωX . For a given X , any (L, b) making (X, L, b) into a spin curve will be called an r-spin structure. Families of smooth r-spin curves are triples (X /T, L, b) of a family X /T of smooth curves, a line bundle L, and an isomorphism b, which induces an r-spin structure on each geometric ?bre of X /T . Example: r = 2. A 2-spin curve is what has classically been called a spin curve (a curve with a ? theta-characteristic L) with an explicit isomorphism b : L?2 - ω . For any choice of m = (m1 , m2 , m3 , . . . , mn ), such that r divides 2g ?2? mi , we may also de?ne an n-pointed r-spin curve of type m as a triple ((X, (p1 , p2 , . . . , pn )), L, b) such that (X, (p1 , p2 , . . . , pn )) is a smooth, n-pointed curve, and b is an isomorphism from L?r to ωX (? mi pi ). Families of n-pointed r-spin curves are de?ned analogously. Two spin structures (L, b) and (L′ , b′ ) are isomorphic if there exists an isomorphism between the bundles L and L′ which respects the homomorphism b and b′ . Over a ?xed curve X , any two r-spin structures (L, b) and (L, b′ ) which di?er only by their isomorphism b or b′ must be isomorphic, provided the base is algebraically 1/r closed. The set Sg [X ] of isomorphism classes of r-spin structures on X is a principal homogeneous space over the r-torsion Jac r X of the Jacobian of X ; thus it has r2g elements in it. Similarly, two spin curves (X, L, b) and (X ′ , L′ , b′ ) are isomorphic if there is an ? ? isomorphism τ : X - X ′ and an isomorphism of spin structures i : L - τ ? L. Example 2: g = 1, r ≥ 2, m = 0. If n ≥ 1 and m = 0, then, up to isomorphism, an n-pointed r-spin curve of genus 1 is just an n-pointed curve of genus 1 with a point of order r on the curve. However, the automorphisms of the underlying curve identify some of these r-spin structures. In particular, when n = 1 and r is odd, the elliptic involution acts freely on all the non-trivial r-spin structures, thus there are only 1 + (r2 ? 1)/2 isomorphism classes of r-spin curves over a generic 1-pointed curve of genus 1.

4 1/r

TYLER J. JARVIS 1/r,m

Let Sg denote the stack of smooth r-spin curves of genus g . And let Sg,n denote the stack of n-pointed spin curves (for a given m). When g = 1, and n = 1, 1/r 1/r,0 we will also write S1 to denote the stack S1,1 . If X has no automorphisms, the sets Sg [X ] and Sg,n [X ], as de?ned above, are just the ?bres of Sg , or 1/r,m Sg,n over the point corresponding to X in Mg , or in Mg,n , respectively. 2.2. Stable Spin Curves. To compactify the moduli of spin curves, it is necessary to de?ne a spin structure for stable curves. To do this we need not just line bundles, but also rank-one torsion-free sheaves over stable curves. Some additional structure, as given in De?nitions 2.2.3 and 2.2.4, is also necessary to ensure that 1/r,m the compacti?ed moduli space Sg,n is separated and smooth. 2.2.1. De?nitions. To begin we need the de?nition of torsion-free sheaves. De?nition 2.2.1. A relatively torsion-free sheaf (or just torsion-free sheaf) on a family of stable or semi-stable curves f : X → T is a coherent OX -module E that is ?at over T , such that on each ?bre Xt = X ×T Spec k (t) the induced Et has no associated primes of height one. We will only be concerned with rank-one torsion-free sheaves. Such sheaves are called admissible by Alexeev [1] and sheaves of pure dimension 1 by Simpson [19]. Of course, on the open set where f is smooth, a torsion-free sheaf is locally free. Note 2.2.2. It is well-known and easy to check that if a rank-one, torsion-free sheaf ?X,p E is not locally free (also called singular) at a node p of X , then the completion O of the local ring of X near p is isomorphic to A = k [[x, y ]]/xy , and E corresponds to an A-module E ? = xk [[x]] ⊕ yk [[y ]] =< ζ1 , ζ2 |yζ1 = xζ2 = 0 > [18, Prop. 11.3]. De?nition 2.2.3. Given an n-pointed, semi-stable curve (X, p1 , . . . , pn ), and a rank-one, torsion-free sheaf K on X , and given an n-tuple m = (m1 , . . . , mn ) of integers, we denote by K(m) the sheaf K ? O(? mi pi ). A dth root of K of type m is a pair (E , b) of a rank-one, torsion-free sheaf E , and an OX -module homomorphism b : E ?d - K(m) with the following properties: 1. d · deg E = deg K ? mi 2. b is an isomorphism on the locus of X where E is locally free 3. for every point p ∈ X where E is not free, the length of the cokernel of b at p is d ? 1. Unfortunately, the moduli space of stable curves with dth roots of a ?xed sheaf K is not smooth when d is not prime, and so we must consider not just roots of a bundle, but rather a coherent net of roots. This additional structure su?ces to make the stack of stable curves with coherent root nets smooth [11]. De?nition 2.2.4. Given a semi-stable n-pointed curve (X, p1 , . . . , pn ), and a rankone, torsion-free sheaf K on X , and an n-tuple m, a coherent net of roots of type m for K is a collection {Ed , cd,d′ } consisting of a rank-one torsion-free sheaf Ed for ?d/d′ - Ed′ for each d′ dividing each d dividing r, and a homomorphism cd,d′ : Ed d with the following properties: ? E1 = K, and cd,d = 1 for each d dividing r.

1/r 1/r,m 1/r

PICARD GROUP OF SPIN CURVE MODULI

5

? For every pair of divisors d′ and d of r such that d′ divides d, let m′ be the n′ ′ tuple (m′ 1 , . . . , mn ) such that mi is the smallest, non-negative integer congru?d/d′ - Ed ′ , ent to mi mod(d/d′ ). The OX -module homomorphism cd,d′ : Ed ′ ′ must make Ed into a d/d -root of Ed′ , of type m , such that all these maps are compatible. That is, the diagram (Ed

?d/d′ ?d′ /d′′

)

(cd,d′ )?d /d -

′

′′

Ed ′

?d′ /d′′

cd

,d ′′

cd′ ,d′′

commutes for every d′′ |d′ |d|r.

? Ed′′

If r is prime, then a coherent net of rth-roots is simply an rth root of K. Moreover, if Ed is locally free, then up to isomorphism Ed uniquely determines all Ed′ and all cd,d′ such that d′ |d. De?nition 2.2.5. A stable, n-pointed, r-spin curve of type m = (m1 , . . . , mn ) is an n-pointed, stable curve (X, p1 , . . . , pn ) and a coherent net of rth roots of ωX of type m, where ωX is the canonical (dualizing) sheaf of X . An r-spin curve is called smooth if X is smooth. Note that this de?nition of a smooth r-spin curve di?ers from that of Section 2.1, in that a spin curve carries the additional data of explicit isomorphisms ?d/d′ - Ed′ ; however, for smooth curves, and indeed, whenever Er is locally Ed free, Er and cr,1 completely determine the spin structure, up to isomorphism. De?nition 2.2.6. An isomorphism of r-spin curves from (X, p1 , . . . , pn , {Ed , cd,d′ }) ′ ′ ′ to (X ′ , p′ 1 , . . . , pn , {Ed , cd,d′ }) is an isomorphism of pointed curves τ : (X, p1 , . . . , pn )

′ - (X ′ , p′ 1 , . . . , pn )

′ - Ed }, with β1 the canonical isomorand a system of isomorphisms {βd : τ ? Ed - ωX (? mi pi ), and such that the βd are compatible phism τ ? ωX ′ (? i mi p′ i) with all of the maps cd,d′ and τ ? c′ d,d′ .

The de?nition of families of r-spin curves is relatively technical and unenlightening. For the details of those de?nitions see [11]. For our purposes, it will su?ce to know the basic properties of the stack of r-spin curves from [11] as given in Section 2.3 2.2.2. Alternate Description of Stable Spin Curves. The following characterization of spin curves in terms of line bundles on a partial normalization of the underlying curve is very useful and helps illustrate the nature of stable spin curves. - X ? π Consider a stable spin curve (X, {Ed, cd,d′ }) and the partial normalization X of X at each of the singularities of Er (i.e., the nodes of X where Er fails to be lo?X,p of the local ring of X near a singularity p of cally free). The completion O Er is isomorphic to A = k [[x, y ]]/xy , and Er corresponds to an A-module E ? = ?r - ω xk [[x]] ⊕ yk [[y ]] =< ζ1 , ζ2 |xζ2 = yζ1 = 0 >. The homomorphism cr,1 : Er r - A of the form ζ1 corresponds to a homomorphism of A-modules E ?r → xu ,

6

TYLER J. JARVIS

r i r ?i ζ2 → y v , and ζ1 ζ2 → 0 for 0 < i < r. The condition on the cokernel (De?nition 2.2.3(3)) implies that u + v = r. The pair {u, v } is called the order of cr,1 at p. If Er is locally free at p, then cr,1 is an isomorphism and the order is {0, 0}. ? = k [[x]] ⊕ k [[y ]] ? ?? + ⊕O ? ? ? , where {p+ , p? } is the inverse image Let A =O X,p X,p ?1 ?, π (p) of the normalized node. The pullback π ? Er of Er corresponds to E ?A A which is no longer torsion-free; but if π ? Er := π ? Er /torsion, which corresponds to a ? ?r - π ? ωX (?up+ ? ?-module E ? , then cr,1 induces an isomorphism c free A ?r,1 : π ? Er ? + ? vp ) = ωX ? ((1 ? u)p + (1 ? v )p ). ? → X with π ?1 (qi ) = {q + , q ? }, Conversely, given a partial normalization π : X i i the inverse images of the singular points qi , and given integers ui ∈ (0, r) and vi ∈ (0, r) for each normalized singularity qi such that ui + vi = r, consider an rth + ? root L of the line bundle π ? ωX ? OX (ui qi + vi qi )) ? ((ui ? = ωX ? (? ? ? OX ? (? + ? 1)qi + (vi ? 1)qi )). Of course for such an L to exist, ui and vi must be chosen to make the degree of the rth power of L divisible by r. From this rth root and partial normalization we can create an rth root of ωX on X by taking Er = π? L, and by taking cr,1 to be the map induced by adjointness from the composite ? - π ? ωX ? O ? (? ui q + + vi q ? ) ?→ π ? ωX . Thus rth roots of ωX are in L?r i i X + ? ? one-to-one correspondence with rth roots of ωX ? (?(u ? 1)qi ? (v ? 1)qi ) on X . ?r - ωX at p, then for each d Moreover, if {u, v } is the order of cr,1 : Er dividing r, the order of (Ed , cd,1 ) is {ud , vd } where ud and vd are the least nonnegative integers congruent to u and v respectively, modulo d. So Ed is locally free at p if and only if d divides u (and hence v ). If u and v are relatively prime, then no Ed is locally free, and thus all Ed are completely determined (up to isomorphism) by (Er , cr,1 ) (or π ? Er ) by ?r/d + ? π ? Ed := π ? Er ? OX ? (1/d(u ? ud )p + 1/d(v ? vd )p )

and Ed = π? π ? Ed . However, if u and v are not relatively prime, but rather have gcd(u, v ) = ? > 1, then the root E? is locally free at p, and hence requires the additional gluing datum of an ?th root of unity (non-canonically determined) to construct E? from π ? E? . The remainder of the spin structure can clearly be recon?r/? - E? and c?,1 : E ?? - ω. structed from the two pieces cr,? : Er ? When the spin structure has no singularity (i.e., Er is locally free) at a node of the underlying curve, this corresponds to E. Witten’s de?nition of the Ramond sector of topological gravity [21, 20]; whereas, when u and v are non-zero, the spin structure is what Witten calls a generalized Neveu-Schwarz sector. If gcd(u, v ) = ? > 1 then we sometimes say that the spin structure is semi-Ramond, corresponding to the fact that E? is locally free (Ramond) but Er is not. 2.2.3. Examples. It is useful to consider a few examples of stable spin curves. Both of the examples in this section are relevant to the study of the Picard group of the stack and will be important in Section 3.2. Example 1: Two irreducible components and one node. First consider a stable curve X with two smooth, irreducible components C and D, of genus k and g ?k respectively, meeting in one double point p. In this case there exists a unique choice of u and v that makes the degree of π ? ωX (?up+ ? vp? ) ? = + ? ωX ? (?(u ? 1)p ? (v ? 1)p ) divisible by r on both components. The resulting rth roots are locally free (Ramond) if and only if ωX has an rth root, which is to say, if and only if u and v can be chosen to be 0, or 2k ? 1 ≡ 0 (mod r). If 2k ? 1 ≡ 0

PICARD GROUP OF SPIN CURVE MODULI

7

(mod r) then the resulting (Neveu-Schwarz) rth root corresponds to an rth root of ωC (?(u ? 1)p+) on C and an rth root of ωD (?(v ? 1)p? ) on D. If gcd(2k ? 1, r) = 1 then all of the dth roots are Neveu-Schwarz. Even if gcd(2k ? 1, r) = ? is bigger than 1, since the dual graph of X is a tree, all gluing data for constructing E? from π ? E? will yield (non-canonically again) isomorphic E? ’s and hence isomorphic nets 1/r,u?1 1/r,v ?1 of roots, and the spin curve corresponds to an element in Sk,1 × Sg?k,1 . Thus spin curves obey something like the splitting axiom of quantum cohomology (see [13, §4.1] for more details). Example 2: One irreducible component and one node. The second example is given by an irreducible stable curve X with one node. In this case there are r di?erent choices of u and v that permit spin structures: either u = v = 0, in which case the resulting spin structure is locally free (Ramond); or u ∈ {1, . . . , r ? 1} and v = r ? u, in which case the rth root structure is not locally free (Neveu-Schwarz). In this second case, if gcd(u, r) = 1, then the r+ spin structure on X is induced by an rth root of the bundle ωX ? (?(u ? 1)p ? 1/r,(u?1,v ?1) ? (v ? 1)p ), and thus corresponds to an element of Sg?1,2 . Since the points + ? p and p of the normalization are not ordered, this gives a degree-2 morphism 1/r,(n?1,v ?1) /r ? - S1 Sg,r g . But if ? = gcd(u, r) > 1, then for a given π E? there are ? distinct choices of gluing data, and ? choices of E? . This gives ?/2 distinct morphisms Sg,2

1/r,(u?1),(v ?1)

into Sg .

1/r

2.3. Properties of the Stack of Spin Curves. 2.3.1. Basic Properties. The main facts we need to know from [11] about the stack of stable spin curves are contained in the following theorem. of n-pointed rTheorem 2.3.1. [11, Thms 2.4.4, and 3.3.1]) The stack Sg,n spin curves of genus g and type m is a smooth, proper, Deligne-Mumford stack 1/r,m - Mg,n is ?nite and over Z[1/r], and the natural forgetful morphism Sg,n surjective. Moreover, if ?g,r (m) is de?ned as ? 1 if g = 0 and r|2 + mi ? ? ? gcd(r, m1 , . . . , mn ) if g = 1 and r| mi ?g,r (m) := gcd(2, r, m1 , . . . , mn ) if g > 1 and r| mi + 2 ? 2g ? ? ? 0 otherwise and if dg,r (m) is de?ned to be the number of (positive) divisors of ?g,r (m) (includ1/r,m 1/r,m

ing 1 and ?g,r (m) itself ), then Sg,n is the disjoint union of dg,r (m) irreducible components. 1/r,m 1/r,m Also, Sg,n contains the stack of smooth spin curves Sg,n as an open dense substack, and the coarse moduli spaces of Sg,n jective (respectively, quasi-projective).

1/r,m

and Sg,n

1/r,m

are normal and pro-

Although the actual details of the de?nition of a family of r-spin curves are not necessary for this paper, we do need the description from [11] of the universal deformation of a stable spin curve. Theorem 2.3.2. ([11, Thm. 2.4.2]): Given a stable spin curve (X, {Ed , cd,d′ }) with singularities qi of order {ui , vi }, the universal deformation space of (X, {Ed , cd,d′ })

8

TYLER J. JARVIS

is the cover Spec o[[τ1 , . . . , τm , tm+1 , . . . , t3g?3+n ]] - Spec o[[t1 , . . . , t3g?3+n ]], where ti = τiri , and ri = r/ gcd(ui , vi ), the scheme Spec o[[t1 , . . . , t3g?3+n ]] is the universal deformation space for the underlying curve X , and the nodes q1 , . . . , qm correspond to the loci of vanishing of t1 , . . . , tm , respectively. 2.3.2. Relations Between the Di?erent Stacks. There are several natural morphisms between the stacks. 1. There is a canonical isomorphism from Sg,n to Sg,n where m′ is an n-tuple whose entries are all congruent to m mod r; namely for any net ′ ′ ′ , cd,d′ } be the net given by Ed {Ed , cd,d′ } of type m, let {Ed = Ed ?O(1/d (mi ? ′ ′ mi )pi ) where pi is the ith marked point, and cd,d′ is the obvious homomorphism. Because of this canonical isomorphism, we will often assume that all the mi lie between 0 and r ? 1 (inclusive). 1/r,m′ /r,m π - S1 2. There’s a morphism Sg,n+1 , where m′ is the (n + 1)-tuple g,n (m1 , . . . , mn , 0); and π is the morphism which simply forgets the (n + 1)st marked point. If mn+1 is not congruent to zero mod r, the degree of ω (m′ ) n+1 n is 2g ? 2 ? i=1 mi and the degree of ω (m) is 2g ? 2 ? 1 mi . Since both cannot be simultaneously divisible by r at least one stack is empty, and there is no morphism. This morphism, sometimes called “forgetting tails,” is not 1/r,m the universal curve over Sg,n , although it is birational to the universal curve. In particular, the two are isomorphic over the open stack Sg,n . 1/r,m - S1/s,m , which 3. If s divides r, then there is a natural map [r/s] : Sg,n g,n forgets all of the roots and homomorphisms in the rth-root net except those associated to divisors of s. 3. Picard group 3.1. De?nitions. Throughout this section we will assume that g ≥ 2 and n = 0, or that g = 1 = n. By the term Picard group we mean the Picard group of the moduli functor; that is to say, the Picard group is the group of line bundles on the stack. By a line 1/r bundle L on the stack Sg , we mean a functor that takes any family of spin curves Q = (X /S, {Ed , cd,d′ }) in Sg and assigns to it a line bundle L(Q) on the scheme S , and which takes any morphism of spin curves f : Q/S → P/T and assigns to it an ? isomorphism of line bundles L(f ) : L(Q) - f ? L(P), with the condition that the isomorphism must satisfy the cocycle condition (i.e., the isomorphism induced by a composition of maps agrees with the composition of the induced isomorphisms). 1/r The groups Pic Sg , Pic Mg , and Pic Mg are de?ned similarly. For more details on Picard groups of moduli problems see [10, pg. 50] or [17, §5]. 3.2. Basic Divisors and Relations. 3.2.1. The Tautological Bundles. Recall that for any family of stable curves π : X → S , the Hodge class λ(X /S ) in Pic Mg is the determinant of the Hodge bundle (the push-forward of the canonical bundle) λ(X /S ) := det π! ωX /S = ∧g π? ωX /S .

1/r 1/r,m 1/r,m 1/r,m′

PICARD GROUP OF SPIN CURVE MODULI

9

It is well-known that Pic Mg is the free Abelian group generated by λ when g > 1 (c.f. [2]) and for g = 1 and g = 2 Pic Mg is also generated by λ, but is cyclic of order 12, and 10, respectively [6, §5.4]. In a similar way we de?ne a bundle ? = ?1/r in Pic Sg as the determinant of the rth root bundle. In particular, if Q = (π : X → S, {Ed , cd,d′ }) is a stable spin curve, then ?(Q) := det π! Er = (det R0 π? Er ) ? (det R1 π? Er )?1 on S . Similarly, de?ne ?1/d := det π! Ed for each d dividing r. Note that the pullback of ? from Sg,s to Sg

1/r 1/r

via [r/s]? is exactly ?1/s .

1/r 1/r

3.2.2. Boundary Divisors Induced from Mg . In addition to λ and ?, there are elements of Pic Sg that arise from the boundary divisors of Sg . Recall that the boundary of Mg consists of the divisors δi where i ∈ {0, . . . , ?g/2?}. Here, when i is greater than zero, δi is the closure of the locus of points in Mg corresponding to stable curves with exactly one node and two irreducible components, one of genus i and the other of genus g ? i. And when i is zero, δ0 is the closure of the locus of points corresponding to irreducible curves with a single node. As we saw in Example 1 of Section 2.2.3, for any curve X with exactly one node and two irreducible components of genera i and g ? i, there is a unique choice of integer u(i) between 0 and r ? 1 that determines a bundle W whose rth roots de?ne the spin structures on X . If 2i ≡ 1 (mod r), then u(i) = v (i) = 0 and the bundle W is just the canonical bundle W = ωX . In this case all the spin structures on X are locally free. If on the other hand, 2i ≡ 1 (mod r), then there is a unique choice of u(i) with r > u(i) > 0, and such that 2i ? 1 ? u(i) ≡ 0 (mod r). In this case v (i) = r ? u(i), and W is not a line bundle on X , but rather a line bundle on the ? → X at the node q . If ν ?1 (q ) = {q + , q ? }, then normalization ν : X

+ ? W = ωX ? ((1 ? u(i))q + (1 ? v (i))q ).

In this case the spin structures on X are constructed by pushing the rth roots of ? down to X , i.e., using the rth roots of W on the irreducible components W on X to make a torsion-free sheaf on the curve X . By Theorem 2.3.1, if i and g ? i are at least 2, and r is odd, there is one irreducible divisor αi lying above δi . And when r is even, there are four divisors lying over δi (even-even, even-odd, odd-even, odd-odd). In general, let Dk,r (m) denote the set of positive divisors of ?k,r (m), as in Theorem 2.3.1. These divisors index the set of irreducible components of Sk,n

(a,b) αi 1/r,m

. For

denote each i ≥ 1, and each a ∈ Di,r (u(i) ? 1) and b ∈ Dg?i,r (v (i) ? 1), let the irreducible divisor consisting of the locus of spin curves lying over δi with a spin structure of index a on the genus-i component and of index b on the genus-(g ? i) component. Over a stable curve in δ0 there are also several choices of spin structure. Indeed, as we saw in Example 2 of Section 2.2.3, for any choice of order {u, v }, there is + ? again a unique bundle Wu = ωX ? ((1 ? u)q + (1 ? v )q ) so that any rth root of ? ? , and thus the entire spin structure except for Wu gives the pullback π Er to X glue. And conversely, any spin structure comes from an rth root of some Wu for 0 ≤ u < r and a choice of glue (where the glue corresponds to an isomorphism ? - E? |q? and ? = gcd(u, v, r)). Again for a particular choice of order η : E? |q+ {u, v } of index a ∈ Dg?1,r (u ? 1, v ? 1), and a particular choice of glue η , the

10

TYLER J. JARVIS

corresponding divisor of spin curves of the given order, index and glue is irreducible. (a) We denote these divisors by γj,η , where j is the smaller of u and v , η is the gluing datum, and a is the index. Of course, since the two points of the normalization are (a) (a) not distinguishable, we have γj,η = γr?j,η?1 . We denote the set of gluings for a given choice of j by gj , and we denote the set of gluings for a ?xed j , modulo the equivalence (r/2, η ) ? (r/2, η ?1 ), by gj /S2 . Let αi denote the sum αi :=

Di (u(i)?1)×Dg?i (v (i)?1)

αi

(a,b)

and let γj denote the sum γj :=

η ∈gj /S2 a∈Dg?1 (j ?1,r ?j ?1)

γj,η .

(a)

Pullback along the forgetful map p : Sg

1/r

1/r

→ Mg induces a map p? from Pic Mg

1/r

to Pic Sg . And it is clear that the image of the boundary divisors (which we will denote by δi , regardless of whether we are working in Pic Mg or Pic Sg ) is still supported on the boundary of Sg . Indeed, for i greater than one, δi is a linear combination of the αi

(a,b) 1/r

’s, and δ0 is a linear combination of the γj,η ’s.

(a)

Proposition 3.2.1. Let ci := gcd(2i ? 1, r) = gcd(u(i), v (i)). If i ≥ 1 then δi = (r/ci )αi And if dj := gcd(j, r), then we have δ0 =

0≤j ≤r/2

(r/dj )γj .

Proof. The universal deformation of a spin curve with underlying stable curve in δi is dependent only upon u and v ; in particular, if c := gcd(u, v ) and r′ := r/c then the forgetful map from the universal deformation of the spin curve to the universal ′ deformation of the underlying curve is of the form Spec o[[s]] → Spec o[[sr ]] (see Theorem 2.3.2). Moreover, for i ≥ 1 the morphism Sg

1/r Sg 1/r

→ Mg is rami?ed along

αi to order r′ = r/ci . Similarly, over δ0 , the map → Mg is rami?ed along γj ′ to order r = r/dj . In either case the proposition follows. 3.2.3. Boundary Divisors Induced from Other-order Spin Curves. If s divides r, 1/r /s - S1 induces a homomorphism say sd = r, then the natural map [d] : Sg g 1/s 1/r 1/s 1/s ? [d] : Pic Sg Pic Sg . Let α , and γ also indicate the images of the corresponding boundary divisors under the map [d]? .

i j

Proposition 3.2.2. Let u(i) indicate the unique integer 0 ≤ u(i) < r such that 1/s 2i ? 1 ? u(i) ≡ 0 (mod r). For each divisor s of r, let ci be gcd(u(i), v (i), s) = gcd(u(i), s) and let di divisors

1/s αi , 1/s

be gcd(j, (r ? j ), s) = gcd(j, s). In Pic Sg

1/r αi ,

1/r

the boundary as follows.

and

1/s γi

are given in terms of the elements

and

1/r γj

PICARD GROUP OF SPIN CURVE MODULI

11

αi

1/s

1/s

=

r ci 1/r αi 1/r s c

i

1/s

And γk behaves similarly, but now the index j behaves like u (mod s), and 1/r there are several choices of k between 0 and r/2 such that k ≡ ±j (mod s) (γk is 1/r 1/r 1/s the same divisor as γr?k ). This gives several divisors γk over each γj : γj

1/s

=

k≡±j (mod s) 0≤k≤r/2

r dk 1/r γk 1/r s d

k

1/s

Proof. The proof is straightforward and very similar to the proof of Proposition 3.2.1. 1/r 1/s Only note that γk lies over γj if k ≡ j (mod s) or if k ≡ ?j (mod s). 3.2.4. Independence of Some Elements of Pic Sg . The following two results are generalizations of Cornalba’s results [3, Proposition 7.2]. Their proofs are essentially the same as those in [3], and so will just be sketched here. Proposition 3.2.3. The forgetful map Sg phism on the Picard groups.

1/r 1/r

→ Mg induces an injective homomor-

Proof. The method of proof is simply to recall from [2] that there are families of stable curves X /S , with S a smooth and complete curve, such that the vectors (degS λ, degS δ0 , . . . , degS δ?g/2? ) are independent. After suitable base change, one can install a spin structure on the families in question. (This follows, for example, from the fact that, after base change, a spin structure can be installed on the generic ?bre of X /S , and since the stack is proper over Mg , this structure can be extended–again after possible base change–to the entire family). Since the e?ect of base change is to multiply the vectors’ entries by a constant, the vectors are still 1/r independent, and thus the elements λ and δi are all independent in Pic Sg . J. Koll? ar pointed out to me the following alternate proof: Note that the stacks in question are both smooth. Thus the result follows from the fact that for any ?nite cover f : X ′ → X of a normal variety X and for any line bundle L on X , the line bundle f? f ? L is a multiple of L (c.f., [7, 6.5.3.2]). In other words, for some n we have f? f ? L = L?n . But since the Picard group of Mg has no torsion, the pullback of any line bundle to Sg

1/r

cannot be trivial.

(a,b)

Proposition 3.2.4. The elements λ, αi when g > 1.

and γj,ρ are independent in Pic Sg

(a)

1/r

Proof. Again the idea is to install a spin structure on a family of curves X /S over a smooth, complete curve, where the degrees of λ, and the δi are known. In particular, given two curves S and T of genera i and g ? i, respectively, ?x t ∈ T and let s vary in S . Consider the family X /S constructed by joining the two curves at the points s and t [10, §7]. Then the degrees of λ and δj are all zero on S , except when

12

TYLER J. JARVIS

j = i, and then degS δi = 2 ? 2i. We can construct an rth root of ωS ((1 ? u(i))s) on S and an rth root of ωT ((1 ? v (i))t) on T , and thus an r-spin structure on X /S . Moreover, the two rth roots can be chosen to be of any index in Di,r (u(i) ? 1) or Dg?i,r (v (i) ? 1), respectively; and thus X /S can be endowed with a spin structure of any index (a, b) in Di,r (u(i) ? i)× Dg?i,r (v (i) ? 1) along every ?bre. Because the (a′ ,b′ ) (a,b) for any other index (a′ , b′ ) must be zero on are disjoint, the degree of αi αi (a,b) S , but degS δi = 0 implies that deg( (a,b) αi ) is non-zero. + cj,ρ γj,ρ , Consequently, in any relation of the form 0 = ?λ + ei αi the coe?cients ei must all vanish. And thus a relation must be of the form 0 = (a) (a) (a) ?λ + cj,ρ γj,ρ . But a similar method shows that the coe?cients cj,ρ must also be zero. In particular, consider the family Y /C constructed by taking a general curve C of genus g ? 1 and identifying one ?xed point p with another, variable point q [10, §7]. Again one may produce an r-spin structure on Y /C of any type. And (a) (b) (b) (a) degC λ = degC γk,ψ = 0 whenever γk,ψ = γj,ρ , but degC δ0 = j,ρ,a r/d(j ) deg γj,ρ is equal to 2 ? 2g , which is non-zero (since g > 1). Thus cj,ρ = 0, and so also ? = 0. 3.3. Less-Obvious Relations and Their Consequences. Another important question about these bundles is what relations exist between the bundles ?, λ, and the various boundary divisors. In this section we provide a partial answer. 3.3.1. Main Relation. The main result of this section is Theorem 3.3.4, which provides relations between ?1/r , ?1/s , λ, and some boundary divisors. The proof will be given in Section 3.3.2. To state the result we need the following de?nitions. Motivation for the somewhat-peculiar notation will be made clear in the following section. De?nition 3.3.1. Let u(i) denote, as in the previous section, the unique integer 1/s 0 ≤ u(i) < r such that 2i ? 1 ≡ u(i) (mod r). Let v (i) = r ? u(i), ci := 1/s gcd(u(i), v (i), s) = gcd(2i ? 1, s), and dj := gcd(j, r ? j, s) = gcd(j, s) and let /r 1/r 1/r ?r , Er > to be the element in Pic S1 ci = ci , dj = dj . Then we de?ne < E g de?ned by the following boundary divisor: ?r , Er >:= <E

1≤i≤g/2 (a) (a,b) (a,b) (a) (a)

(u(i)v (i)/ci )αi +

0≤j ≤r/2

(j (r ? j )/dj )γj

Similarly, for s dividing r we make the following de?nitions. De?nition 3.3.2. Let u′ (i) denote the unique integer 0 ≤ u′ (i) < s such that ?1/s , Es > denote the 2i ? 1 ≡ u′ (i) (mod s). Let v ′ (i) = s ? u′ (i). And let < E following boundary divisor in Pic Sg : ?1/s , Es > = <E

1≤i≤g/2 1/r

u′ (i)v ′ (i)

1/s ci ′ ′

αi

1/s

+

0≤j ≤s/2 1/r

j (s ? j )

1/s dj

γj

1/s

=

r s

(u (i)v (i))

1≤i≤g/2 1/r ci

αi

+

r s

(j (s ? j ))

1≤j ≤s/2 1≤k≤r/2 k≡±j (mod s) 1/r dk

γk

1/r

PICARD GROUP OF SPIN CURVE MODULI

13

De?nition 3.3.3. Let δ indicate the element of Pic Sg boundary divisors pulled back from Pic Mg :

g/2

1/r

de?ned by the sum of the

δ=

i=0

δi (r/dj )γj +

0≤j ≤r/2 1≤i≤g/2

=

(r/ci )αi s

1≤i≤g/2 1/s ci 1/s

=

0≤k≤s/2

s

1/s dk

γk

1/s

+

αi

With all of this notation in place we can now state the main theorem. Theorem 3.3.4. In terms of the notation de?ned above, the following relations 1/r hold in Pic Sg : ?r , Er > = (2r2 ? 12r + 12)λ ? 2r2 ? + (r ? 1)δ r<E ?s , Es > = (2s2 ? 12s + 12)λ ? 2s2 ?s + (s ? 1)δ s<E

?r , Er >, < E ?s , Es >, This may also be written in terms of α and γ instead of < E and δ . But the ?nal expression is greatly simpli?ed if we de?ne σk

1/s

:=

1≤i≤g/2 i≡k (mod s)

αi .

1/s

Let σk = σk , and if k is not an integer, then let σk = 0. 1/r 1/s Also, we will continue to use the notation cj = gcd(2i ? 1, s), ci = ci , dj

1/s

1/r

1/s

= gcd(j, s), and dj = dj .

1/r

Theorem 3.3.4.bis. In terms of the notation de?ned above, the following relation 1/r holds in Pic Sg : (2r2 ? 12r + 12)λ ? 2r2 ? = (1 ? r)(σ r+1 + γ0 )

2

+

1<k< r 2

2(r/ck )(rk ? 2k 2 + 2k ? r)σk 2(r/ck )(3rk ? 2k 2 + 2k ? 2r ? r2 )σk

r 2 +1<k<r

+ +

(r/dj )(j (r ? j ) ? (r ? 1))γj

1<j ≤r/2

14

TYLER J. JARVIS

Similarly, (2s2 ? 12s + 12)λ ? 2s2 ?1/s = (1 ? s)(σ s+1 + γ0 )

2

1/s

+

s 1<k< 2

2(s/ck )(sk ? 2k 2 + 2k ? s)σk

1/s

1/s

1/s

+

s 2 +1<k<s

2(s/ck )(3sk ? 2k 2 + 2k ? 2s ? s2 )σk (s/dj )(j (s ? j ) ? (s ? 1))γj

1<j ≤s/2 1/s 1/s

1/s

+

The proof of Theorem 3.3.4 will be postponed until the next section. The proof of Theorem 3.3.4.bis is a straightforward, but tedious calculation, accomplished simply by writing out the de?nitions of all of the di?erent terms, and applying Theorem 3.3.4. The main thing to note about the relations of Theorems 3.3.4 and 3.3.4.bis is 1/r 1/r that they hold in Pic Sg and not just in Pic Sg ? Q—that is to say, not just modulo torsion. We also have the following immediate corollaries: Corollary 3.3.5. If s divides r the following relations hold in Pic Sg : (2r2 ? 12r + 12)λ = 2r2 ?1/r (2s2 ? 12s + 12)λ = 2s2 ?1/s 2r2 (s2 ? 6s + 6)?1/r = 2s2 (r2 ? 6r + 6)?1/s Corollary 3.3.6. The following special cases of the relations in Theorems 3.3.4 and 3.3.4.bis hold: r=2 r=3 r=4 r=5 r=6 4λ + 8?1/2 6λ + 18?1/3 4λ + 32?1/4 4λ + 8?1/2 2λ ? 50?1/5 12λ ? 72?1/6 4λ + 8?1/2 6λ + 18?1/3 26λ ? 98?1/7 44λ ? 128?1/8 4λ ? 32?1/4 4λ ? 8?1/2 = = = = = = = = = = = = γ0 2γ0 + 2σ2 3γ0 ? 2γ2 γ0 + 2γ2 ?4γ0 + 10(γ2 + σ2 + σ4 ) ? 4σ3 ?5γ0 + 9γ2 + 8(γ3 + σ2 + σ5 ) γ0 + 3γ2 2γ0 + 4γ3 + 4(σ2 + σ5 ) ?6(γ0 + σ4 ) + 28(γ2 + σ3 + σ5 ) + 42(γ3 + σ2 + σ6 ) ?7γ0 + 20γ2 + 18γ4 + 64(γ3 + σ2 + σ3 + σ6 + σ7 ) 3γ0 + 6γ4 ? 4γ2 γ0 + 4γ2 + 2γ4

1/r

r=7 r=8

Note that in the case when r = 2, the relation 4λ + 8?1/2 = γ0 is exactly the content of Theorem (3.6) in [4]. 3.3.2. Proof of the Main Relation and Some Other Relations. Although some of the proof of Theorem 3.3.4 resembles that of Theorem (3.6) in [4], the proof of

PICARD GROUP OF SPIN CURVE MODULI

15

Theorem 3.3.4 requires many involved calculations for the case r > 2 that do not arise when r = 2. To accomplish the proof without annihilating torsion elements, we need some tools other than the usual Grothendieck-Riemann-Roch; in particular, we’ll use the following construction of Deligne [5]. De?nition 3.3.7. Given two line bundles L and M on a family of semi-stable curves f : X → S , de?ne a new line bundle, the Deligne product < L, M > on S , as < L, M >:= (det f! LM) ? (det f! L)?1 ? (det f! M)?1 ? det f! OX . Here, as earlier, det f! L indicates the line bundle (det R0 f? L) ? (det R1 f? L)?1 . We will write this additively as < L, M >= det f! L, M ? det f! L ? det f! M + det f! O. Deligne shows that this operation is symmetric and bilinear. It is also straightforward to check that when the base S is a smooth curve then the degree on S of the Deligne product < L, M > is just the intersection number of L with M; that is to say, if (?.?) is the intersection pairing on X , then degS < L, M >= (L.M). Using this notation, and writing ω for the canonical bundle of X /S , Serre duality takes the form < ω, M >= det f! (M?1 ) ? (det f! M), or det f! (Lω ) = det f! (L?1 ). Deligne proves the following variant of Grothendieck-Riemann-Roch that holds with integer, rather than just rational coe?cients. Lemma 3.3.8. (Deligne-Riemann-Roch) Let f : X → S be a family of stable curves, and let L be any line bundle on X /S . If ω denotes the canonical sheaf of X /S , then 2(det f! L) =< L, L > ? < L, ω > +2(det f! ω ). We also need some additional line bundles associated to a family of r-spin curves (X /S, {Ed , cd,d′ }) that intuitively amount to measuring the di?erence between Er and a “real” rth root of the canonical bundle. To construct these bundles we ?rst recall from [12] some canonical constructions related to any rth root (Er , cr,1 ) on a curve X /S ?Er → X , uniquely determined by X and Er , 1. There is a semi-stable curve π : X ? is X . so that the stable model of X ?, there is a unique line bundle O ? (1), and a canonically-determined, 2. On X X ?r injective map β : OX → ωX ? (1) ? /S , the push forward π? OX ? (1) is Er , and the map b is induced from β by adjointness. 3. Moreover, the degree of OX ? (1) is one on any exceptional curve in any ?bre. ?r . And we de?ne a new bundle E by To simplify, we will denote O ? (1) by E

XEr

???r . Er = ωX ? ? Er The advantage locus, and thus line bundles on of using Er is that it is completely supported on the exceptional it is easy to describe explicitly; and its Deligne product with other ? easy to compute. X

16

TYLER J. JARVIS

Similarly, if s divides r, then there is a family (Es , cs,1 ) of sth roots on X . And ?s on X ?Es which pushes down corresponding to this family, there is a line bundle E ′ ??s ? . to Es , and another line bundle Es = ωX ?s ? (E ) ?r , Er > and < E ?s , Es > agree with Theorem 3.3.9. The Deligne products < E De?nition 3.3.1. Also, the Deligne product of the canonical bundle ω with E is trivial. In particular, we have ?r , Er > is exactly 1. The Deligne product < E (u(i)v (i)/ci )αi

1≤i≤g/2 1/r

+

(j (r ? j )/dj )γj .

0≤j ≤r/2

1/r

?s , Es > is exactly 2. The Deligne product < E r/s

1≤i≤g/2

(u′′ (i)v ′′ (i)/ci )αi

1/r

1/r

+ r/s

1≤j ≤s/2 1≤k≤r/2 k≡j (mod s)

j (s ? j )/dj γk1/r .

1/r

3. And < ωX ?s , Es >. ?r , Er >= O =< ωX Proof. For X smooth, Er is canonically isomorphic to O, and thus in the smooth ?r , Er > and < ω ? , Er > are trivial. case, < E X In the general case these products are all integral linear combinations of boundary divisors. To compute the coe?cients we evaluate degrees on families of curves X /S parameterized by a smooth curve S and having smooth generic ?bre. In this case, ? , whereas the canonical bundle since E is supported on the exceptional locus of X has degree zero on the exceptional locus, the product < ωX ? , Er > must be trivial. ? Computing the coe?cients of < Er , Er > requires that we consider the local structure of Er near an exceptional curve. Recall from [11] that for any singularity where E has order {u, v } with u, v > 0, and such that if c := gcd(u, v ), u′ := u/c, v ′ := v/c, and r′ := r/c, the underlying singularity of X /S is analytically ?S,s [[x, y ]]/xy ? τ r′ ), where τ is an element of O ?S,s . Er is isomorphic to Spec (O ′ v′ generated by two elements, say ν and ξ , with the relations xν = τ ξ and yξ = τ u ν . ? → X is locally given as Moreover, over such a singularity, π : X π : Proj A (A[ν, ξ ]/(νx ? τ v ξ, ντ u ? ξy )) → Spec A, where A is the local ring of X at the singularity. The exceptional curve, call it D, is de?ned by the vanishing of x and y , and we have a situation like that depicted in Figure 1. We need to express Er in terms of a divisor, but this is easy since it is supported completely on the exceptional locus. Er is locally of the form OX ? (nD ), for some integer n. And any Weil divisor of the form nD is Cartier if and only if u′ and v ′ both divide n. Moreover, it is easy to see that if nD is Cartier, then when restricted to the exceptional curve D, the degree of nD is ?n/u′ ? n/v ′ . Finally, since Er has degree ?r on D, we have n = u′ v ′ c = uv/c so that Er = OX ? ((uv/c)D ). ?r , Er > just note that for families f : X → Now, to compute the coe?cients of < E ?r , Er > S over a smooth base curve S with smooth generic ?bre, the degree of < E

′ ′

PICARD GROUP OF SPIN CURVE MODULI

17

D

~ X

y ξ/ν = τ u’

x ν/ξ = τ v’

π

X

xy= τ

r’

Generic Fibre (τ=0) /

Special Fibre (τ=0)

? → X over a singularity of E . Figure 1. Local structure of π : X ?r , Er ). Thus if Dp indicates the exceptional curve is just the intersection number (E over a singularity p, and if {up , vp } indicates the order of (Er , cr,1 ) near p, then ?r , Er > degS < E =

p a singsingularity of E

?r (up vp /cp ) degDp E (u(i)v (i)/ci ) +

p of type αi p of type γj

=

(j (r ? j )/dj ).

?r = 1 and because over γj we have u = j The second line follows because degDp E ?r , Er >, and the result for s is just and v = r ? j . This proves the theorem for < E 1/s ? the pullback of the relation for < Es , Es > from Pic Sg . Now we can prove the main theorem.

?r ?r Proof. (of Theorem 3.3.4) Since Er ? E = ω , we have

?r , Er > r<E

??r , E?1 > ? < E ??r , ω > + < E ??r , ω > = ?<E r r r r ?r ?r ??r ?r ?r = <E ,ω > ? < E , Er >

?r ?r ?r , E ?r > = <E , ω > ?r 2 < E

?r now give Deligne-Riemann-Roch and ? := det f! E

18

TYLER J. JARVIS

= = = = =

?r ?r <E , ω > ?r2 (2? ? 2λ+ < E , ω >) ??r , ω > 2r2 λ ? 2r2 ? ? (r ? 1) < E r 1 2r2 λ ? 2r2 ? ? (r ? 1) < ω ? E? r ,ω >

2r2 λ ? 2r2 ? ? (r ? 1) < ω, ω > 2r2 λ ? 2r2 ? ? (r ? 1)(12λ ? δ )

Here the last equality follows from the well-known Mumford isomorphism: < ω, ω >= 12λ ? δ (see [10]). 3.4. Torsion in Pic Sg . The Picard group of Mg is known to be freely generated by λ when g is greater than 2 (see [2]). And Harer [9] has shown that for r = 2, the 1/2 rational Picard group Pic Sg ? Q has rank one for g ≥ 9. So one might expect 1/2 that Pic Sg is freely generated by ? or λ, but Cornalba showed in [4] that Pic Sg,2 has 4-torsion, and one of the consequences of Theorem 3.3.4 is that whenever 2 or 1/r 3 divides r, there are torsion elements in Pic Sg . In particular, the following proposition holds. Proposition 3.4.1. If r is not relatively prime to 6 and g > 1, then Pic Sg has torsion elements: 2 2 1. If r is even, then r2 ? ? (r2 ? 6r + 6)λ = 0, and thus 1 2 (r ? ? (r ? 6r + 6)λ) 1/r is an element of order 4 in Pic Sg . 1 2 2 2 2 (r ? ? (r2 ? 6r +6)λ) 2. If 3 divides r, then 3 (r ? ? (r ? 6r +6)λ) = 0, and thus 3 is an element of order 3 or 6. 3. If r = sd and d is even, then r2 (?s ? ?r ) ? 6(d2 ? rd + r + 1)λ = 0, and thus 1 2 2 2 r (?s ? ?r ) ? 3(d ? rd + r + 1)λ has order 4. 2 2 r (?s ? ?r ) ? 4(d2 ? rd + r + 1)λ = 0, and 4. If r = sd and 3 divides d, then 3 1 2 thus 3 r (?s ? ?r ) ? 2(d2 ? rd + r + 1)λ has order 3 or 6.

2 2 Corollary 3.4.2. If 6 divides r then 1 6 (r ? ? (r ? 6r + 6)λ) is an element of order 12. 1/r 1/r

Proof of Proposition 3.4.1. If the proposition were false and the element in question 1/r were zero, then in Pic Sg this element would be a sum of boundary divisors. In + ck,ρ γk,ρ . particular, the element in question would be of the form ei αi In the ?rst case, multiplication by two, and in the second case, multiplication by three, allows us to replace the element in question with a sum (from Theorem 3.3.4) consisting exclusively of boundary divisors. Thus for the ?rst case we have a relation between boundary divisors where the sum on the right has all coe?cients divisible by two: (a) (a) (a,b) (a,b) ?, E >= 2 +2 ck,ρ γk,ρ . (1 ? r)δ + r < E ei αi And for the second case, the sum on the right has all coe?cients divisible by three: ?, E >= 3(boundary divisors) (1 ? r)δ + r < E Thus by Theorem 3.2.4 the coe?cients on the left must also be divisible by 2 (a) or 3, respectively. However, in both cases the coe?cient of γ0,ρ for any ρ and a is 1 ? r, which has no divisors in common with r; a contradiction.

(a,b) (a,b) (a) (a)

PICARD GROUP OF SPIN CURVE MODULI

19

The third and fourth cases are similar, but ?rst we must subtract d2 times the second equation of Theorem 3.3.4.bis from the ?rst. Now reduction mod 2 and mod 3 give the necessary contradictions in the third and fourth cases, respectively.

4. Examples 4.1. Genus 1 and Index 1. In the case of g = 1, the only boundary divisors 1/r 1/r,0 of the stack S1 := S1,1 are those lying over δ0 ; that is, the Ramond divisors γ0,ρ , corresponding to the di?erent gluings {ρ} of OP1 at the unique node, and the Neveu-Schwarz divisors γj . In particular, if we write S1 its irreducible components S1 where a then the

1/r 1/r

as the disjoint union of

=

d |r

S1

1/r,(d)

1/r,(d) generic geometric point of S1 1/r,(1) only boundary divisor in S1 is

has Er isomorphic to a dth root of O, γ0,1 corresponding to the trivial bundle 1/r,(1) - M1,1 is unrami?ed, and

1/r,(1) ?, E > reduces to . The boundary divisor < E so δ = δ0 = γ0 = γ0,1 in Pic S1 zero in this case, and so Theorem 3.3.4 gives

OX = Er on the singular curve X . Moreover, S1

2r2 ? = (2r2 ? 12r + 12)λ + (r ? 1)δ = (2r2 ? 12r + 12)λ + (r ? 1)γ0 . We can give a much more complete description of the Picard group in this case using Edidin and Graham’s equivariant intersection theory [6] and the following 1/r,(1) explicit construction of S1 . Proposition 4.1.1. The stack S1 is isomorphic to the quotient A2 c4 ,c6 ? ? := 3 2 ?4r {(c4 , c6 )|c4 ? c6 = 0} by the action of Gm , given by v · (c4 , c6 ) = (v c4 , v ?6r c6 ). Similarly, S1 of Gm .

1/r,(1) 1/r,(1)

is A2 c4 ,c6 ? D := {(c4 , c6 )|c4 = 0 = c6 } modulo the same action

2 Proof. First recall that if ? is the locus {c3 4 ? c6 } and D is the locus {c4 = c6 = 0}, then the stacks M1,1 and M1,1 have a representation as the space of cubic forms 2 {y 2 = x3 ? 27c4 x ? 54c6 }, that is A2 c4 ,c6 ? ? and Ac4 ,c6 ? D respectively, modulo the “standard” Gm action v · (c4 , c6 ) = (v ?4 c4 , v ?6 c6 ) [6, Remark following §5.4]. - A2 and the action of the proposition We denote this action by s : Gm × A2 2 2 by b : Gm × A A . We have commutative diagrams of stacks 1/r,(1) (A2 ? ?)/b - S1

? ?- ? (A ? ?)/s M1,1

2

and

20

TYLER J. JARVIS

1/r,(1) (A2 ? D)/b - S1

? ? ?(A2 ? D)/s M1,1 where the top morphism is given by the fact that there is a b-equivariant choice of a line bundle Er on the curve y 2 = x3 ? 27c4 ? 54c6 ? P2 × (A2 ? D) and a ? ?r - ω . The bundle Er is generated by an rth root b-equivariant isomorphism Er of dx/y , the invariant di?erential. This makes sense because dx/y has no zeros or poles. Alternately, we may simply take the trivial line bundle N on A2 generated by an element ζ , with the action b de?ned as v · ζ = v ?1 ζ . If π : {y 2 = x3 ? 27c4 ? - A2 is the projection to A2 , then we de?ne Er to be π ? N , and the 54c6 } ?r - ω to be ζ r → dx . This homomorphism is b-equivariant homomorphism Er y dx ?r dx since v · ( dx y ) = v y ; and it is well-known that for this family y generates the relative dualizing sheaf ωπ . The proposition now follows since both the left and right-hand vertical morphisms are ? etale of degree 1/r, and the bottom morphism is an isomorphism. Thus the top morphism is ? etale of degree 1. It is clearly an isomorphism on geometric points, thus also an isomorphism of stacks. It is easy to see that the line bundle N induces the pushforward π? Er on S1 1/r,(1) and S1 . We denote this bundle by ?+ . Similarly the bundle ?? := R1 π? Er = ?1 ?π? (ω ?Er ) can be seen to be ?λ + ?+ , thus ? = ?+ ? ?? = λ, which is compatible with 12λ = δ and with Theorem 3.3.4. Moreover, the explicit description of N 1/r,(1) shows that r?+ = λ. So the order of ?+ is 12r in Pic S1 . Corollary 4.1.2. The Chow rings A? (S1 ) and A? (S1 ) are isomorphic 1/r,(1) 2 2 to Z[t]/12rt and Z[t]/24r t , respectively. Consequently, Pic S1 =< ?+ >? = 1/r,(1) + ? Z/12rZ and Pic S =< ? >= Z.

1 1/r,(1) 1/r,(1) 1/r,(1)

Proof. By [6, Prop. 18 and 19] for any smooth quotient stack F = [X/G] the Chow 1 ? G ring A? (F ) is the equivariant Chow ring A? G (X ) = A? (X ), and Pic F = AG (X ). 2 G 2 Thus it su?ces to compute A? (X ), where X is the Ac4 ,c6 ? ? or Ac4 ,c6 ? D, and G in Gm with the action b : Gm × X - X of Proposition 4.1.1. Choosing an N + 1-dimensional representation V of G with all weights ?1, and letting U = V ? {0} be the open set where G acts freely, then the diagonal action of G 2 G on (A2 c4 ,c6 ? {0}) × U is free, and Ai (Ac4 ,c6 ? {0}) is de?ned [6, Defn 1]to be the 2 usual Chow group Ai ((A ?{0}) × U/G) of the quotient scheme ((A2 ?{0}) × U )/G, which is isomorphic to the complement of the zero section of the vector bundle 2 2 2 O(4r) ⊕ O(6r) over PN . Thus A? G (Ac4 ,c6 ? {0}) = Z[t]/24r t . 3 2 Moreover, since the form c4 ? c6 has weighted degree ?12r with respect to the 2 action, the G equivariant fundamental class [?]G of ? is 12rt, and AG ? (A ? ?) = G 2 A? ((A ? {0} × U )/[?]G = Z[t]/12rt. Similarly, the class [D]G is the intersection of [c4 = 0]G and [c6 = 0] = (4rt) · (6rt) = 0. The theorem follows.

PICARD GROUP OF SPIN CURVE MODULI

21

4.2. Other Components in Genus 1. Because we have no explicit representation 1/r,(d) of S1 as a quotient stack for d > 1, this case is more di?cult. Moreover, Er has no global sections, nor any higher cohomology, so the bundle ? is trivial (and ?+ = ?? = 0). The only other obvious bundles on the stack are those induced by 1/r,(d) - Sd/r,(1) from Pic Sd/r,(1) . In particular, we have pullback along [d] : S1 1 1 d/r,+ the bundles ? and λ. With λ = r?d/r,+ /d and since 12λ = δ , the relation of Theorem 3.3.4 gives 2r2 λ = 0 in Pic S1

1/r,(d)

.

d/r,(1) - Pic S1/r,(d) In particular, if 6 does not divide r, then the homomorphism [d]? : Pic S1 1 is not injective. In the case of d = 2 or 3 we can follow Mumford [17] and construct r-spin curves of index d which have non-trivial automorphisms, and these will give homomor1/r,(d) - Gm . In particular, in the case that d = 2, consider the phisms Pic S1 2 curve E1728 : y = x(x2 ? 1), and the two-torsion point p = (0, 0). Associated to p is the line bundle Er := {f ψ |f ∈ k (E1728 ), (f ) ≥ p ? ∞}, and the isomorphism ? cr,1 : E ?r - ω de?ned by r

gdx . xr/2 y This is an isomorphism because x is a global section of O(?2p + 2∞) giving an isomorphism to O, dx y is a global section of ω giving an isomorphism to O , and 2 divides r. An automorphism σ of order 4 of the underlying curve E1728 can be de?ned as σ (x, y ) = (?x, iy ), and σ can be extended to an automorphism of the entire spin curve (E1728 , (Er , cr,1 )) by letting ζ be a primitive 4rth root of unity such that ζ r = (?1)r/2 i and then de?ning gψ r → σ (ψ ) = ζψ.

dx dx r cr,1 is preserved by σ since = (?1)r/2 i( yx r/2 ) = ζ ( yxr/2 ). (Er , cr,1 ) is clearly of index (2), and although π? Er , R1 π? Er , and π! Er are all 1/r,(2) , the bundle Er/2 = Er is always isomorphic to OE , trivial (zero or O), on S1 and so the sheaves ?2/r,+ := π? Er/2 and ?2/r,? := R1 π? Er/2 are always line bundles dx σ ( yx r/2 )

on S1 (actually they are just the pullbacks of ?+ and ?? , respectively, from 2/r,(1) 1/r - S2/r ). S1 via the morphism [2] : S1 1 1/r,(2) For every line bundle L ∈ Pic S1 , the geometric point (E1728 , (Er , cr,1 )) of 1/r,(2) S1 associates a one-dimensional vector space L(E1728 , (Er , cr,1 )) ? = k , and σ induces an automorphism L(σ ) ∈ Gm of L(E1728 , (Er , cr,1 )). Moreover, L(σ ) has order dividing 4r (the order of σ ). Thus we have a homomorphism Pic S1

1/r,(2)

1/r,(2)

- < ζ >? = Z/4r.

Moreover, the explicit description of Er shows that ?2/r,+ maps to ζ 2 and ?2/r,? maps to ζ 2?r . In the case of d = 3 we can use a similar argument, applied to the curve E0 : y 2 + y = x3 , the line bundle Er = {f ψ |f ∈ k (E0 ), (f ) ≥ p ?∞} and the isomorphism cr,1 : ψ r → dy . x · y r/3

22

TYLER J. JARVIS

Choose a primitive 3rth root of unity ξ , and de?ne the automorphism τ : (x, y ) → (ξ ?r x, y ) ψ → ξψ. This is compatible with cr,1 , and so is an automorphism of (E0 , (Er , cr,1 )). This 1/r,(3) - < ξ >? gives a homomorphism Pic S1 = Z/3r and the elements ?3/r,+ := ? + 3/r,? 1 π? Er/3 (= [3] ? ) and ? := R π? Er/3 map to ξ 3 and ξ 3?r , respectively. Thus we have the following commutative diagrams. Pic S1

3/r,(1)

? - < ?3/r,+ >= Z/12r

?

1/r,(3) Pic S1

? - Z/3r

and Pic S1

1/r,(1)

? - < ?2/r,+ >= Z/12r

? 1/r,(2) Pic S1

? - Z/4r

where the right-hand vertical morphisms take ?3/r,+ to 3 and ?2/r,+ to 2, respectively, and λ = r?3/r,+ or r?2/r,+ , respectively. For d > 3 this strategy does not work as well, since no automorphisms of the underlying curve preserve a spin structure of type d. Nevertheless, we still have for any (E, (Er , cr,1 )) the automorphism de?ned by taking Er to η Er , where η is any rth root of unity. This gives, for η a primitive rth root of unity, Pic S1

1/r,(d)

- < η >= Z/r

and the bundle ?d/r,+ := π? Er/d maps to η d . This inspires the following conjecture: Conjecture 4.2.1. Pic S1

1/r,(d)

=< ?d/r,+

4.3. General genus, r = 2. For g > 2, and r = 2, we have that 2? + λ is an element of order 4, and λ (and hence ?) is an element of in?nite order. Moreover, 1/2,even 1/2,odd Harer has proved for g > 9 that H1 (Sg , Z) = H1 (Sg , Z) = Z/4, and 1 / 2 ,even 1 / 2 ,odd 2 2 H (Sg , Q) = H (Sg , Q) = Q, so for g > 9 we have 1/2,odd Pic S = Pic S1/2,even ? = Z × Z/4Z.

g g

? ? ?Z/2r >= Z/r ? ? Z/(r/d)

if d = 2 if d = 3 if d > 3

What is not yet completely clear, but seems reasonable to expect, is the following presentation for that group.

PICARD GROUP OF SPIN CURVE MODULI

23

Conjecture 4.3.1. For all g > 2,

/2,even /2,odd Pic S1 = Pic S1 =< ?, λ | 8? + 4λ = 0 > . g g

Conclusion We have worked out many relations between the elements of Pic Sg and Pic Sg . This generalizes the work of Cornalba [3, 4], whose results hold in the case where r = 2. One of the interesting consequences of these relations is the existence of 1/r elements in Pic Sg of elements of order 4 if 2 divides r, and elements of order 3, if 3 divides r. Somehow 2 and 3 seem to be special, however, and when r is relatively 1/r prime to 6 and g > 2, there do not appear to be any torsion elements in Pic Sg . Corresponding results for Pic Sg,n will appear in [13], where they are used to prove the genus-zero case of the generalized Witten conjecture [21]. Acknowledgments I wish to thank Takashi Kimura, J? anos Koll? ar, Bill Lang, and Arkady Vaintrob for helpful discussions and suggestions regarding this work. I am also grateful to Heidi Jarvis for help with typesetting. References

[1] Valery Alexeev, Compacti?ed Jacobians, alg-geom 9608012, preprint, 1996. [2] Enrico Arbarello and Maurizio Cornalba, The Picard groups of the moduli spaces of curves, Topology 26 (1987), no. 2, 153–171. [3] Maurizio Cornalba, Moduli of curves and theta-characteristics, Lectures on Riemann Surfaces (M. Cornalba, X. Gomez-Mont, and A. Verjovsky, eds.), World Scienti?c, 1989, pp. 560–589. , A remark on the Picard group of spin moduli space, Atti Accademia Nazionale Lincei. [4] Classe Scienze, Fisiche, Matematiche Naturali. Rend. Lincei (9) Mat. Appl. 2 (1991), no. 3, 211–217. [5] Pierre Deligne, Le d? eterminant de la cohomologie, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 93–177. [6] Dan Edidin, Notes on the construction of the moduli space of curves, math.AG/9805101, preprint, 1998. ? ements de g? ? [7] Alexandre Grothendieck and Jean Dieudonn? e, El? eom? etrie alg? ebrique II: Etude globale ? el? ementaire de quelques classes de morphismes, vol. 8, Publications Math? ematiques IHES, 1961. [8] John L. Harer, The second homology group of the mapping class group of an orientable surface, Inventiones Mathematicae 72 (1983), no. 2, 221–239. [9] , The rational Picard group of the moduli space of Riemann surfaces with spin structure, Mapping class groups and moduli spaces of Riemann surfaces (G¨ ottingen, 1991/Seattle, WA, 1991), Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp. 107–136. [10] Joe Harris and David Mumford, On the Kodaira dimension of the moduli space of curves, Inventiones Mathematicae 67 (1982), no. 1, 23–88. [11] Tyler J. Jarvis, Geometry of the moduli of higher spin curves, math.AG/9809138, preprint, 1998. , Torsion-free sheaves and moduli of generalized spin curves, Compositio Mathematica [12] 110 (1998), no. 3, 291–333. [13] Tyler J. Jarvis, Takashi Kimura, and Arkady Vaintrob, Moduli spaces of higher spin curves and integrable hierarchies, math.AG/9905034, preprint, 1999. [14] Maxim L. Kontsevich, Intersection theory on the moduli space of curves, Funktsional. Anal. i Prilozhen. 25 (1991), no. 2, 50–57, 96. [15] , Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, 1–23.

1/r,m 1/r 1/r

24

TYLER J. JARVIS

[16] Eduard Looijenga, Intersection theory on Deligne-Mumford compacti?cations (after Witten and Kontsevich), Ast? erisque (1993), no. 216, Exp. No. 768, 4, 187–212, S? eminaire Bourbaki, Vol. 1992/93. [17] David Mumford, Picard groups of moduli problems, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, pp. 33–81. [18] C. S. Seshadri, Fibr? es vectoriels sur les courbes alg? ebriques, Soci? et? e Math? ematique de France, ? Paris, 1982, Notes written by J.-M. Drezet from a course at the Ecole Normale Sup? erieure, June 1980. [19] Carlos T. Simpson, Moduli of representations of the fundamental group of a smooth projective ? variety. I, Inst. Hautes Etudes Sci. Publ. Math. (1994), no. 79, 47–129. [20] Edward Witten, The N-matrix model and gauged WZW models, Nuclear Phys. B 371 (1992), no. 1-2, 191–245. , Algebraic geometry associated with matrix models of two-dimensional gravity, Topo[21] logical methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 235–269. Department of Mathematics, Brigham Young University, Provo, UT 84602 E-mail address : jarvis@math.byu.edu

相关文章:

更多相关标签:

- Geometry of the moduli of higher spin curves
- On the geometry of the moduli space of spin curves
- The Q-Picard group of the moduli space of curves in positive characteristic
- MODULI OF TWISTED SPIN CURVES
- Gravitational Descendants and the Moduli Space of Higher Spin Curves
- The Picard group of the Moduli of G-bundles on a Curve
- The Least Action Principle And The Spin Of Galaxies In The Local Group
- NéRON MODELS AND COMPACTIFIED PICARD SCHEMES OVER THE MODULI STACK OF STABLE CURVES
- Neron models and Compactified Picard schemes over the moduli stack of stable curves
- Moduli Spaces of Higher Spin Curves and Integrable Hierarchies