dd -ELLIPTIC LOCI IN GENUS 2 AND 3 CARL LIAN
Abstract.
We consider the loci of curves of genus 2 and 3 admitting a d -to-1 map to agenus 1 curve. After compactifying these loci via admissible covers, we obtain formulasfor their Chow classes, recovering results of Faber-Pagani and van Zelm when d = 2 . Theanswers exhibit quasimodularity properties similar to those in the Gromov-Witten theory ofa fixed genus 1 curve; we conjecture that the quasimodularity persists in higher genus, andindicate a number of possible variants. Introduction
Let H g/ ,d denote the moduli space of degree d covers f : C → E , where C is a smoothcurve of genus g , E is a smooth curve of genus 1, and f is simply branched at marked points x , . . . , x g − ∈ C mapping to distinct points y , . . . , y g − ∈ E . We then have a diagram H g/ ,d π g/ ,d (cid:47) (cid:47) ψ g/ ,d (cid:15) (cid:15) M g M , g − where the map π g/ ,d remembers the source curve, and ψ g/ ,d remembers the target curvewith the branch points. The degree of ψ g/ ,d is given by a Hurwitz number , which countsmonodromy actions of π ( E − { y , . . . , y g − } , y ) on the d -element set f − ( y ) . We have thefollowing groundbreaking result of Dijkgraaf: Theorem 1.1 ([Dijk95]) . For g ≥ , the generating series (cid:88) d ≥ deg( ψ g/ ,d ) q d is a quasimodular form of weight g − . Okounkov-Pandharipande [OP06] give a substantial generalization to the Gromov-Wittentheory of an elliptic curve with arbitrary insertions.In this paper, we consider instead the enumerative properties of the map π g/ ,d . Here, thegeometry is more subtle, as we no longer have a combinatorial model as in Hurwitz theory.For enumerative applications, one needs to compactify the moduli spaces involved; we passto the Harris-Mumford stack A dm g/ ,d of admissible covers, see [HM82] or §2.3. We have the Date : April 16, 2020. a r X i v : . [ m a t h . AG ] A p r iagram: A dm g/ ,d π g/ ,d (cid:47) (cid:47) ψ g/ ,d (cid:15) (cid:15) M g M , g − We prove:
Theorem 1.2.
For g = 2 , , we have: (cid:88) d ≥ [( π g/ ,d ) ∗ (1)] q d ∈ A g − ( M g ) ⊗ Qmod , where Qmod is the ring of quasimodular forms.
More precisely, we have the following formulas, where σ k ( d ) is the sum of the k -th powersof the divisors of d : Theorem 1.3.
The class of ( π / ,d ) ∗ (1) in A ( M ) is: (2 σ ( d ) − dσ ( d )) δ + (4 σ ( d ) − σ ( d )) δ . Theorem 1.4.
The class of ( π / ,d ) ∗ (1) in A ( M ) is: (cid:0) ( − d + 6780 d − σ ( d ) + (5592 d − σ ( d ) + 252 σ ( d ) (cid:1) λ + (cid:0) (1224 d − d + 156) σ ( d ) + ( − d + 840) σ ( d ) (cid:1) λδ + (cid:0) (2160 d − d + 216) σ ( d ) + ( − d + 240) σ ( d ) (cid:1) λδ + (cid:0) ( − d + 39 d − σ ( d ) + (51 d − σ ( d ) (cid:1) δ + (cid:0) ( − d + 36 d − σ ( d ) + (192 d ) σ ( d ) (cid:1) δ δ + (cid:0) ( − d − d + 36) σ ( d ) + (192 d + 120) σ ( d ) (cid:1) δ + (cid:0) (216 d − d + 60) σ ( d ) + ( − d + 360) σ ( d ) (cid:1) κ . As a check, both formulas become zero after substituting d = 1 . When d = 2 , we recoverthe main results of Faber-Pagani [FP15]. Indeed, the morphism π / , is generically 4-to-1,where one factor of 2 comes from the complement C → E to any bielliptic map C → E (see[Ku88, §2]), and another comes from the labelling of the two ramification points. Thus, ouranswer differs from [FP15, Proposition 2] by a factor of 4. In genus 3, the morphism π / , has degree , coming from the ways of labelling the ramification points of a bielliptic cover,and our answer differs from the correction to [FP15, Theorem 1] given by van Zelm [vZ18a,(3.5)] by a factor of 24.We are then led to conjecture: Conjecture 1.
The statement of Theorem 1.2 holds for all g ≥ . In fact, our method, which we now outline, suggests a number of possible refinements ofConjecture 1. For g = 2 , , it suffices to intersect π g/ ,d with test classes of complementarydimension, all of which lie in the boundary of M g . The test classes may then be moved togeneral cycles in a boundary divisor of M g . The intersection of general boundary cycles withthe admissible locus can be expressed in terms of contributions from admissible covers of a mall number of topological types, and we compute these contributions in terms of branchedcover loci in lower genus. This leads naturally to a number of auxiliary situations in whichquasimodularity phenomena also occur, for example: • Loci of d -elliptic curves with marked points having equal image under the d -ellipticmap, see §3.2 • Loci of curves covering a fixed elliptic curve, see §5.1 and also [OP06] • Correspondence maps ( π g/ ,d ) ∗ ◦ ψ ∗ g/ ,d , see §5.2 • Loci of d -elliptic curves with marked ramification points, see §5.3 and Appendix ALet us mention one consequence of Conjecture 1. Building on work of Graber-Pandharipande[GP03], van Zelm [vZ18b] has shown that the class ( π g/ , ) ∗ (1) ∈ H g − ( M g ) is non-tautological for g ≥ , and that ( π g/ , ) ∗ (1) ∈ H g − ( M g ) is non-tautological for g = 12 .Assuming Conjecture 1, we have that for these values of g , the generating functions (cid:88) d ≥ ( π g/ ,d ) ∗ (1) q d in the quotients of A g − ( M g ) and A g − ( M g ) by their tautological subgroups are nonzeroquasimodular forms (note that the tautological part of A g − ( M g ) is zero by Looijenga’s result[Loo95]). In particular, we would get infinitely many non-tautological classes from d -ellipticloci on M g for g ≥ , and on M .The structure of this paper is as follows. We collect preliminaries in §2, recording the neededfacts about intersection theory on M g,n and recalling the definitions of admissible coversand quasimodular forms. In §3, we carry out some enumerative calculations for branchedcovers that we will need when considering d -elliptic loci. In §4, we prove Theorem 1.3 on the d -elliptic loci in genus 2; it is here where we explain our method in the most detail. In §5, weestablish variants of Theorem 1.3 suggesting possible variants of Conjecture 1. Finally, weput together all of the previous results to prove Theorem 1.4 on the d -elliptic loci in genus 3in §6.We remark in Appendix A that we have quasimodularity for d -elliptic loci on M , , wherethe ramification points of a d -elliptic cover are marked. However, we explain a new feature:not all contributions to the classes of the d -elliptic loci from admissible covers of individualtopological types are themselves quasimodular.1.1. Acknowledgments.
I am grateful to my advisor Johan de Jong, who suggested theinitial research directions and offered invaluable guidance throughout. I also thank JimBryan and Georg Oberdieck for pointing out the quasi-modularity in the initial results, IzzetCoskun for encouraging me to work out the genus 3 case, and Dawei Chen, Raymond Cheng,Bong Lian, Nicola Pagani, Renata Picciotto, and Michael Thaddeus for helpful comments ondrafts of this work. This work was completed in part with the support of an NSF GraduateResearch Fellowship. 2.
Preliminaries
Conventions.
We work over C . Fiber products are over Spec( C ) unless otherwisestated. All curves, unless otherwise stated, are assumed projective and connected with onlynodes as singularities. The genus of a curve X refers to its arithmetic genus and is denoted p a ( X ) . A rational curve is an irreducible curve of geometric genus 0. All moduli spaces re understood to be moduli stacks, rather than coarse spaces. In all figures, unlabelledirreducible components of curves are rational, and all other components are labelled withtheir geometric genus.If X is a nodal curve, its stabilization , obtained by contracting rational tails and bridges(that is, non-stable components), is denoted X s . We use similar notation for pointed nodalcurves.All Chow rings are taken with rational coefficients and are denoted A ∗ ( X ) , where X is a variety or Deligne-Mumford stack over C . When referring to Chow groups, we usesubscripts (recording the dimensions of cycles) and superscripts (recording their codimensions)interchangeably when X is smooth. We will frequently refer to the Chow class of a properand generically finite morphism f : Y → X , by which we mean f ∗ ([ Y ]) . The class of f in A ∗ ( X ) is denoted [ f ] . When there is no opportunity for confusion, we sometimes refer to thesame class by “the class of Y ” or [ Y ] ∈ A ∗ ( X ) .” If X is proper and f : X → Spec( C ) is thestructure morphism, we denote the proper pushforward map f ∗ by (cid:82) X .We deal throughout this paper with boundary classes on moduli spaces of curves. We willfind it more convenient to carry out intersection-theoretic calculations using classes obtainedas pushforwards of fundamental classes from (products of) moduli spaces of curves of lowergenus, and label these classes using upper case Greek letters. For example, when g ≥ , wedenote by ∆ ∈ A ∗ ( M g ) the class of the morphism M g − , → M g that glues together the twomarked points. We reserve lower-case letters for substack classes (also known as Q -classes ):for example, δ ∈ A ∗ ( M g ) is the class of the substack of curves with a non-separating node.We have δ = 12 ∆ . In general, the denominator is the order of the automorphism group of the stable graphassociated to the boundary stratum, which in this case is the graph consisting of a singlevertex and a self-loop.2.2.
Intersection numbers on moduli spaces of curves.
Here, we collect notation forvarious classes on moduli spaces of curves, and intersection numbers of these classes. Wewill frequently abuse notation: for instance, ∆ will always denote the class of the locus ofirreducible nodal curves on any M g,n , but it will be clear in context the spaces on whichthese classes are defined. The intersection numbers given here can be verified using the admcycles.sage package, [DSvZ].2.2.1. M , . The rational Picard group A ( M ,n ) is freely generated by boundary divisors,see [AC87]. When n = 2 , we have the boundary divisors ∆ , parametrizing irreducible nodalcurves, and ∆ , parametrizing reducible curves, see Figure 1. The intersection pairing is asfollows: ∆ ∆ ∆ − M , . In A ( M , ) , we have the boundary divisor ∆ parametrizing irreducible nodalcurves, and the boundary divisors ∆ ,S , where S ⊂ { , , } , parametrizing reducible nodalcurves with the marked points corresponding to elements of S lying on the rational component,see Figure 2. (We require | S | ≥ .) igure 1. Boundary classes in A ( M , ) For the same S , let ∆ ,S ∈ A ( M , ) be the class of curves in the boundary divisor ∆ ,S whose genus 1 component is nodal; if | S | = 2 , let ∆ ,S ∈ A ( M , ) be the class of curvesconsisting of a chain of three components, where the rational tail contains the two markedpoints corresponding to the elements of S , see Figure 3. Figure 2.
Some boundary classes in A ( M ) Figure 3.
Some boundary classes in A ( M ) We will not need the intersection numbers of the boundary divisors with all curve classes in A ( M , ) , but we record the intersections of the boundary divisors with those defined above: ∆ ∆ , { , , } ∆ , { , } ∆ , { , } ∆ . { , } ∆ , { , } − ∆ , { , } − ∆ , { , } − ∆ , { , } − M . Mumford has computed A ∗ ( M ) in [Mum83]. A ( M ) is generated by the bound-ary classes ∆ , ∆ , parametrizing irreducible nodal curves and reducible curves, respectively, ee Figure 4. A ( M ) is generated by the boundary classes ∆ , ∆ , parametrizing irre-ducible binodal curves and reducible curves where one component is a rational nodal curve,respectively, see Figure 5. Figure 4.
Boundary classes in A ( M ) Figure 5.
Boundary classes in A ( M ) The intersection pairing is as follows: ∆ ∆ ∆ − − M , . It follows from [Fab90] that A ( M , ) has dimension 5, with a basis given bythe boundary classes ∆ , ∆ a , ∆ b , Ξ , ∆ , shown in Figure 6.The intersection pairing is as follows: ∆ ∆ a ∆ b Ξ ∆ ∆ − ∆ a − ∆ b − − Ξ − ∆ − We will also need the classes ∆ ∈ A ( M , ) and Γ (5) , Γ (6) , Γ (11) ∈ A ( M , ) , shown inFigure 7.The subscripts in the classes Γ ( i ) ∈ A ( M , ) are chosen in such a way that Γ ( i ) × M , =∆ ( i ) ∈ A ( M ) , see §2.2.5. We have the following intersection numbers: Γ (5) Γ (6) Γ (11) ∆ − igure 6. Boundary classes in A ( M , ) Figure 7.
Some boundary classes in A ( M , ) and A ( M , ) M . Faber has computed A ∗ ( M ) in [Fab90]. We have that A ( M ) and A ( M ) both have dimension 7 and pair perfectly. To describe bases of these groups, we first recallthe definitions of the λ and κ classes. Let u : C g → M g be the universal curve, let ω C g / M g bethe relative dualizing sheaf, and let K ∈ A ( C g ) denote the divisor class of ω C g / M g . Then, bydefinition: λ i = c i ( u ∗ ω C g / M g ) ∈ A i ( M g ) κ i = u ∗ ( K i +1 ) ∈ A i ( M g ) We will only need the class λ in this paper, so we write λ = λ in A ( M ) , with no risk ofconfusion. Then, a basis for A ( M ) is given by the seven classes λ , λδ , λδ , δ , δ δ , δ , κ .A basis for A ( M ) is given by surface classes ∆ [ i ] for i ∈ { , , , , , , } , retainingthe indexing from [Fab90], see Figure 8.We have the following intersection numbers: igure 8. Boundary classes in A ( M ) λ λδ λδ δ δ δ δ κ ∆ [1] − ∆ [4] − ∆ [5] −
112 124 − − ∆ [6] − −
12 112 ∆ [8] −
112 124 −
116 12 −
124 124 ∆ [10] − −
12 18 124 ∆ [11] 1288 124 − −
124 1288
Admissible covers.
We recall the definition of [HM82]:
Definition 2.1.
Let
X, Y be curves. Let b = (2 p a ( X ) − − d (2 p a ( Y ) − , and let y , . . . , y b ∈ Y be such that ( Y, y , . . . , y b ) is stable. Then, an admissible cover consists ofthe data of the stable marked curve ( Y, y , . . . , y b ) and a finite morphism f : X → Y suchthat: • f ( x ) is a smooth point of Y if and only if x is a smooth point of X , • f is simply branched over the y i and étale over the rest of the smooth locus of Y , and • at each node of X , the ramification indices of f restricted to the two branches areequal. Remark 2.2.
It is clear that non-separating nodes of X must map to non-separating nodesof Y . Hence, the preimage of a smooth component of Y must be a disjoint union of smoothcomponents of X .Admissible covers of degree d from a genus g curve to a genus h curve are parametrized bya proper Deligne-Mumford stack A dm g/h,d , see [HM82, Mo95, ACV03]. A dm g/h,d containsthe Hurwitz space H g/h,d parametrizing simply branched covers of smooth curves as a denseopen substack.Let b = 2 g − − d (2 h − . Then, we have a forgetful map ψ g/h,d : A dm g/h,d → M h,b remembering the target, and another π g/h,d : A dm g/h,d → M g,b remembering the stabilization f the source. We will often abuse notation and write π g/h,d : A dm g/h,d → M g,r for r < b ,obtained by post-composing with the map M g,b → M g,r forgetting the last b − r points. Lemma 2.3.
The morphism ψ g/h,d : A dm g/h,d → M h,b is quasifinite (and hence finite).Proof. Over the open locus M h,b , this is classical: the number of points in any fiber is givenby a Hurwitz number, counting isomorphism classes of monodromy actions of the finitelygenerated group π ( Y − { y , . . . , y b } ) on a general fiber of f : X → Y . Over a general pointof any boundary stratum of M h,b parametizing admissible covers f : X → Y , there are afinite number of possible collections of ramification profiles above the nodes of Y , each ofwhich leads to finitely many collections of covers of the individual components of Y , which inturn can be glued together in finitely many ways. (cid:3) We also recall from [HM82] the explicit local description of A dm g/h,d . Let [ f : X → Y ] be a point of A dm g/h,d . Let y (cid:48) , . . . , y (cid:48) n be the nodes of Y , and let y , . . . , y b ∈ Y be thebranch points of f . Let C [[ t , . . . , t h − b ]] be the deformation space of ( Y, y , . . . , y b ) , so that t , . . . , t n are smoothing parameters for the nodes y (cid:48) , . . . , y (cid:48) n . Let x i, , . . . , x i,r i be the nodesof X mapping to y (cid:48) i , and denote the ramification index of f at x i,j by a i,j . Proposition 2.4 ([HM82]) . The complete local ring of A dm g/h,d at [ f ] is C (cid:2)(cid:2) t , . . . , t h − b , { t i,j } ≤ i ≤ n ≤ j ≤ r i (cid:3)(cid:3) / (cid:0) t = t a , , = · · · = t a ,r ,r , . . . , t n = t a n, n, = · · · = t a n,rn n,r n (cid:1) . Here, the variable t i,j is the smoothing parameter for X at x i,j . In particular, A dm g/h,d isCohen-Macaulay of pure dimension h − b . Moreover, if the a i,j are all equal to 1, thatis, f is unramified over the nodes of Y (or if Y is smooth to begin with), then A dm g/h,d issmooth at [ f ] .One can readily extend the theory to construct stacks of admissible covers with arbitraryramification profiles; we use this in §3.3. Even in this more general setting, we always requirethe target curve, marked with branch points, to be stable.In this paper, we primarily study the case h = 1 , that is, the moduli of covers of ellipticcurves. The space A dm g/ ,d is reducible when d is composite, due to the existence of covers C → E → E factoring through a non-trivial isogeny. However, the open and closedsubstack A dm prim g/ ,d parametrizing primitive covers, that is, those that do not factor througha non-trivial isogeny (more generally, through a non-trivial admissible cover of genus 1curves), is irreducible. In fact, this is already true for a fixed elliptic target, see [GK87] or[Buj15, Theorem 1.4]. For our enumerative results, however, the individual components of A dm g/ ,d play no essential role: we consider the entire moduli space, including the componentsparametrizing non-primitive covers.2.4. Quasimodular forms.
A possible reference for this section is [R12]. For positiveintegers d, k , define σ k ( d ) = (cid:88) a | d a k ecall that the ring Qmod of quasimodular forms is generated over C by the Eisenstein series E = 1 − ∞ (cid:88) d =1 σ ( d ) q d E = 1 + 240 ∞ (cid:88) d =1 σ ( d ) q d E = 1 − ∞ (cid:88) d =1 σ ( d ) q d , where we take q to be a formal variable, and that more generally, for integers k ≥ , we have E k = 1 − kB k ∞ (cid:88) n =1 σ k − ( n ) q n ∈ Qmod , where B k is a Bernoulli number. The weight of E k is k , and Qmod is a graded C -algebraby weight.We have the Ramanujan identities q dE dq = E − E q dE dq = E E − E q dE dq = E E − E , so in particular ∞ (cid:88) d =1 P ( d ) σ k − ( d ) q d ∈ Qmod for any P ( d ) ∈ C [ d ] . Thus, Theorem 1.2 will be an immediate consequence of Theorems 1.3and 1.4.The Ramanujan identities also give the convolution formulas (cid:88) d + d = d σ ( d ) σ ( d ) = (cid:18) − d + 112 (cid:19) σ ( d ) + 512 σ ( d ) (cid:88) d + d = d d σ ( d ) σ ( d ) = (cid:18) − d + 124 d (cid:19) σ ( d ) + 524 dσ ( d ) (cid:88) d + d + d = d σ ( d ) σ ( d ) σ ( d ) = (cid:18) d − d + 1192 (cid:19) σ ( d ) + (cid:18) − d + 596 (cid:19) σ ( d ) + 7192 σ ( d ) Auxiliary computations
In this section, we record a number of enumerative results for branched covers that we willtake as inputs in the main computation. .1. Counting isogenies.Lemma 3.1.
Let ( E, p ) be an elliptic curve and d be a positive integer. Then, the numberof isomorphism classes of isogenies E → F of degree d is σ ( d ) . Likewise, the number ofisomorphism classes of isogenies F → E of degree d is σ ( d ) .Proof. We see that these two numbers are equal by taking duals, so it suffices to countisogenies E → F of degree d , i.e., quotients of E by a subgroup of order d , which is thenumber of index d sublattices of Z . A sublattice of Z is determined by a Z -basis ( a, , ( b, c ) ,where a, c are positive and as small as possible; b is uniquely determined modulo a . As ac = d ,the number of such sublattices is exactly σ ( d ) . (cid:3) Corollary 3.2.
The degrees of the morphisms ψ / ,d : A dm / ,d → M , and π / ,d : A dm / ,d → M , remembering the target and (contracted) source, respectively, of a cover,are both σ ( d ) . Moreover, both morphisms are unramified over M , .Proof. The first statement is exactly the content of Lemma 3.1. To see that both morphismsare unramified over M , , note that the open locus H / ,d ⊂ A dm / ,d parametrizing coversof smooth curves is smooth, and that the set-theoretic fibers of both morphisms over anypoint of M , all have the same size. (cid:3) The 2-pointed d -elliptic locus on M , .Lemma 3.3. Let ( E, p ) be an elliptic curve and d a positive integer. Then, the number ofpairs (up to isomorphism) ( f, q ) where f : E → F is an isogeny and q (cid:54) = p is a pre-image ofthe origin of F is ( d − σ ( d ) .Proof. We give a bijection between the set of ( f, q ) and the set of pairs ( G, g ) where G ⊂ E [ d ] is a subgroup of order d and (cid:54) = g ∈ G . The claim then follows from Lemma 3.1. In onedirection, given ( f, q ) , we take G = ker( f ) and g = q . In the other, let F = E/G and f bethe quotient map, and take q = g . (cid:3) Let A dm / ,d, be the moduli space of triples ( f, x , x ) , where f : X → Y is a degree d cover of a marked genus 1 curve ( Y, y ) by a genus 1 curve X , and x , x ∈ X are distinctpoints with f ( x ) = f ( x ) = y . We have a finite morphism ψ / ,d, : A dm / ,d, → M , remembering the target curve, and a morphism π / ,d, : A dm / ,d, → M , remembering thestabilized source curve. Proposition 3.4.
Adopting the notation of §2.2.1, we have: (cid:90) M , [ π / ,d, ] · ∆ = ( d − σ ( d ) , (cid:90) M , [ π / ,d, ] · ∆ = 0 . Proof.
Let ( E, p ) be a general elliptic curve. Any two geometric points of M , are equivalent,so the boundary class ∆ is equivalent to the class of the morphism t E : E → M , sending q (cid:55)→ ( E, p, q ) . Then, the first statement follows from Lemma 3.3 provided the intersection of t E and π / ,d, is transverse. This is easy to see: at an intersection point ( E, p, q ) , a tangentvector from E fixes ( E, p ) but moves q to first order, while a tangent vector from A dm / ,d, moves ( E, p ) to first order, owing to Corollary 3.2. he second statement follows from the fact that no (pointed) admissible cover f : X → Y in A dm / ,d, has the property that X contracts to a curve in ∆ . Indeed, Y would need tobe singular, in which case X must be a cycle of m rational curves, each of which maps to thenormalization of Y via the map x (cid:55)→ x d/m , totally ramified at the nodes. (To see this, onecan follow the method of §4.1.2.) The contraction of such a curve does not lie in ∆ . (cid:3) Corollary 3.5.
We have [ π / ,d, ] = ( d − σ ( d ) (cid:18)
124 ∆ + ∆ (cid:19) in A ( M , ) .Proof. Immediate from §2.2.1. (cid:3)
Doubly totally ramified covers of P . The following is an easy special case of [L19,Theorem 1.4].
Lemma 3.6.
Let ( E, x ) be a general elliptic curve and d a positive integer. Then, thenumber of tuples ( f, x , x , x , x ) , where x , . . . , x ∈ E are distinct and f : E → P is adegree d morphism (considered up to automorphism of the target) totally ramified at x , x and simply ramified at x , x , is d − .Proof. The linear system defining f must be a 2-dimensional subspace W of V = H ( E, O ( d · x )) . In order for W to be totally ramified at x , we need O ( d · x ) ∼ = O ( d · x ) , that is, x ∈ E [ d ] − { x } . For such an x , there are unique (up to scaling) sections in V vanishing tomaximal order at x , x ; thus W is uniquely determined by the d − possible choices of x .Moreover, f will be simply branched over two distinct points x , x of P unless it has twosimple ramification points over the same point of P or a triple ramification point; however,this will only happen of E admits a degree d cover of P branched over 3 points, which isimpossible for E general. There are two ways to label the simple ramification points, so theconclusion follows. (cid:3) Corollary 3.7.
Let ( E, x ) be a general elliptic curve and d a positive integer. Then, thenumber of tuples ( f, x , x , x , x ) , where x , . . . , x ∈ E are distinct and f : E → P is adegree d morphism simply ramified at x , x and totally ramified at x , x , is d − .Proof. Pullback by translation by x − x defines a bijection with the objects counted hereand those in Lemma 3.6. (cid:3) Let A dm d,d, , / ,d be the moduli space of degree d admissible covers f : X → Y , where X has genus 1, Y has genus 0, and f is ramified at four points x , x , x , x to orders d, d, , ,respectively. We consider the map π d,d, , / ,d : A dm d,d, , / ,d → M , sending f to the stabilizationof ( X, x , x , x ) . Proposition 3.8.
Adopting the notation of §2.2.2, we have: (cid:90) M , [ π d,d, , / ,d ] · ∆ = 2( d − , (cid:90) M , [ π d,d, , / ,d ] · ∆ ,S = 0 , for all S ⊂ { , , } with | S | ≥ . roof. Similarly to the proof of Proposition 3.4, we replace ∆ with the class of M E, , asdefined by the Cartesian diagram M E, (cid:47) (cid:47) (cid:15) (cid:15) M , (cid:15) (cid:15) [ E ] (cid:47) (cid:47) M , where [ E ] is the geometric point corresponding to a general elliptic curve E . Then, to checkthat [ M E, ] and π d,d, , / ,d intersect transversely, note that A dm d,d, , / ,d is unramified over a generalpoint of M , , so a tangent vector from A dm d,d, , / ,d will deform E to first order, whereas atangent vector from M E, will fix E and deform the marked points to first order. The firststatement then follows from Lemma 3.6.On the other hand, it is straightforward to check that the image of π d,d, , / ,d is disjoint fromevery ∆ ,S : if Y is singular and [ f : X → Y ] ∈ A dm d,d, , / ,d , then X is either a union of anelliptic curve and rational tails, all of which will be contracted, or a union of smooth rationalcurves. The second statement follows. (cid:3) Proposition 3.9.
Let C be a general curve of genus 2. Then, up to automorphisms of thetarget, there are d − tuples ( f, x , x , x , x , x , x ) , where x , . . . , x ∈ C are distinctpoints, and f : C → P is a degree d morphism totally ramified at x , x and simply ramifiedat x , x , x , x .Proof. The fact that such an f is simplify ramified at four other points follows from adimension count: the dimension of the space of covers C → P branched over 5 points orfewer is 2, so a general point of M admits no such covers.We appeal to the degeneration technique described in [L19]: it suffices to count limit linearseries V on the reducible curve C formed by attaching general elliptic curves E , E to acopy of P , where each elliptic component contains three of the x i . Of the (cid:0) (cid:1) = 20 possibledistributions of the x i onto the elliptic components, there are 12 ways for x , x to lie ondifferent components, and 8 for them to lie on the same component. In the first case, itfollows from Lemma 3.6 that there are d − possible aspects of V on each E i , and theaspect of V on P must be the unique pencil with vanishing sequence (0 , d ) at both markedpoints. In the second, we have, by Corollary 3.7, d − possible aspects on the componentcontaining x , x and −
1) = 6 on the other, and the aspect of V on P must be theunique pencil with vanishing sequences (0 , , ( d − , d ) at the two marked points.Thus, our answer is · (2( d − + 8 · · d − d − , as desired. (cid:3) Remark 3.10.
One can also recover Proposition 3.9 using Tarasca’s formula for the closureof the locus of [( C, p, q )] ∈ M such that C admits a cover of P totally ramified at p and q ,see [T15]. . The d -elliptic locus on M In this section, we prove Theorem 1.3. It suffices to compute the intersection of themorphism π / ,d : A dm / ,d → M with an arbitrary curve class in A ( M ) . It is possibleto simplify the computation by specializing to particular test curves and computing theseintersection numbers with these classes directly, but instead we explain a more abstractapproach that we will employ later for pointed genus 2 curves and in genus 3.It follows from [Mum83] that A ( M ) = 0 , and hence that any curve class on M comesfrom the boundary. In particular, any curve class may be represented as a rational linearcombination of morphisms from a smooth, connected scheme C of dimension 1 to one of thetwo boundary divisors:( ∆ ) C → M , → M ( ∆ ) C → M , × M , → M We may furthermore take C to be general, in the sense that C intersects any given finitecollection of subvarieties of its associated boundary divisor as generically as possible. Now,consider the intersection of such a C with the admissible locus π / ,d : A dm / ,d → M . Notethat we get a stratification of A dm / ,d by pulling back the stratification of M , by boundarystrata under the finite morphism ψ / ,d , see Lemma 2.3. Then, C may be chosen to avoid thezero-dimensional strata of A dm / ,d .Thus, when intersecting C with the admissible locus, we need only consider admissiblecovers f : X → Y where Y has at most one node; in fact, because X must be singular, Y must have exactly one node. We classify such covers in §4.1, then in §4.2 and §4.3 computethe contributions of the covers of each topological type to the intersection numbers.4.1. Classification of Admissible Covers.
Let f : X → Y be a point of A dm / ,d where Y has exactly one single node, and the stabilization X s of X lies in one of the boundarydivisors ∆ i . We consider the cases i = 0 , separately.4.1.1. [ Y ] ∈ ∆ . Let Y i be the component of Y of genus i , and let y = Y ∩ Y . By assumption,both Y i are smooth. The pre-image of Y i is then a union of smooth curves; we may ignorethe case where one of the components has genus 2, by the assumption on X s . Thus, f − ( Y ) either consists of a single genus 1 curve X , or two disjoint elliptic curves X , X (cid:48) .In both cases, the f must be unramified over y , and simply ramified over two points of Y . Thus, the pre-image of Y consists of smooth rational curves attached to the Y at thepre-images of y , all of which map isomorphically to Y except one, which has degree 2 over Y . In the case that f − ( Y ) has two components, the degree 2 component must be a bridge X and X (cid:48) in order for X to be connected.We thus get covers of two topological types, which we denote by (∆ , ∆ ) (Figure 9)and (∆ , ∆ ) (Figure 10); the coordinates are the boundary strata in which X s and Y lie,respectively. igure 9. Cover of type (∆ , ∆ ) Figure 10.
Cover of type (∆ , ∆ ) [ Y ] ∈ ∆ . Let [ f : X → Y ] ∈ A dm / ,d be an admissible cover, where Y is anirreducible nodal curve of genus 1. We have a diagram (cid:101) X ν X (cid:47) (cid:47) (cid:101) f (cid:15) (cid:15) X f (cid:15) (cid:15) P ν Y (cid:47) (cid:47) Y where the maps ν X , ν Y are normalizations. Let y ∈ Y denote the node, and let y (cid:48) , y (cid:48)(cid:48) ∈ P be its pre-images under ν . Let y , y , ∈ Y be the branch points of f ; by abuse of notation, wealso let y , y ∈ P denote their preimages under ν Y . Then, (cid:101) f is simply branched over y , y ,possibly branched over y (cid:48) , y (cid:48)(cid:48) , and unramified everywhere else.Let (cid:101) X , . . . , (cid:101) X n be the components of (cid:101) X , and let d i be the degree of (cid:101) X i over P . Let s i bethe total number of points of (cid:101) X i lying over y (cid:48) and y (cid:48)(cid:48) . Then, the number of points of f − ( y ) is t = (cid:80) s i . We have p a ( (cid:101) X ) ≥ − n . On the other hand, (cid:101) X is is the blowup of X at t nodes,so p a ( (cid:101) X ) = 2 − t ≥ − n , hence t ≤ n + 1 . On the other hand, s i ≥ for each i , so t ≥ n .Because t is an integer, the three possibilities for the s i (up to re-indexing) are: s i = 2 forall i (type (∆ , ∆ ) , Figure 11), s = 4 and s i = 2 for all i ≥ (type (∆ , ∆ ) , Figure 12),and s = s = 3 and s i = 2 for all i ≥ . In the last case, it is easy to check that X s will lie inone of the zero-dimensional boundary strata of M , so we may disregard covers of this type.Suppose f is a cover of type (∆ , ∆ ) . Then, X must consist of a smooth genus 1 component X attached at two points to a chain of m − rational curves, each of which maps to Y via x (cid:55)→ x a . (Note that a ≥ , as X has degree a over P .) The map f | X : X → P is totallyramified at the two nodes on X , and is simply ramified at two other points on X . We mayalso have m = 1 , in which case X is irreducible with a single node, and its normalization (cid:101) X = X maps to P as above.Finally, suppose f is a cover of type (∆ , ∆ ) . We have a single component X ⊂ X withfour points mapping to y ∈ Y ; all other components of X have two points mapping to y .As X is connected, we see that X must consist of two disjoint chains of curves X , . . . , X m − and X (cid:48) , . . . , X (cid:48) n − attached at two points to X . We allow one or both of m, n to be equal to ; in this case, X has a non-separating node. We first assume m, n > , in which case allcomponents of X have genus 0.Each of the components of X other than X is unramified over P away from the two nodes;thus, each X i → P is of the form x (cid:55)→ x a for some a (independent of i ), branched over y (cid:48) , y (cid:48)(cid:48) .Similarly each X (cid:48) j → P is of the form x (cid:55)→ x b for some b (independent of j ), branched over y (cid:48) , y (cid:48)(cid:48) .Now, X has degree a + b over P , and each of y (cid:48) , y (cid:48)(cid:48) has two points in its pre-image,of ramification indices a and b . By Riemann-Hurwitz, there are two additional simpleramification points on X mapping to y , y ∈ Y .The situation is similar when at least one of m, n is equal to 1: the chains of smoothrational curves attached to X are replaced with a non-separating node on X , and thenormalization of X maps to P as before. Figure 11.
Cover of type (∆ , ∆ ) Figure 12.
Cover of type (∆ , ∆ ) Intersection numbers: the case [ C ] ∈ ∆ . Suppose that we have a general curveclass C → M , × M , . In the Cartesian diagram A C (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,dπ / ,d (cid:15) (cid:15) C (cid:47) (cid:47) M , × M , (cid:47) (cid:47) M we wish to compute the degree of A C has a 0-cycle on M .Suppose d , d are positive integers satisfying d + d = d . We also form the diagram d ,d C (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,d × ∆ A dm / ,d (cid:47) (cid:47) (cid:15) (cid:15) M , (cid:15) (cid:15) A dm / ,d × A dm / ,d ψ / ,d × ψ / ,d (cid:47) (cid:47) π / ,d × π / ,d (cid:15) (cid:15) M , × M , C (cid:47) (cid:47) M , × M , where both squares are Cartesian. Lemma 4.1.
We have a bijection of sets A C (Spec( C )) ∼ = (cid:97) d + d = d A d ,d C (Spec( C )) In particular, the groupoids of geometric points of A C and A d ,d C are in fact sets, i.e., haveno non-trivial automorphisms.Proof. By §4.1, a geometric point of A C consists of a point x ∈ C and a cover f : X → Y oftype (∆ , ∆ ) , along with the data of an isomorphism of X s with the curve corresponding tothe image of x in M . It is clear that this data has no non-trivial automorphisms.From f , we may associate an ordered pair of covers of the same elliptic curve, whosedegrees are integers a, b satisfying d + d = d . We thus get a geometric point of A d ,d C , andit is again easy to see that such points have no non-trivial automorphisms.The construction of the inverse map is clear: we need only note that C may be chosento avoid the point ([ E ] , [ E ]) ∈ M , × M , , where E denotes a singular curve of genus 1;thus, all covers in A d ,d C are covers of (smooth) elliptic curves. (cid:3) Lemma 4.2.
The intersection multiplicity at all points of A d ,d C is 1, and the intersectionmultiplicity at all points of A C is 2.Proof. Analytically locally near a point of A d ,d C , the map A dm / ,d × ∆ A dm / ,d → M , ×M , is the inclusion of a smooth curve in a smooth surface, by Corollary 3.2 and thesmoothness of M , . Thus, C ⊂ M , × M , may be chosen to intersect A dm / , ( d ,d ) transversely.Now, let f : X → Y be an admissible cover of type (∆ , ∆ ) . The complete local ringof A dm / ,d at [ f ] is isomorphic to C [[ s, t ]] , where t is a smoothing parameter for the nodeof Y , and the quotient C [[ s, t ]] → C [[ s ]] corresponds to the universal deformation of theelliptic component of Y . Let C [[ x, y, z ]] be the complete local ring of M at [ X s ] , where z isa smoothing parameter for the node of X s , and the variables x, y are deformation parametersfor the two elliptic components.Consider the induced map T : C [[ x, y, z ]] → C [[ s, t ]] . We have the following, up to harmlessrenormalizations of the coordinates: • T ( z ) ≡ t mod ( s , st, t ) . To see this, consider any 1-parameter deformation of Y that smooths the node to first order. The corresponding deformation of f smoothsthe nodes of X to first order, and the induced deformation of X s is obtained bycontracting the rational bridge of X in the total space, introducing an ordinarydouble point. It follows that the node of X s is smoothed to order 2. T ( x ) ≡ T ( y ) ≡ s mod ( t, s ) . Indeed, varying the elliptic component of Y to firstorder varies the elliptic components of X to first order.The map on complete local rings at [ X s ] induced by C (cid:55)→ M , × M , → M is of theform C [[ x, y, z ]] (cid:55)→ C [[ x, y ]] (cid:55)→ C [[ u ]] , where x, y map to power series with generic linearleading terms. It is straightforward to check that the complete local ring of A C at ([ f ] , [ X s ]) is isomorphic to C [ t ] / ( t ) . (cid:3) Proposition 4.3.
The degree of A C as a 0-cycle on M is: (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) (cid:90) M , ×M , [ C ] · [∆] where [∆] = [ p × M , ] + [ M , × p ] is the class of the diagonal in M , × M , , and p is theclass of a geometric point in M , .Proof. By the previous two lemmas, it suffices to show that the degree of A d ,d C is: σ ( d ) σ ( d ) (cid:90) M , ×M , [ C ] · [∆] Let E be any elliptic curve. Writing [∆] = [ p × M , ] + [ M , × p ] , where we take p to bethe class of [ E ] ∈ M , , the contribution of the first summand to A dm / ,d × ∆ A dm / ,d is A dm / ,d ( E ) × A dm / ,d , where the first factor is the moduli space of degree d coversof the fixed curve E . By §3.1, this class pushes forward to σ ( d ) σ ( d )[ p × M , ] on M , ×M , . Similarly, the second summand gives the class σ ( d ) σ ( d )[ M , × p ] , so adding thesecontributions and intersecting with C yields the result. (cid:3) Intersection numbers: the case [ C ] ∈ ∆ . Given C → M , general, we wish tocompute the degree of A C , as defined below: A C (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,dπ / ,d (cid:15) (cid:15) C (cid:47) (cid:47) M , (cid:47) (cid:47) M By §4.1, we have three topological types of covers contributing to A C : type (∆ , ∆ ) , type (∆ , ∆ ) , and type (∆ , ∆ ) . We now consider each contribution separately.4.3.1. Contribution from type (∆ , ∆ ) . Consider the Cartesian diagram A (∆ , ∆ ) C (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,d, π / ,d, (cid:15) (cid:15) C (cid:47) (cid:47) M , where A dm / ,d, and its forgetful map to π / ,d, : A dm / ,d, → M , are as defined in §3.2. Proposition 4.4.
The contribution to A C from covers of type (∆ , ∆ ) is: (cid:90) M , [ C ] · [ π / ,d, ] . roof. It is easy to check that A (∆ , ∆ ) C (Spec( C )) is a set, and is isomorphic to the subgroupoidof A C (Spec( C )) consisting of covers of type (∆ , ∆ ) . If C is general, it intersects π / ,d, transversely (see, for example, the proof of Proposition 3.4) on M , . On the other hand,an argument analogous to the proof of Lemma 4.2 shows that a general C intersects π / ,d with multiplicity 2 at any cover f : X → Y of type (∆ , ∆ ) , due to the contraction of therational bridge of X after applying π / ,d . (cid:3) Contribution from type (∆ , ∆ ) . Fix positive integers a, m satisfying am = d . Considerthe Cartesian diagram A (∆ , ∆ ) ,aC (cid:47) (cid:47) (cid:15) (cid:15) A dm a,a, , / ,aπ a,a, , / ,a (cid:15) (cid:15) C (cid:47) (cid:47) M , where A dm a,a, , / ,a is as defined in §3.3, and the map π a,a, , / ,a : A dm a,a, , / ,a → M , remembersthe (stabilized) source curve with the two total ramification points. Thus, the points of A (∆ , ∆ ) ,aC record the main data of a cover of type (∆ , ∆ ) , namely the restriction to thegenus 1 component, which is a cover of P totally ramified at two points. For a general C , allpoints of A (∆ , ∆ ) ,aC correspond to covers of smooth curves. Proposition 4.5.
The contribution to A C from covers of type (∆ , ∆ ) is: (cid:88) am = d (cid:32) m (cid:90) M , [ C ] · [ π a,a, , / ,a ] (cid:33) . Proof.
It is easy to check that the geometric points of A C (Spec( C )) are in bijection with thegeometric points of A (∆ , ∆ ) ,aC , where a ranges over the positive integer factors of d . However,a cover f : X → Y of type (∆ , ∆ ) has automorphism group of order a m − , as the groupof a -th roots of unity acts on each rational component of X . On the other hand, each A (∆ , ∆ ) ,aC (Spec( C )) is a set. If C is general, it intersects π a,a, , / ,a transversely (see, for example,the proof of Proposition 3.8).It now suffices to show that the intersection multiplicity (on the level of complete localrings) of C with π / ,d at a cover f : X → Y of type (∆ , ∆ ) is ma m − ; after dividing by theorder of the automorphism group, we get the factor of m in the statement, and the conclusionfollows.By Proposition 2.4, the complete local ring at [ f ] is C [[ s, t, t , . . . , t m ]] / ( t = t a = · · · = t am ) , which is canonically a C [[ s, t ]] -algebra via ψ / ,d : A dm / ,d → M , . Here, t is a smoothingparameter for the node of Y and t i is a smoothing parameter for the node x i ∈ X . Thequotient C [[ s ]] corresponds to the deformation of the target that moves the marked points y , y ∈ Y apart.Let C [[ x, y, z ]] be the complete local ring of M at [ X s ] , where the coordinates are chosenas follows. The coordinate z is the smoothing parameter for the node, so that C [[ x, y ]] is thedeformation space of the marked normalization ( X , x , x m ) of X s (the points x , x m are the odes along X ⊂ X ). Then, y is the deformation parameter of the elliptic curve ( X , x ) ,and the quotient C [[ x ]] corresponds to the deformation of X moving x and x m apart.Consider the induced map on complete local rings T : C [[ x, y, z ]] → C [[ s, t, t , . . . , t m ]] / ( t = t a = · · · = t am ) , We have the following, up to harmless renormalizations of the coordinates: • T ( z ) ≡ t m mod ( s, t − t , . . . , t − t n ) . Indeed, in the 1-parameter deformation of f smoothing the nodes of X to first order, consider the total space of the associateddeformation of X . Contracting the rational components of X produces an A m -singularity, so the node of X s is smoothed to order m in its induced deformation. • T ( z ) ≡ t i ) , as a deformation that keeps the node x i in X also keeps thenode in X s . • T ( y ) ≡ s mod ( t , . . . , t n ) . The content here is that first-order deformation of [ f ] that moves y .y apart varies the elliptic curve ( X , x ) to first order. Indeed, themap π a,a, , / ,a is unramified over a general point of of M , , and ψ a,a, , / ,a is unramifiedover M , , so we have the claim for C general.The first two claims imply that T ( z ) = ( t · · · t m ) u , where u is a unit. Then, it isstraightforward to check that the complete local ring of A C at ([ f ] , [ X s ]) is C [ t, t , . . . , t m ] / ( t − t ai , t · · · t m ) , which has dimension ma m − as a C -vector space (for example, a basis is given by monomials t e t e · · · t e m m , where ≤ e ≤ m − and ≤ e i ≤ a − ). This completes the proof. (cid:3) Contribution from type (∆ , ∆ ) . Lemma 4.6.
Let x , x , x , x ∈ P be four distinct points, and let a, b ≥ be integers. Then,up to scaling on the target, there is a unique cover g : P → P of degree a + b such that f has zeroes of orders a, b at x , x , respectively, and poles of orders a, b at x , x , respectively.If the x i are general, then g is simply ramified over two other distinct points.Proof. The unique such map is the meromorphic function g ( x ) = ( x − x ) a ( x − x ) b ( x − x ) a ( x − x ) b . The second half of the statement follows from Riemann-Hurwitz and a dimension count, asthere are finitely many covers of P branched over 3 points. (cid:3) Consider the Cartesian diagram A (∆ , ∆ ) C (cid:47) (cid:47) (cid:15) (cid:15) M , (cid:15) (cid:15) C (cid:47) (cid:47) M , where the map M , → M , glues the third and fourth marked points; its class is ∆ ∈ A ( M , ) . For generic C , the points of A (∆ , ∆ ) C correspond to the points on C whose imagesin M , are irreducible nodal curves. roposition 4.7. The contribution to A C from covers of type (∆ , ∆ ) is: (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) (cid:90) M , [ C ] · ∆ Proof.
It is clear that A (∆ , ∆ ) C (Spec( C )) is a set. Given positive integers a, b, m, n satisfying am + bn = d , a geometric point of A (∆ , ∆ ) C gives rise to a unique cover g as in Lemma 4.6,which, for general C , will be simply ramified over two points distinct from each other andfrom , ∞ . Then, g gives rise to two admissible covers of type (∆ , ∆ ) , distinguished bythe labelling of the two simple ramification points; all such covers are obtained (uniquely) inthis way.Each cover f : X → Y of type (∆ , ∆ ) has automorphism group of order a m − b n − ,coming from the actions of roots of unity on the rational components of X . Now, considerthe complete local rings of A C at ([ f ] , X s ) .As in the proof of Proposition 4.5, consider the map on complete local rings T : C [[ x, y, z ]] → C [[ s, t, t , . . . , t m , u , . . . , u n ]] / ( t − t ai , t − u bj ) , induced by π / ,d . We take y to be the smoothing parameter of the node obtained from thegluing map M , → M , z to be that coming from the map M , → M , , and x such thatthe quotient C [[ x ]] corresponds to the deformation of X s along M , . We verify that, up torenormalizing coordinates, • T ( x ) = s mod ( t , . . . , t m , u , . . . , u n ) , • T ( y ) = ( t . . . t m ) v , and • T ( z ) = ( u . . . u n ) v (cid:48) ,where v, v (cid:48) are units. The complete local ring of A (∆ , ∆ ) C at [ f ] is thus C [ t, t , . . . , t m , u , . . . , u n ] / ( t − t ai , t − u bj , u · · · u n ) , which has a C -basis given by monomials t e ( t e · · · t e m − m − )( u f · · · u f n n ) , where ≤ e ≤ m − , ≤ e i ≤ a − , and ≤ f j ≤ b − . The total contribution to A C of each f is therefore ( ma m − b n ) / ( a m − b n − ) = mb .Summing over all a, m, b, n , we obtain that each point of intersection of C and ∆ contributes (cid:32) (cid:88) am + bn = d mb (cid:33) = 2 (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) to A C . (cid:3) The class of the admissible locus.
We are now ready to compute the class of the d -elliptic locus in genus 2, that of π / ,d : A dm / ,d → M . Proposition 4.8.
We have: (cid:90) M [ π / ,d ] · ∆ = 4( d − σ ( d ) (cid:90) M [ π / ,d ] · ∆ = 2 (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) roof. Note first that the formulas of Propositions 4.3, 4.4, 4.5, and 4.7 hold for any curveclass defined on the relevant boundary divisor, as such a class may be written as a linearcombination of general curves, and the formulas are all linear in [ C ] .We first take ∆ as the pushforward of [ p × M , ] ∈ A ( M , × M , ) to M . Wehave (cid:82) M , ×M , ∆ · [∆] = 1 , so the formula for (cid:82) M [ π / ,d ] · ∆ follows immediately fromProposition 4.3.As a check, ∆ may also be expressed as the pushforward of ∆ ∈ A ( M , ) to M , so wecan apply Propositions 4.4, 4.5, and 4.7. Combining these with Proposition 3.4, Proposition3.8, and §2.2.1, respectively, we find that the first two contributions are zero, and the third is (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) , so we obtain the same result.Finally, ∆ is the pushforward of ∆ ∈ A ( M , ) to M . Applying the same formulas asabove, we get a contribution of d − σ ( d ) in type (∆ , ∆ ) , a contribution of (cid:88) am = d a − m = 2( d − σ ( d ) in type (∆ , ∆ ) (where we have applied the projection formula for the forgetful morphism M , → M , , under which ∆ pulls back to ∆ ), and zero in type (∆ , ∆ ) . (cid:3) Proof of Theorem 1.3.
Immediate from of §2.2.3 and the fact that δ i = ∆ i for i = 0 , , alongwith the identity (cid:88) d + d = d σ ( d ) σ ( d ) = (cid:18) − d + 112 (cid:19) σ ( d ) + 512 σ ( d ) , see §2.4. (cid:3) Variants in genus 2
Covers of a fixed elliptic curve.
Fix a general elliptic curve E ; we consider genus 2curves covering E . Define the space of such covers A dm / ,d ( E ) by the Cartesian diagram A dm / ,d ( E ) (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,d (cid:15) (cid:15) [ E ] (cid:47) (cid:47) M , , where the map A dm / ,d → M , is the composition of ψ / ,d : A dm / ,d → M , with themap forgetting the second marked point.By post-composing with π / ,d , we get a map π / ,d ( E ) : A dm / ,d ( E ) → M ; we wishto compute its class in A ( M ) . We do so by intersecting with the boundary divisors ∆ and ∆ . If f : X → Y is an admissible cover appearing in one of these intersections, then Y = E ∪ P , where both components are attached at a node and both marked points are onthe rational component, and f is of type (∆ , ∆ ) or (∆ , ∆ ) . roposition 5.1. We have (cid:90) M π / ,d ( E ) · ∆ = ( d − σ ( d ) (cid:90) M π / ,d ( E ) · ∆ = 2 (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) Proof.
The points of intersection of π / ,d ( E ) and the gluing morphism M , → M consistof admissible covers of type (∆ , ∆ ) with target Y , which correspond to isogenies E (cid:48) → E with a second marked point in the kernel. However, note that each intersection is countedtwice in this way, as ( E (cid:48) , , x ) and ( E (cid:48) , , − x ) correspond to the same point of M , (if x is2-torsion, then multiplication by − defines an automorphism of the cover of order 2). As inProposition 4.4, each cover appears with multiplicity 2, so the first claim follows.The points of intersection of π / ,d ( E ) and the gluing morphism M , × M , → M consist of admissible covers of type (∆ , ∆ ) with target Y , which correspond to orderedpairs of isogenies E → E, E → E . As in Lemma 4.2, each such admissible cover appearswith multiplicity 2, and it is easy to check that the geometric points have no non-trivialautomorphisms. We are now done by Lemma 3.1. (cid:3) Using the fact that δ = ∆ and δ = ∆ , and applying again the identity (cid:88) d + d = d σ ( d ) σ ( d ) = (cid:18) − d + 112 (cid:19) σ ( d ) + 512 σ ( d ) we conclude: Theorem 5.2.
The class of π / ,d ( E ) in A ( M ) is: (cid:18)(cid:18) − d + 25 (cid:19) σ ( d ) + 4 σ ( d ) (cid:19) δ + (cid:18)(cid:18) − d − (cid:19) σ ( d ) + 4 σ ( d ) (cid:19) δ . Interlude: quasimodularity for correspondences.Theorem 5.3.
Consider the correspondence ( π / ,d ) ∗ ◦ ψ ∗ / ,d : A ∗ ( M , ) → A ∗ ( M ) . Then, for a fixed α ∈ A ∗ ( M , ) , we have ( π / ,d ) ∗ ◦ ψ ∗ / ,d ( α ) ∈ A ∗ ( M ) ⊗ Qmod . Proof.
It suffices to check the claim on a basis of A ∗ ( M , ) . Note that all classes of geometricpoints on M , or M are rationally equivalent, as both spaces are unirational. When α is the class of a point, we have Theorem 1.1. When α is the fundamental class, the claimfollows from Theorem 1.2 (in genus 2). When α = ∆ , we may replace the locus of covers ofa nodal curve of genus 1 with that of covers of a fixed smooth curve, in which case we aredone by Theorem 5.2.It remains to consider α = ∆ . Consider the intersection of a general divisor D → M withthe cycle ( π / ,d ) ∗ ◦ ψ ∗ / ,d ∆ . The contribution from covers of type (∆ , ∆ ) to the intersectionof D with the admissible locus is the intersection of D with the pushforward of π / ,d, to M , which is quasimodular by Proposition 3.4. The contribution from type (∆ , ∆ ) is the ntersection of D with A dm / ,d × ∆ A dm / ,d , where d , d range over integers satisfying d + d = d ; this is also quasimodular by Corollary 3.2. (cid:3) The d -elliptic locus on M , . Here, we compute the class in A ( M , ) of the mor-phism π / ,d : A dm / ,d → M , , whose image is the closure of the locus of pointed curves ( C, p ) admitting a degree d cover of an elliptic curve, ramified at p . We do so by intersectingwith test surfaces, following the same method as in §4. Because A ( M , ) = 0 (see §2.2.4), itsuffices to consider test surfaces in the boundary of M , , which is the union of the boundarydivisors (∆ ) M , → M , (∆ ) M , × M , → M , Here, the map M , → M , glues together the second and third marked points, and themap M , × M , → M , glues the second marked point on the first component to themarked point of the second. As in the unpointed case, a general surface S mapping to one ofthese boundary classes will intersect the admissible locus at covers of one of the four typesdescribed in §4.1.5.3.1. The case [ S ] ∈ ∆ . All admissible covers f : X → Y in the intersection of S →M , × M , → M , and π / ,d : Adm / ,d → M , have type (∆ , ∆ ) . Let s : M , → M , be the map attaching a 2-pointed rational curve to an elliptic curve at its origin. Then, the1-pointed curve X is obtained by gluing a point of M , in the image of s , at its first markedpoint, to a point of M , , at its origin.For integers d , d satisfying d + d = d , we have a diagram A d ,d S (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,d × ∆ A dm / ,d (cid:47) (cid:47) (cid:15) (cid:15) M , (cid:15) (cid:15) A dm / ,d × A dm / ,d π / ,d × π / ,d (cid:47) (cid:47) M , × M , s × id (cid:15) (cid:15) S (cid:47) (cid:47) M , × M , where both squares are Cartesian. Taking the union over all possible ( d , d ) , the geometricpoints of A d ,d S are in bijection with those of the intersection of A dm / ,d and S , and thereare no non-trivial automorphisms on either side. In the map π / ,d : A dm / ,d → M , , therational bridge of X in a cover f : X → Y of type (∆ , ∆ ) does not get contracted, so infact the argument of Lemma 4.2 shows that both intersections are transverse for general S .Using the fact that ∆ = [ p × M , ] + [ M , × p ] and applying Corollary 3.2, we find, as inProposition 4.3: Proposition 5.4.
For S → M , × M , , we have: (cid:90) M , [ S ] · [ π / ,d ] = (cid:32)(cid:90) M , ×M , ([ p × M , ] + [∆ × p ]) · [ S ] (cid:33) · (cid:88) d + d = d σ ( d ) σ ( d ) . .3.2. The case [ S ] ∈ ∆ . In the intersection A S (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,dπ / ,d (cid:15) (cid:15) S (cid:47) (cid:47) M , we have contributions from covers of types (∆ , ∆ ) , (∆ , ∆ ) , and (∆ , ∆ ) .First, consider a cover f : X → Y of type (∆ , ∆ ) . The 1-pointed curve X may beobtained by identifying the points x , x of [( X , x , x , x )] ∈ M , lying in either of theboundary divisors ∆ , { , } , ∆ , { , } . Let s { , } , s { , } : M , → M , be the maps definingthese two boundary divisors, we have the Cartesian diagrams A (∆ , ∆ ) , { , } S (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,d, π / ,d, (cid:15) (cid:15) M , s { , } (cid:15) (cid:15) S (cid:47) (cid:47) M , A (∆ , ∆ ) , { , } S (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,d, π / ,d, (cid:15) (cid:15) M , s { , } (cid:15) (cid:15) S (cid:47) (cid:47) M , Following the proof of Proposition 4.4, both intersections above are seen to be transverse (notethat we no longer contract the rational bridge of X ) and have no non-trivial automorphisms.We conclude: Proposition 5.5.
The contribution to A S from covers of type (∆ , ∆ ) is: (cid:90) M , [ S ] · (cid:0) [ s { , } ◦ π / ,d, ] + [ s { , } ◦ π / ,d, ] (cid:1) The analysis of covers of type (∆ , ∆ ) is essentially identical to that of Proposition 4.5.We find: Proposition 5.6.
The contribution to A S from covers of type (∆ , ∆ ) is: (cid:88) am = d (cid:32) m (cid:90) M , [ S ] · [ π a,a, , / ,a ] (cid:33) . Finally, consider covers of type (∆ , ∆ ) . Fix a, b, m, n with am + bn = d , and let A dm ( a,b ) , ( a,b ) , , / ,a + b be the space of tuples ( f, x , . . . , x ) , where f : X → Y is a degree a + b admissible cover of genus 0 curves, with six marked points x , . . . , x ∈ X such that f ( x ) = f ( x ) , f ( x ) = f ( x ) , and the ramification indices at x , . . . , x are a, b, a, b, , , respectively.As usual, there is a canonical morphism π ( a,b ) , ( a,b ) , , / ,a + b : A dm ( a,b ) , ( a,b ) , , / ,a + b → M , rememberingthe pointed source curve. Let r : M , → M , be the map sending [( X, x , . . . , x )] (cid:55)→ [( X/ ( x ∼ x ) , x , x , x ) s ] We have a Cartesian diagram (∆ , ∆ ) S (cid:47) (cid:47) (cid:15) (cid:15) A dm ( a,b ) , ( a,b ) , , / ,a + bπ ( a,b ) , ( a,b ) , , / ,a + b (cid:15) (cid:15) M , r (cid:15) (cid:15) S (cid:47) (cid:47) M , It is routine to check, following the proof of Proposition 4.7:
Proposition 5.7.
The contribution to A S from covers of type (∆ , ∆ ) is (cid:88) am + bn = d (cid:32) mb (cid:90) M , [ S ] · [ r ◦ π ( a,b ) , ( a,b ) , , / ,a + b ] (cid:33) . In order to complete the calculation, we will need to compute the class [ r ◦ π ( a,b ) , ( a,b ) , , / ,a + b ] ∈ A ( M , ) . Proposition 5.8.
We have: (cid:90) M , [ r ◦ π ( a,b ) , ( a,b ) , , / ,a + b ] · ∆ = 0 (cid:90) M , [ r ◦ π ( a,b ) , ( a,b ) , , / ,a + b ] · ∆ ,S = 1 for S = { , } , { , , } (cid:90) M , [ r ◦ π ( a,b ) , ( a,b ) , , / ,a + b ] · ∆ ,S = 0 for S = { , } , { , } Proof.
For the first statement, we may replace ∆ with the locus of pointed curves with a fixedunderlying elliptic curve ( E, x ) , which clearly has empty intersection with [ r ◦ π ( a,b ) , ( a,b ) , , / ,a + b ] .Now, consider a cover in A dm ( a,b ) , ( a,b ) , , / ,a + b whose image in M , has a non-separating node.We claim that the only such cover, up to isomorphism, is constructed as follows. Let Y bethe union of two copies Y , Y of P , attached at a node, with two marked points on eachcomponent. Then, X contains two copies of P mapping to Y via the maps x (cid:55)→ x a and x (cid:55)→ x b , respectively. These two components are connected by a copy of P mapping to Y via x (cid:55)→ x , and the rest of the components of X map to Y isomorphically.The cover f : X → Y constructed above gives a single point of intersection of the admissiblelocus with ∆ , { , } and ∆ , { , , } ; it is now standard to check that the multiplicity is 1. (cid:3) Final computation.
We are now ready to intersect [ π / ,d ] ∈ A ( M , ) with the bound-ary test surfaces. roposition 5.9. We have: (cid:90) M , [ π / ,d ] · ∆ = 4( d − σ ( d ) (cid:90) M , [ π / ,d ] · ∆ a = (cid:88) d + d = d σ ( d ) σ ( d ) (cid:90) M , [ π / ,d ] · ∆ b = (cid:88) d + d = d σ ( d ) σ ( d ) (cid:90) M , [ π / ,d ] · Ξ = −
124 ( d − σ ( d ) (cid:90) M , [ π / ,d ] · ∆ = − (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) Proof.
First, we consider the classes ∆ a , ∆ b , and ∆ contained in M , × M , ; these arethe push-forwards of the boundary divisors M , × p , ∆ × M , , and ∆ × M , , respectively.Thus, (cid:90) M , ×M , ([ p × M , ] + [∆ × p ]) · [∆ a ] = 1 (cid:90) M , ×M , ([ p × M , ] + [∆ × p ]) · [∆ b ] = 1 (cid:90) M , ×M , ([ p × M , ] + [∆ × p ]) · [∆ ] = − , applying §2.2.1. The formulas for the intersections of π / ,d with these three classes nowfollow from Proposition 5.4.Now, we consider the classes ∆ , ∆ a , ∆ b , Ξ contained in M , ; these are the push-forwards of the boundary divisors ∆ , ∆ , { , } , ∆ , { , , } , and ∆ , { , } , respectively. (We haveincluded the middle two classes, which arose earlier, as a check.)By Propositions 3.8 and 5.6, the only class for which covers of type (∆ , ∆ ) contribute is ∆ , in which we get a contribution of d − σ ( d ) . By Propositions 5.7 and 5.8, coversof type (∆ , ∆ ) contribute (cid:80) d + d = d σ ( d ) σ ( d ) to each of ∆ a , ∆ b and nothing to theothers.Finally, consider covers of type (∆ , ∆ ) . By Corollary 3.5, we have [ s { , } ◦ π / ,d, ] = ( d − σ ( d ) (cid:18)
124 ∆ , { , } + ∆ , { , } (cid:19) [ s { , } ◦ π / ,d, ] = ( d − σ ( d ) (cid:18)
124 ∆ , { , } + ∆ , { , } (cid:19) Applying Proposition 5.5 and §2.2.2, we get no contributions to ∆ a and ∆ b , a contributionof d − σ ( d ) to ∆ , and a contribution of − ( d − σ ( d ) to Ξ . Combining all of theabove yields the needed intersection numbers. (cid:3) heorem 5.10. The class of π / ,d in A ( M , ) is: (cid:18) − dσ ( d ) + 112 σ ( d ) (cid:19) δ + (cid:18) σ ( d ) − σ ( d ) (cid:19) δ a + (cid:18)(cid:18) − d − (cid:19) σ ( d ) + 1312 σ ( d ) (cid:19) δ b + (2 σ ( d ) − dσ ( d )) ξ + (4 σ ( d ) − σ ( d )) δ . In particular, (cid:88) d ≥ [ π / ,d ] q d ∈ Qmod ⊗ A ( M , ) . Proof.
Immediate from §2.2.4; note that the dual graphs associated to all five boundaryclasses have automorphism group of order 2, except ∆ , which has automorphism group oforder 8. (cid:3) The classes δ , δ a , δ b push forward to zero on M , and the classes ξ , δ push forwardto δ , δ , respectively. Thus, Theorem 5.10 recovers Theorem 1.3. In addition, taking d = 2 in Theorem 5.10 recovers [vZ18a, Proposition 3.2.9].6. The d -elliptic locus on M We now carry out the methods developed earlier and use the results above to compute the(unpointed) d -elliptic locus in genus 3, [ π / ,d ] ∈ A ( M ) . As A ( M ) = 0 [Fab90, Theorem1.9], any test surface lies in one of the two boundary divisors: (∆ ) M , → M (∆ ) M , × M , → M Classification of Admissible Covers.
We will compute the intersection of the ad-missible locus with a general test surface S in one of the two boundary divisors. The samearguments from before show that we need only consider the codimension 1 strata in A dm / ,d ,parametrizing covers whose targets have exactly one node. Moreover, the same dimensioncount shows that we may disregard the strata whose images in M have dimension 2 or less,or equivalently whose general fiber under the map π / ,d is positive-dimensional.A similar analysis as in genus 2 yields seven topological types of covers in A dm / ,d thatthat give nonzero contributions to general test surfaces, shown in Figures 13, 14, 15, 16, 17,18, 19.6.2. The case [ S ] ∈ ∆ . Define the intersection A S by the Cartesian diagram A S (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,dπ / ,d (cid:15) (cid:15) S (cid:47) (cid:47) M , × M , (cid:47) (cid:47) M We consider the contributions to A S from covers of the three possible types: (∆ , ∆ , ) , (∆ , ∆ , ) , and (∆ , ∆ , ) . igure 13. Cover of type (∆ , ∆ , ) Figure 14.
Cover of type (∆ , ∆ , ) Figure 15.
Cover of type (∆ , ∆ , ) Figure 16.
Cover of type (∆ , ∆ , ) Figure 17.
Cover of type (∆ , ∆ ) Figure 18.
Cover of type (∆ , ∆ ) igure 19. Cover of type (∆ , ∆ ) Type (∆ , ∆ , ) . For any d ≥ , we define the space A dm / ,d as a functor by theCartesian diagram A dm / ,d (cid:47) (cid:47) π / ,d (cid:15) (cid:15) A dm / ,dπ / ,d (cid:15) (cid:15) M , u (cid:47) (cid:47) M Here, u : M , → M is the map forgetting the marked point. Generically, A dm / ,d parametrizes covers f : X → Y along with an arbitrary point of X . Define the map ψ / ,d bythe composition A dm / ,d → A dm / ,d → M , → M , where the middle map is ψ / ,d and the last map forgets the second point.As in §4.2, we have a diagram A (∆ , ∆ , ) , ( d ,d ) S (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,d × ∆ A dm / ,d (cid:47) (cid:47) (cid:15) (cid:15) M , (cid:15) (cid:15) A dm / ,d × A dm / ,d ψ / ,d × ψ / ,d (cid:47) (cid:47) π / ,d × π / ,d (cid:15) (cid:15) M , × M , S (cid:47) (cid:47) M , × M , where both squares are Cartesian. A geometric point of A S corresponding to a cover f : X → Y of type gives rise to a geometric point of A (∆ , ∆ , ) , ( d ,d ) S in an obvious way. Note,however, that the image of [ f ] in M , × M , is of the form ([ Y , q ] , [ Y , y ]) , where Y ⊂ Y isthe elliptic component, q ∈ Y is one of the branch points of f , and y ∈ Y is the node. While q (cid:54) = y , [( Y , q )] and [( Y , y )] are isomorphic via translation.Owing to the contraction of the rational bridge of any admissible cover of type (∆ , ∆ , ) ,each cover f of type (∆ , ∆ , ) appears with multiplicity 2 in A S . In addition, each point (∆ , ∆ , ) , ( d ,d ) S comes from (cid:0) (cid:1) = 6 points of A S , due to the possible labelings of the branchpoints of the d -elliptic map.Using the fact that ∆ = [ p × M , ] + [ M , × p ] , and applying the projection formula, wefind: Proposition 6.1.
The contribution to A S from covers of type (∆ , ∆ , ) is: (cid:32)(cid:90) M ×M , u ∗ ([ S ]) · (cid:32) (cid:88) d + d = d σ ( d ) (cid:0) [[ π / ,d ( E )] × M , ] + [[ π / ,d ] × p ] (cid:1)(cid:33)(cid:33) Type (∆ , ∆ , ) . We have a Cartesian diagram A (∆ , ∆ , ) S (cid:47) (cid:47) (cid:15) (cid:15) A dm / ,dπ / ,d (cid:15) (cid:15) S (cid:47) (cid:47) M , × M , (cid:47) (cid:47) M , where pr : M , × M , → M , is the projection. Given a point ([( C, q )] , [( E, p )]) of M , × M , , the curve C ∪ p ∼ q E is the image under π / ,d of a cover of type (∆ , ∆ , ) ifand only if there exists a d -elliptic map g : C → E ramified at q , in which we may glue g to the unique double cover E → P ramified at the origin (and attach additional rationaltails) to form an admissible cover whose source contracts to C ∪ p ∼ q E . The transversality isstraightforward, and we find that: Proposition 6.2.
The contribution to A S from covers of type (∆ , ∆ , ) is: (cid:32)(cid:90) M , pr ∗ ([ S ]) · [ π / ,d ] (cid:33) The factor of 24 comes from the ways to label the ramification points.6.2.3. Type (∆ , ∆ , ) . For d , d with d + d = d , we have a diagram A (∆ , ∆ , ) S (cid:47) (cid:47) (cid:15) (cid:15) M , × A dm / ,d × ∆ A dm / ,d (cid:15) (cid:15) (cid:47) (cid:47) M , × A dm / ,d × A dm / ,d id × π / ,d × π / ,d (cid:15) (cid:15) M , × M , × ∆ (cid:47) (cid:47) M , × M , × M , ξ × id (cid:15) (cid:15) S (cid:47) (cid:47) M , × M , where ξ : M , × M , → M , is the map defining the boundary divisor ∆ in M , , andboth squares above are Cartesian. Proposition 6.3.
The contribution to A S from covers of type (∆ , ∆ , ) is: (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) · (cid:32)(cid:90) M , ×M , [ S ] · ([∆ a × M , ] + [∆ × p ]) (cid:33) roof. Given covers E → E and E → E of degrees d , d , respectively, and a 2-pointedcurve ( E (cid:48) , p , p ) of genus 1, we construct an admissible cover of type (∆ , ∆ , ) by attaching E (cid:48) the E i at their origins along the p i , mapping E (cid:48) → P via the complete linear series |O ( p + p ) | , and labelling the ramification points in one of
4! = 24 ways. The transversalityis straightforward. Decomposing the class of ∆ as usual, we get the desired result; here theclass ∆ a ∈ A ( M , ) arises as the pushforward of M , × p under ξ . (cid:3) The case [ S ] ∈ ∆ . The only such S we will need is defined as follows. Let C be a general curve of genus 2, and take S = C × C . The map S (cid:55)→ M , is defined by ( x, y ) (cid:55)→ [( C, x, y )] , where if x = y , we interpret the image as the reducible curve with a2-pointed rational curve attached at x . Clearly, covers of types (∆ , ∆ ) and (∆ , ∆ ) do not appear along S , so we do not give a general formula for contributions from suchcovers. Moreover, if C is general, then it is not d -elliptic, so we also do not see covers of type (∆ , ∆ , ) . Therefore, the only contributions to the intersection of S with the admissiblelocus come from covers of type (∆ , ∆ ) . Proposition 6.4.
We have (cid:90) M [ C × C ] · [ π / ,d ] = 48( dσ ( d ) − σ ( d )) . Proof.
This amounts to enumerating covers of type (∆ , ∆ ) where the genus 2 componentin the source is isomorphic to C ; the result is then immediate from Proposition 3.9 and alocal computation identical to that of Proposition 4.5. Indeed, we have (cid:88) am = d a − m = 48 d (cid:32) (cid:88) am = d a (cid:33) − (cid:32) (cid:88) am = d m (cid:33) = 48( dσ ( d ) − σ ( d )) . (cid:3) In the final computation, we will use the following:
Proposition 6.5.
We have [ C × C ] = 2(∆ (1) + ∆ (4) ) in A ( M ) , where the classes on theright hand side are defined as in §2.2.5.Proof. Because any two geometric points of M are rationally equivalent, we may replacethe general genus 2 curve C by the reducible genus 2 curve C obtained by gluing twonodal curves of arithmetic genus 1 together at a separating node. Then, the space M C , parametrizing two points on C (that is, the fiber over [ C ] of the forgetful map M , → M )has four components, corresponding to the choices of components of C on which the markedpoints can lie. The two components of M C , for which the marked points lie on the samecomponent of C each contribute ∆ (1) to the class of M C , , and the two components forwhich the marked points lie on opposite components each contribute ∆ (4) . (cid:3) The class of the admissible locus. roposition 6.6. We have: (cid:90) M [ π / ,d ] · ∆ (1) = 96( d − σ ( d ) (cid:90) M [ π / ,d ] · ∆ (4) = 24( dσ ( d ) − σ ( d )) − d − σ ( d ) (cid:90) M [ π / ,d ] · ∆ (5) = 24 (cid:32) (cid:88) d + d = d (2 d − σ ( d ) σ ( d ) (cid:33)(cid:90) M [ π / ,d ] · ∆ (6) = 0 (cid:90) M [ π / ,d ] · ∆ (8) = 12 (cid:32) (cid:88) d + d = d ( d + 1) σ ( d ) σ ( d ) (cid:33) − ( d − σ ( d ) (cid:90) M [ π / ,d ] · ∆ (10) = 48 (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33)(cid:90) M [ π / ,d ] · ∆ (11) = 24 (cid:32) (cid:88) d + d + d = d σ ( d ) σ ( d ) σ ( d ) (cid:33) − (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) Proof.
We first deal with the surface classes ∆ ( i ) for i = 1 , , , , , , which factor through M , × M , . These are pushed forward from the following classes:( ∆ (1) ) ∆ × p ( ∆ (5) ) Γ (5) × M , ( ∆ (6) ) Γ (6) × M , ( ∆ (8) ) Ξ × p ( ∆ (10) ) ∆ a × p ( ∆ (11) ) (a) ∆ (11) × p or (b) Γ (11) × M , We summarize the contributions from covers of the three possible types in the table below,where we have applied Propositions 6.1, 6.2, and 6.3, along with the intersection numbersof §2.2.4 and the intersection numbers with d -elliptic loci in genus 2 from Propositions 4.8,5.1, and 5.9. The last two rows correspond to the computation of the intersection of theadmissible locus with ∆ (11) , computed as the pushforwards of the two classes labelled (a) and(b) above. ype (∆ , ∆ , ) Type (∆ , ∆ , ) Type (∆ , ∆ , )∆ (1) d − σ ( d ) 0∆ (5) (cid:32) (cid:88) d + d = d ( d − σ ( d ) σ ( d ) (cid:33) (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) ∆ (6) ∆ (8) (cid:32) (cid:88) d + d = d ( d − σ ( d ) σ ( d ) (cid:33) − ( d − σ ( d ) (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) ∆ (10) (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) ∆ (11) (a) (cid:32) (cid:88) d + d + d = d σ ( d ) σ ( d ) σ ( d ) (cid:33) − (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) ∆ (11) (b) (cid:32) (cid:88) d + d + d = d σ ( d ) σ ( d ) σ ( d ) (cid:33) − (cid:32) (cid:88) d + d = d σ ( d ) σ ( d ) (cid:33) Combining the above yields six of the intersection numbers claimed; the seventh, of [ π / ,d ] with ∆ (4) , now follows from Propositions 6.4 and 6.5. (cid:3) Remark 6.7.
One can also implement the following check: the class ∆ (7) ∈ A ( M ) isrationally equivalent to ∆ (6) , so its intersection with the admissible locus should be zero.Using the fact that ∆ (7) is the pushforward of ∆ b × p from M , × M , , we indeed finda contribution of 0 from type (∆ , ∆ , ) , a contribution of (cid:80) d + d = d σ ( d ) σ ( d ) from type (∆ , ∆ , ) , and a contribution of − (cid:80) d + d = d σ ( d ) σ ( d ) from type (∆ , ∆ , ) . Proof of Theorem 1.4.
The result now follows from Proposition 6.6, along with the intersectionnumbers of §2.2.5 and the convolution formulas of §2.4. (cid:3)
Appendix A. Quasi-modularity on M , The quasimodularity for d -elliptic loci in genus 2 found in Theorems 1.3 (forgetting markedbranch points) and 5.10 (with one branch point) can in fact be upgraded to M , , rememberingboth branch points of a d -elliptic cover. That is: Theorem A.1.
We have (cid:88) d ≥ [ π / ,d ] q d ∈ Qmod ⊗ A ( M , ) . This result will propagate to quasimodular contributions to the d -elliptic locus on M ,providing further evidence for Conjecture 1. The method of proof is the same as above, usingthe fact that A ( M , ) = 0 [Fab90, Lemma 1.14]; we do not carry out the full calculation.However, we point out one new aspect, that the contributions from admissible covers of certaintopological types are not individually quasimodular, but the non-quasimodular contributionscancel in the sum.Let T → M , be a general boundary cycle of dimension 3. Consider the contributionsto the intersection of T with π / ,d : A dm / ,d → M , from covers of types (∆ , ∆ ) and (∆ , ∆ ) .The following purely combinatorial lemma is not difficult: emma A.2. Let H ( d, λ , λ , λ ) denote the Hurwitz number counting covers (weighted byautomorphisms) f : C → P branched over 3 points with ramification profiles ( λ , λ , λ ) ,where we require C connected. We have: (a) H ( d ; ( d ) , ( d ) , (3 , d − )) = ( d − d − (b) H ( d ; ( a, b ) , ( a, b ) , (3 , d − )) = (cid:40) if a (cid:54) = b if a = b First, consider covers of type (∆ , ∆ ) . As in Propositions 4.5 and 5.6, we have: Proposition A.3.
The contribution to the intersection of T and π / ,d from covers of type (∆ , ∆ ) is (cid:88) am = d (cid:32) m (cid:90) M , [ T ] · [ π a,a, , / ,a ] (cid:33) . We now apply Proposition A.3 with T = ∆ , . The intersection [ T ] · [ π a,a, , / ,a ] includesadmissible covers formed by gluing a degree d map E → P branched over three points withramification indices d, d, to a degree 3 map P → P with ramification indices , , , at thetriple points in the source and target. Applying Lemma A.2(a) and Proposition A.3, we finda contribution to (cid:82) M , [ T ] · [ π / ,d ] of (cid:88) am = d ( a − a − · m = (cid:18) d + 13 (cid:19) σ ( d ) − dτ ( d ) , where τ ( d ) denotes the number of divisors of d ; the generating function for dτ ( d ) is notquasimodular.On the other hand, consider contributions along T from covers of type (∆ , ∆ ) . Weget a quasimodular contribution analogous to that of Proposition 5.7, but we get a newcontribution from admissible covers formed by gluing a degree d map P → P branched overthree points with ramification profiles ( a, b ) , ( a, b ) , (3 , d − ) to a degree 3 map P → P withramification indices , , , at the triple points in the source and target. By Lemma A.2(b)and the usual local computation, we get an additional contribution of (cid:88) am + bn = d mb − (cid:88) b ( m + n )= d mb = (cid:88) d + d = d σ ( d ) σ ( d ) − (cid:88) bn (cid:48) = d n (cid:48) − (cid:88) i =1 n (cid:48) b = (cid:88) d + d = d σ ( d ) σ ( d ) − (cid:88) bn (cid:48) = d b (cid:18) n (cid:48) ( n (cid:48) − (cid:19) = (cid:88) d + d = d σ ( d ) σ ( d ) − dσ ( d ) + 12 dτ ( d ) In particular, the last term cancels out the non-quasimodular term from type (∆ , ∆ ) . eferences [ACV03] Dan Abramovich, Alessio Corti, Angelo Vistoli, Twisted bundles and admissible covers . Commun.Algebra (2003), 3547-3618 2.3[AC87] Enrico Arbarello and Maurizio Cornalba, The Picard groups of the moduli spaces of curves , Topology, (1987), 153âĂŞ171 2.2.1[Buj15] Gabriel Bujokas, Covers of an elliptic curve E and curves in E × P , Ph.D. Thesis, Harvard University(2015) 2.3[DSvZ] Vincent Delecroix, Johannes Schmitt, Jason van Zelm, admcycles.sage , arXiv 2002.01709 2.2[Dijk95] Robbert Dijkgraaf, Mirror symmetry and elliptic curves , Prog. Math. , (1995), 149-163 1.1[Fab90] Carel Faber,
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Department of Mathematics, Columbia University, New York, NY, 10027, USA
E-mail address : [email protected] URL : http://math.columbia.edu/~clian/http://math.columbia.edu/~clian/