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Counting hyperelliptic curves in Abelian surfaces with quasi-modular forms Rose, Simon Charles Florian 2012-12-31

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Counting Hyperelliptic Curves in Abelian Surfaces with Quasi-modular Forms by Simon Charles Florian Rose B.Sc. Honours, The University of Alberta, 2005 M.Sc., The University of British Columbia, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2012 c© Simon Charles Florian Rose 2012Abstract In this thesis we produce a generating function for the number of hyper- elliptic curves (up to translation) on a polarized Abelian surface using the crepant resolution conjecture and the Yau-Zaslow formula. We present a formula to compute these in terms of P. A. MacMahon’s generalized sum- of-divisors functions, and prove that they are quasi-modular forms. iiPreface The material from Chapter 6 is taken from [3], which is to appear in Journal fu¨r die Reine und Angewandte Mathematik (Crelle’s Journal). iiiTable of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Abelian surfaces and Kummer surfaces . . . . . . . . . . . . 7 2.2 Gromov-Witten theory of hyperelliptic curves . . . . . . . . 12 2.3 The case A ∼= S × F . . . . . . . . . . . . . . . . . . . . . . . 22 3 Main Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 Computation of the invariants on Km(An) . . . . . . . . . . 24 iv3.2 Application of the crepant resolution conjecture . . . . . . . 32 3.3 Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . 39 3.4 A simple consequence . . . . . . . . . . . . . . . . . . . . . . 42 4 Proofs for Low Genera . . . . . . . . . . . . . . . . . . . . . . 45 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Genus 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Genus 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4 Proof of Go¨ttsche’s genus 2 formula . . . . . . . . . . . . . . 52 5 Structure of the Moduli Space . . . . . . . . . . . . . . . . . 55 6 MacMahon’s Generalized Sum-of-Divisors Functions . . . 72 6.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . 72 6.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.2 Further research . . . . . . . . . . . . . . . . . . . . . . . . . 82 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 vList of Tables 1.1 Coefficients for genus 3 curves . . . . . . . . . . . . . . . . . . 5 viList of Figures 2.1 A map in a component of M ([A/± 1];k;n) . . . . . . . . . . 20 4.1 A collapsing component joining two non-collapsing ones . . . 46 viiAcknowledgements I would like to thank my advisor Jim Bryan for introducing me to orb- ifolds, Gromov-Witten theory, and for helping bring me back to a proper appreciation of geometry. I would like to thank my family, friends, and everyone who has been there to help me with, or to distract me from what I should be doing. viiiDedication To my grandfather, Dr. J. C. F. MacDonald. ixChapter 1 Introduction Let (Ah−1, Lh−1) be a polarized abelian surface with polarization of type (1, h − 1). Up to translation in Ah−1, there is an (h − 2)-dimensional fam- ily of curves of arithmetic genus h in the homology class c1(Lh−1)∨. The codimension of the hyperelliptic locus in Mh,0 is h− 2, and so the following natural question arises: Question 1.1. Given a polarized abelian surface (Ah−1, Lh−1), how many curves (up to translation in Ah−1) of geometric genus g in the class c1(Lh−1)∨ are hyperelliptic? We will often write Ah−1 or Lh−1 simply as A or L if there is no possibility of confusion, and when the degree of the polarization is not important. We will also for convenience make the substitution n = h− 1 which works to make most of the formulæ cleaner. Furthermore, throughout this thesis, whenever we say “the number of curves in A...” we will always be referring to the number of curves in the class c1(L)∨ up to translation in A. Let Ng,h denote the number of hyperelliptic curves of geometric genus g 1and arithmetic genus h in a fixed (A,L). Let Fg(u) = ∞∑ h=g Ng,hu h−1 be the generating function for these numbers. We will give an explicit for- mula for Fg(u) in terms of quasi-modular forms. Remark 1.2. Note that in the case g = 2, all curves are hyperelliptic. In [11], it is shown that F2(u) = ∞∑ d=0 σ1(d)u d = E2(u) + 1 24 where σ1(d) = ∑ k|d k, and E2 is the Eisenstein series of weight 2. Thus we see that F2 is in the ring of quasi-modular forms. The goal of this thesis is to transform this natural enumerative problem into the language of orbifold Gromov-Witten theory, and to use the crepant resolution conjecture [5] and the Yau-Zaslow formula [22] to compute the generating functions Fg for all g. We should remark that this number Ng,h is not necessarily well defined— that is, independent of the choice of A—nor necessarily finite. In Section 2.2 we will interpret it in terms of Gromov-Witten invariants which will be defined for all polarized A with c1(L)∨ primitive; in the case that A is sufficiently generic, we expect that this will coincide with the honest count of hyperelliptic curves of geometric and arithmetic genera g and h, respectively. In fact, we provide a refinement of this count. Let A[2] denote the collection of 2-torsion points in A. As we will see in Section 2.2, we can 2translate a hyperelliptic curve so that all of its Weirstrass points all lie on points of A[2]. We can then use the number of Weirstrass points lying over each v ∈ A[2] to refine our count, as follows. Let k : A[2] → Z≥0 be a function, denote by |k| = ∑ v∈A[2] k(v), and let g be such that 2g + 2 = |k|. Let Nk,h denote the number of curves C ⊂ A of geometric genus g and arithmetic genus h whose normalizations C are hyperelliptic and with k(v) Weirstrass points lying over v for each v ∈ A[2] (See Section 2.2, equation (2.4) for a precise definition). Let P be the collection of v ∈ A[2] so that k(v) is odd. The main theorem of this thesis is the following. Theorem 1.3. Assume the Gromov-Witten crepant resolution conjecture for the resolution Km(A) → A/± 1 (See Section 3.2). Then the generating function Fg,k(u) = ∑∞ h=gNk,hu h−1 is given by Fg,k(u) = E(u) 1 2 |S|−2 ∏ v∈S Ak(v)−1 2 (u4) ∏ v/∈S Ck(v) 2 (u2) (1.1) when P satisfies an easily-verified condition (see Remark 3.2), and is zero otherwise. The functions E(q), Ai(q), and Ci(q) are given by E(q) = ∞∑ k=0 σ1(2k + 1)q 2k+1 Ai(q) = ∑ 0<m1<···<mi qm1+···+mi (1− qm1)2 · · · (1− qmi)2 Ci(q) = ∑ 0<m1<···<mi q2m1+···+2mi−i (1− q2m1−1)2 · · · (1− q2mi−1)2 which are all quasi-modular forms. 3We provide for reference a few computations of these series, all pertaining to genus three curves. From the description above, the only functions which will contribute to curves of genus three are A2(u 4) C2(u 2) E(u)2 A1(u 4)C1(u 2) C1(u 2)2 E(u)A1(u 4) E(u)C1(u 2) a few of whose coefficients are given in Table 1.1. In particular, the function F3(u) counting all curves of genus 3 is given by F3(u) = A2(u 4) + 3C2(u 2) + 12A1(u 4)C1(u 2) + 21C1(u 2)2 + 10E(u)C1(u 2) + 6E(u)A1(u 4) + 3E(u)2 which is also included in Table 1.1. 4q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14 q15 q16 E(u)2 1 8 28 64 126 224 344 512 E(u)C1(u2) 1 6 18 40 75 126 198 C1(u2)2 1 4 12 24 44 72 112 E(u)A1(u4) 1 4 9 20 35 52 A1(u4)C1(u2) 1 2 7 10 22 28 C2(u2) 1 2 4 8 14 A2(u4) 1 3 F3(u) 3 10 45 66 180 204 471 454 972 870 1729 1470 2832 2292 4269 Table 1.1: Some coefficients of the generating functions for genus 3 curves. 5The structure of the thesis will be as follows. In Chapter 2, we review some preliminary material regarding the Kummer surface of an Abelian surface and in particular the Kummer lattice K ⊂ H2 ( Km(A) ) . We also discuss the basic construction in orbifold Gromov-Witten theory which al- lows us to study hyperelliptic curves with genus 0 invariants. We provide a partial description of the relevant moduli space in the case that the Picard number of A is 1, and we explain how to obtain enumerative invariants from decidedly non-enumerative ones. Chapter 3 consists of a proof of Theorem 1.3 obtained by computing a restricted form of the Gromov-Witten potential (see Definition 3.4), followed by applying the crepant resolution conjecture to obtain the corresponding potential function on [A/ ± 1]. Lastly, we simplify this by accounting for collapsing components to prove Theorem 1.3. Chapter 4 consists of a proof of the genus one and two case independent of the crepant resolution conjecture. This involves specializing to the case that A ∼= S×F for generic elliptic curves S and F . From there the problem is reduced to counting covers of an elliptic curve, which is classically known. Chapter 5 consists of an analysis of the moduli space of genus 0 twisted stable maps into [A/ ± 1], as well as a discussion of its reduced virtual fundamental class. Chapter 6 is a version of the results appearing in the paper [3]. In this chapter, a number of relationships satisfied by the generating functions Ak(q) and Ck(q) (given in Theorem 1.3) are proven. In particular, a proof of their relationship to Chebyshev polynomials is given, as well as a proof that they lie in the ring of quasi-modular forms. 6Chapter 2 Preliminaries 2.1 Abelian surfaces and Kummer surfaces For the duration of this thesis, unless otherwise noted, all coefficients are integral. The majority of the results in this section follows [4], and as such, proofs are omitted. Let A be an Abelian surface. Then A is a complex torus C2/Γ with Γ of rank 4. As an Abelian group, A has an involution given by multiplication by ±1. This has as fixed points the collection A[2] ∼= Γ/2Γ of sixteen 2-torsion points. The quotient by this action has these as its only singularities, and so by blowing them up we obtain the (crepant) resolution Km(A) which is a smooth K3 surface called the Kummer surface of A. If instead we blow up A at the sixteen 2-torsion points to produce A, we can take the quotient of this space by the lifted involution to obtain the diagram A σ d47d47 s d15d15 Km(A) p d15d15 A pi d47d47 A/± 1 . (2.1) There are a few facts that this yields, all of which are connected to the 7affine F2-geometry of A[2]. We begin by introducing some notation. 1. Let Ev ∈ H2 ( Km(A) ) denote the class of the (−2)-curve lying over a given v ∈ A[2]. 2. Let Λ denote the sublattice of H2 ( Km(A) ) generated by the classes Ev. 3. Let K denote the minimal primitive sublattice of H2 ( Km(A) ) which contains Λ. This is called the Kummer lattice. Furthermore, let P(A[2]) denote the power set of A[2] (which is a group under the operation S + S′ = S ∪ S′ \ (S ∩ S′)) and let Πk for 0 ≤ k ≤ 4 denote the subgroups generated by all of the affine k-planes in A[2]. Then we have that Z/2 = Π4 ⊂ Π3 ⊂ Π2 ⊂ Π1 ⊂ Π0 = P(A[2]). Remark 2.1. We will use throughout this thesis the letters η and ε (possibly with subscripts) to denote elements of Πk. Note also that for each element η ∈ Πk we can think of η as an element of 12Λ via the correspondence η ↔ η̂ = ∑ v∈η 1 2Ev. We will also throughout use the notation |η| to denote the number of ele- ments in η. Remark 2.2. We should note that there are two notions of summation at play here—summation in P(A[2]), and summation in 12Λ. When we write 8η1 + η2 we will always mean the former, and when we write η̂1 + η̂2 we will always mean the latter, so no confusion should arise. As Km(A) is a smooth real 4-manifold, the group H2 ( Km(A) ) comes with a natural intersection form 〈 , 〉 which turns it into a unimodular lattice. When restricted to Λ this is (−2)Id, and so we have that Λ∨ = 12Λ. Thus we have that Λ ⊂ K ⊂ K∨ ⊂ 12Λ and so every w ∈ K (and in K∨) can be written as w = ∑ v∈A[2] av 2 Ev. From this we have a natural map r : 12Λ → Π0 given by ∑ v∈A[2] av 2 Ev mapsto→ { v ∈ A[2] | av ≡ 1 (mod 2) } . Note that for w = ∑ v∈A[2] av 2 Ev, that r̂(w) is nothing but the reduction of the coefficients av of w mod 2. Remark 2.3. It follows further that the intersection form on K can be ex- tended linearly to an intersection form on 12Λ which we also denote by 〈 , 〉. In particular, using the correspondence of Remark 2.1, we can define for any two subsets η1, η2 of A[2] the pairing 〈η̂1, η̂2〉 by 〈η̂1, η̂2〉 = 〈∑ v∈η1 1 2 Ev, ∑ v∈η2 1 2 Ev 〉 = −12 |η1 ∩ η2|. 9The condition that w ∈ K can now be stated as the following. Proposition 2.4 ([4, Proposition 5.5, Chapter VIII]). An element w = ∑ v∈A[2] av 2 Ev ∈ 1 2Λ is in K if and only if r(w) ∈ Π3. That is, w ∈ K if and only if the collection of those v ∈ A[2] such that av is odd is either 1. empty 2. an affine 3-plane 3. all of A[2]. This then yields a description of the Kummer lattice as follows. Corollary 2.5. There is a short exact sequence 0 d47d47 Λ d47d47 K r d47d47 Π3 d47d47 0 . Following prior remarks, this permits us to consider each element η ∈ Π3 as an element η̂ ∈ K. We next describe the relationship between H2(A) and H2 ( Km(A) ) . Consider again the diagram (2.1), and in particular, consider the map α = σ∗ ◦ s ! : H2(A) → H2 ( Km(A) ) . We have the following proposition which relates the two intersection forms. Proposition 2.6 ([4, Proposition 5.1, Chapter VIII]). The map α multiplies the intersection form by 2. That is, α(a)·α(b) = 2a·b for every a, b ∈ H2(A). Moreover, for each v ∈ A[2] and for each class a ∈ H2(A), Ev · α(a) = 0. 10It follows from this proposition that the map α embeds H2(A) as a sublattice of H2 ( Km(A) ) which is orthogonal to the Kummer lattice K. We examine now one of the important properties of this map. Recall that A = C2/Γ, let u : C2 → A be the quotient map, and let λ1, λ2 be basis elements of Γ. Define V = u ( 〈λ1, λ2〉⊗R ) (the image of the 2-plane spanned by λ1, λ2), and let C = [V ]. Then α(C) is the class of the proper transform of pi(V + t) in Km(A) for some generic t ∈ A (i.e. such that V + t does not intersect A[2]). Next, consider the rational curve V/±1 ⊂ A/±1, and let β ∈ H2 ( Km(A) ) be the class of the proper transform of V/ ± 1. Let ε denote the collection of 2-torsion points in V . We have the following relation between the classes α(C) and β. Proposition 2.7. Let V , C, and β be as above. The classes α(C) and β satisfy the relationship β = 1 2 α(C)− ε̂ = 1 2 α(C)− 1 2 ∑ v∈ε Ev Proof. As α(C) is the class of the proper transform in Km(A) of a generic translate of V , we see that in Km(A) we must have that β = 1 2 α(C)− 1 2 ∑ v∈A[2] avEv for some integers av. Since β · Ev = av, we see that av = 1 if v ∈ ε and is zero otherwise, as claimed. 11Recall that there is a canonical isomorphism H2(A) ∼= ∧2H1(A)∨, and so we can regard elements of H2(A) as alternating forms on H1(A). Suppose now that (An, Ln) is a polarized abelian surface with polarization of type (1, n). That is, there is a basis e1, f1, e2, f2 of H1(An) so that c1(Ln) (when viewed as an alternating form) can be written as          0 0 n 0 0 0 0 1 −n 0 0 0 0 −1 0 0          (2.2) In this basis, we can write c1(Ln)∨ = (e1 ∧ f1) + n(e2 ∧ f2). 2.2 Gromov-Witten theory of hyperelliptic curves We aim to compute the number of hyperelliptic curves in A via orbifold Gromov-Witten theory following the ideas of [20, 12, 9]. LetX be a smooth Deligne-Mumford stack with projective coarse moduli space X. We will use the notation M (X ; 2g + 2;β) to denote the moduli stack of twisted stable maps of genus 0 curves into the stack X in the curve class β ∈ H2(X) with (2g + 2) Z/2-stacky points. In the case where A is a polarized Abelian surface with polarization of type 12(1, n), we will use the notation M ([A/± 1]; 2g + 2;n) to denote the moduli stack where the class β is the class 12 pi∗c1(Ln) ∨. As with ordinary Gromov-Witten theory, there are evaluation maps from the moduli stack of twisted stable maps. However, they do not lie inX , but in its rigidified inertia stack, IX (see [1, 2]). In the case that X = [X/G] is a global quotient, this has a particularly simple description. Definition 2.8 Let X = [X/G] be a global quotient stack. We define the rigidified inertia stack to be IX = ∐ (g)⊂G [Xg/Hg] where the disjoint union is taken over all conjugacy classes (g) ⊂ G, where Xg is the fixed-point set of g, and where Hg = C(g)/〈g〉 is the quotient of the centralizer of g in G by the subgroup generated by g. The component corresponding to (e) ⊂ G is called the untwisted sector, while all others are called twisted sectors. For the case of the the quotient stack [A/± 1], it is easy to see that I[A/± 1] = A coproductA[2] and so the twisted sector is identified with A[2], the set of 2-torsion points of A. 13As before, let (A,L) be a polarized Abelian surface. We have a map A d47d47 ∆ d54d54 A×A d47d47 Sym2A where the first map is given by a mapsto→ (a,−a). Next, we consider the moduli space M ( [Sym2A]; 2g + 2; 12∆∗c1(L) ∨) (where the factor of 12 comes from the fact that a curve in A is a double cover of the corresponding curve in Sym2A). This parameterizes genus 0, twisted stable maps into the stack [Sym2A]. As these maps are representable, we may complete them to a diagram C˜ d15d15 d47d47 A×A d15d15 C d47d47 [Sym2A] with C˜ a scheme and the top map equivariant. If C is smooth, then C˜ is a smooth hyperelliptic curve and the projection onto either factor yields a hyperelliptic curve in A in the class c1(L)∨. It follows that the moduli space M ( [Sym2A]; 2g + 2; 12∆∗c1(L) ∨ ) is a compactification of the moduli space of smooth hyperelliptic curves in A. So far this follows very closely [20, 9]. In our case, we can simplify 14matters significantly. As above, we look at the diagram A a mapsto→(a,−a) d47d47 pi d15d15 A×A d15d15 [A/± 1] ι d47d47 [Sym2A] where the map ι : [A/± 1] → [Sym2A] is given by [a] mapsto→ [a,−a]. Theorem 2.9. We have the following isomorphism of stacks. M ([A/± 1]; 2g + 2;β)×A ∼= M ( [Sym2A]; 2g + 2; ι∗β ) Proof. The map in one direction is easy to produce. A family of objects in M ([A/± 1]; 2g + 2;β) over a base scheme B consists of a diagram C˜ d47d47 d15d15 A d15d15 C d47d47 d15d15 A/± 1 B si d71d71 with C of genus 0 and with (2g + 2) sections si (thus if C is smooth, C˜ is hyperelliptic of genus g). Given an element a0 of A we can construct a map A→ A×A given by a mapsto→ (12a0 + a, 1 2a0 − a) 15which we then complete to C˜ d47d47 d15d15 A d15d15 d47d47 A×A d15d15 C d47d47 d15d15 A/± 1 d47d47 Sym2A B si d71d71 which yields the first half. For the second half, note that there is a map + : Sym2A→ A given by [a, b] mapsto→ a+ b. Thus given the diagram C˜ d47d47 d15d15 A×A d15d15 C d47d47 d15d15 Sym2A + d47d47 A B si d71d71 with C a rational curve. Since there are no rational curves in Abelian surfaces, the composition C → Sym2A → A must be constant, and so the diagram factors through the inclusion of a fibre of the map to A. As these are all isomorphic to A/± 1, the claim follows. We can interpret this theorem as saying that counting hyperelliptic curves in A is equivalent to counting certain stacky rational curves in the orbifold [A/±1]. Using the crepant resolution conjecture, this should be the same as counting certain rational curves in the smooth K3 surface Km(A). This has been studied in [22, 7]. 16In [6], it is shown that the (reduced) Gromov-Witten invariants for an Abelian surface only depend on the divisibility and square of the class β. This follows because of the fact that the moduli space A2n of polarized Abelian surfaces whose polarizations have square 2n is connected, and from the deformation invariance of Gromov-Witten invariants. We can use this fact to show that the same holds true in our case; indeed, there is a surjective map from A2n to the moduli space of singular Kummer surfaces with polarizations with square 2n (given by taking the quotient by ±1), and so the same result holds. This justifies our notation of M ([A/± 1]; 2g + 2;n) (specifically its lack of dependence on the class β). As twisted stable maps are representable, each of the (2g+2) evaluation maps evi : M ([A/± 1]; 2g + 2;n) → I[A/ ± 1] must all lie in the twisted sector, which is A[2]. Let ev denote the map ev : M ([A/± 1]; 2g + 2;n) → A[2]2g+2 and let v1, . . . , v16 denote an arbitrary labeling of A[2]. Given k : A[2] → Z≥0 with |k| = 2g + 2, we define M ([A/± 1];k;n) = ev−1(v1, . . . , v1 ︸ ︷︷ ︸ k(v1) , . . . , v16, . . . , v16 ︸ ︷︷ ︸ k(16) ). This is the moduli space of those orbifold maps with k(v) stacky points whose image lies on each v ∈ A[2]. This space has a degree 0 reduced 17virtual fundamental class (see Chapter 5), and so we define GWk,n = deg [ M ([A/± 1];k;n) ]red . It should be noted that in the definition of GWk,n, the specific labelling of the element in A[2]|k| is not relevant; any rearrangement of those terms comes simply from a permutation of the labelling of the marked points, and will yield the same Gromov-Witten invariant. We would like to use this invariant to count hyperelliptic curves, but it is not enumerative; it includes contributions from collapsing components. In the case that A is suitably generic (i.e. has Picard number 1), we can use this to determine the enumerative count as follows (cf. [20, 12]). Definition 2.10 An n-marked comb curve is a genus 0 twisted stable map f : Σ → [A/ ± 1] with 0 ordinary marked points and n Z/2-marked points such that there is a unique irreducible component Σ0 which has nonzero degree; this component is called the handle. All other components are called teeth. It is clear that the locus of (2g + 2)-marked comb curves U2g+2,n is a closed substack of M ([A/± 1]; 2g + 2;n). Similarly, define Uk,n = U|k|,n ∩ M ([A/± 1];k;n). In the case that the Picard number of A is 1, it turns out that all curves are comb curves, as the following proposition shows. Proposition 2.11. If the Picard number of A is 1, then the moduli spaces U2g+2,n and M ([A/± 1]; 2g + 2;n) are equal, and similarly for Uk,n. 18Proof. Suppose that there were more than one component with non-zero degree. Since the class in [A/±1] is primitive, this would immediately yield that the Picard number is greater than 1. As discussed in Chapter 5, this moduli space splits up into components based on how the marked points partition among the teeth (see figure 2.1). From Lemma 5.6, we see that all partitions with even parts contribute zero to the Gromov-Witten invariant, and so for each comb curve Σ and each v ∈ A[2] we obtain a partition λv = (λv1, . . . , λ v rv) of k(v) (where rv denotes the length of the partition λv) into odd parts based on how the marked points are split up among the teeth of Σ (with λvi = 1 being interpreted as there being no collapsing component—that is, it represents a stacky marked point on the handle). Let Uλ,n denote the component consisting of those comb curves with partition type λ = (λv)v∈A[2]. If we define kλ(v) = rv, then since the smoothing of any node is obstructed (see Proposition 5.5) and since all collapsing components must have image a stacky point in [A/± 1], it is clear that Uλ,n = M ◦ ([A/± 1];kλ;n)× ∏ v∈A[2] rv∏ i=1 M (BZ/2;λvi + 1; 0) where M◦ ([A/± 1];k;n) denotes the component consisting of those curves with no collapsing components. Let pλ denote the projection Uλ,n → M◦ ([A/± 1];kλ;n).We have the following theorem, whose proof we defer to Chapter 5. Theorem 2.12. Let λ = (λv)v∈A[2] be a collection of partitions of k, all of 19Figure 2.1: A map in the component of M ([A/± 1];k;n) corresponding to the partitions λ = (λ1, λ2, λ3, λ4) = ( [3], [3 5], [1], [1 3] ) 20which consist of odd parts. Then the virtual degree of pλ is ( − 14 ) 1 2 (|k|−|kλ|). That is, (pλ)∗[Uλ,n] red = ( − 1 4 ) 1 2 (|k|−|kλ|)[ M◦ ([A/± 1];kλ;n) ]red . We note that Uk,n is the disjoint union of the Uλ,n taken over all partition types λ. In particular, there is a component M◦k which consists of those curves with no collapsing components (corresponding to the partitions 1k(v)). We can now define GW ◦k,n = deg[M ◦ k] red. (2.3) From this number we obtain our expected count of hyperelliptic curves. More precisely, in the generic setting we expect that this counts the number of hyperelliptic curves together with the extra data of an ordering on the marked points which collapse to a given 2-torsion point. As such, define Nk,n = GW ◦k,n ∏ v∈A[2] k(v)! (2.4) Our main theorem of this section describes this relationship. Define the generating functions Fn(zv) = ∑ k:A[2]→Z≥0 GWk,n ∏ v∈A[2] zk(v)v k(v)! F ◦n(xv) = ∑ k:A[2]→Z≥0 GW ◦k,n ∏ v∈A[2] xk(v)v k(v)! Theorem 2.13. The two generating functions Fn and F ◦n are equal after the substitution xv = 2 sin(zv/2). 21A proof of this theorem will be provided in Chapter 5. However, from the equation (2.4), we immediately obtain the following. Corollary 2.14. Let k : A[2] → Z≥0 be a function and let g be such that |k| = 2g+2. Then the function Fg,k of theorem 1.3 is given by the coefficient of ∏ v∈A[2] x k(v) v in ∞∑ n=0 F ◦n(xv)u n = ∞∑ n=0 ∑ k:A[2]→Z≥0 GW ◦k,n ∏ v∈A[2] xk(v)v k(v)! un = ∞∑ n=0 ∑ k:A[2]→Z≥0 Nk,n ∏ v∈A[2] xk(v)v u n. Remark 2.15. In the case that the Picard number of A is 1, we can define GW ◦k,n and Nk,n directly via equations (2.3) and (2.4). However, they are defined for all Abelian surfaces A via the relationships given by Theorem 2.13. 2.3 The case A ∼= S × F We will sometimes have need to specialize to the case that A ∼= S × F for generic elliptic curves S and F . In such a case, the quotient A/ ± 1 comes equipped with an elliptic fibration A/ ± 1 → S/ ± 1 whose general fibre is the elliptic curve F . This has four distinguished fibres over S[2] which are isomorphic to F/± 1. So let f : Σ → [A/ ± 1] be a genus 0 twisted stable map in the class [S/±1]+n[F/±1] (which we will assume without loss of generality collapses no components). In such a case, the source must be a tree of rational curves 22Σ0 ∪ Σ1 ∪ · · · ∪ Σk. We can label these so that f∗[Σ0] = [S/ ± 1] and f∗[Σi] = ni[F/± 1] for i ≥ 1. In particular, f |Σ0 is an isomorphism (and so must contain exactly 4 stacky points), while f |Σi is a ramified cover of one of the distinguished fibres, i.e. the rational orbi-curve [F/± 1]. As a (representable) ramified cover of an orbi-curve, it follows that 1. The image of each stacky point is stacky. 2. The pre-image of each stacky point is a collection of stacky points (with odd ramification) and some non-stacky points (with even ramification). In such a case, the moduli space M ([A/± 1]; 2g + 2;n) will be isomor- phic to a product of spaces of Hurwitz covers of the fibre F/±1, a fact which will be necessary in order to compute the orbifold Gromov-Witten theory of [A/± 1] without using the crepant resolution conjecture. 23Chapter 3 Main Work 3.1 Computation of the invariants on Km(An) The goal of this section is to compute the relevant part of the Gromov- Witten potential function for Km(An). Let f : Σ → An be a hyperelliptic curve representing the class c1(Ln)∨ and such that f(w) ∈ A[2] for all Weirstrass points w. Let βn be the class in H2 ( Km(An) ) of the proper transform of (pi ◦ f)(Σ); note that this is a rational curve. Let Hn = α ( c1(Ln)∨ ) . If we choose a basis of H1(An) as in section 2.1, then we can write Hn = S + nF where S = α(e1 ∧ f1) and F = α(e2 ∧ f2). Recall now that, given two basis elements λ1, λ2 of Λ, we can consider the image V in A of the real 2-plane spanned by λ1, λ2. We can further consider the homology class βV of the proper transform of V/±1 in Km(A), which by Proposition 2.7 can be written as βV = 1 2 α ( [V ] ) − 1 2 ∑ v∈εV Ev. We see that to each pair of basis elements we can associate en element εV ∈ Π0 (since λ1, λ2 span a plane, this will in fact be in Π2). 24Let ε0 be the element so obtained from the basis elements e1, f1, let ε∗1 be the element so obtained from the basis elements e2, f2, and let finally ε1 = ε0 + ε∗1 (where, recall, the summation is done in P(A[2]) and is hence the symmetric difference of the two elements). Proposition 3.1. We have the following congruences. β2k ≡ 1 2 H2k − ε̂0 (mod K) β2k+1 ≡ 1 2 H2k+1 − ε̂1 (mod K) Proof. We assume as per Section 2.2 that the image curve is fixed by ±1 and so descends to a genus 0 map f˜ : Σ → [A/ ± 1]. We assume further (by deformation invariance) that A ∼= S ×F , which puts us in the situation described in Section 2.3. We will further assume that n is even, the odd case being similar. Let Σ = Σ0∪Σ1∪ · · · ∪Σk be the source curve. Since n is even, we must have that either the degree of each Σi is even, or that any odd ones come in pairs. As in the proof of Proposition 2.7, we have that Hn is the class of a double cover of the proper transform of the image of f˜ in Km(A), and so we have that βn = 1 2 Hn − 1 2 ∑ v∈A[2] avEv where av is the intersection multiplicity of the proper transform with the exceptional curve Ev. This is given by the sum of all the ramification indices over all points p ∈ Σ which map to v ∈ A[2]. Since Ev ∈ K, this only depends (mod K) on the ramification indices mod 2. Since f˜∗[Σi] = ni[F/± 251], it follows that the image of all the stacky points on a given Σi lie in ε∗1, or some translation thereof. Recall from Section 2.3 that the ramification must be even at each non- stacky point in the pre-image, and is odd at each stacky point in the pre- image, and hence av ≡ k(v) (mod 2). Consider first a component Σi with f˜ |Σi of even degree. Due to these ramification considerations, it follows that over each v ∈ A[2] the number of stacky pre-images must be even, and so (mod K) this contributes zero to βn. Consider next components Σi,Σj with f˜ |Σi and f˜ |Σj of odd degree (recall that these must come in pairs). We see that for both of these the number of stacky pre-images of a given v must be odd, and so one of two things occur. 1. Σi and Σj map via f˜ to the same fibre. 2. Σi and Σj map to different fibres. In the first case, the contribution to the number of stacky pre-images winds up being even, and so (mod K) contributes zero. In the second case, the stacky points form an affine 3-plane η in A[2]—but since η̂ ∈ K, it follows that (mod K) these also contribute nothing. As such, all that contributes (mod K) are the stacky points coming from the curve Σ0. But this is exactly ε0, which completes the proof. Remark 3.2. The condition on P so that the generating function Fg,k(u) is non-zero can be now explained as follows. Recall that P is the collection of 26those v ∈ A[2] such that k(v) is odd. It will arise naturally during the proof of Theorem 1.3 that we must have P ≡ εi (mod K). One consequence of this is that if P ≡ ε0 (mod K), then only even polarizations can occur (and conversely for P ≡ ε1 (mod K)). We are now ready to compute the potential function. We first recall the definition of the Gromov-Witten potential of a smooth projective variety. Definition 3.3 Let X be a smooth projective variety, and let γ0, . . . , γa be an additive basis of H∗(X). The genus 0 Gromov-Witten potential function is defined by FX(y0, . . . , ya, q) = ∞∑ m0,...,ma=0 ∑ β∈H2(X) 〈γm00 · · · γ ma a 〉 X β ym00 m0! · · · ymaa ma! qβ. As stated in the introduction, we are only concerned with a restricted form of this function. More specifically, we are only concerned with homol- ogy classes β such that p∗β = 12 pi∗c1(Ln) ∨, and so we make the following definition. Definition 3.4 Define the restricted genus 0 potential function as Fn := F Km(An)(y0, . . . , ya, q) = ∞∑ m0,...,ma=0 ∑ w∈K 〈γm00 · · · γ ma a 〉 Km(An) βn+w ym00 m0! · · · ymaa ma! qβn+w. Remark 3.5. Due to dimension considerations, the only classes that will produce non-zero invariants will be divisor classes. As such, we choose as a basis of H2 ( Km(An) ) the classes γv which are dual to the exceptional curve 27classes Ev, as well as classes γS , γF dual to S and F , respectively. As such, if we define m = (mv1 , . . . ,mv16 ,mS ,mF ) then we can write the function Fn as Fn(yv1 , . . . , yv16 , yS , yF , q) = ∑ m ∑ w∈K 〈γmSS γ mF F ∏ v∈A[2] γmvv 〉βn+w ymSS mS ! ymFF mF ! ∏ v∈A[2] ymvv mv! qβn+w where we omit for simplicity of notation the superscript on the brackets 〈· · · 〉Km(An)β . Moreover, define the generating function F := ∞∑ n=0 Fn(yv1 , . . . , yv16 , yS , yF , q). A priori this does not make sense, but as we will see, the formal variable q used to define Fn permits this to be well defined as a formal power series in the variables yα and q. It is this function that we will use (with the crepant resolution conjecture) to compute the number of hyperelliptic curves in A. Theorem 3.6. The restricted genus 0 potential function F is given by F = λ0 u2 ∆(u2) ∑ w∈K u−〈w,w−2bε0〉 ∏ v∈A[2] exp ( ( ∫ w γv)yv ) qw + λ1 u2 ∆(u2) ∑ w∈K u−〈w,w−2bε1〉+1 ∏ v∈A[2] exp ( ( ∫ w γv)yv ) qw where 〈 , 〉 is the intersection form on K ⊂ Km(An), where u and λi are 28given by u = ( exp(yF )q F )1/2 λi = ( exp(yS)q S)1/2 ∏ v∈A[2] exp ( ( ∫ −bεi γv)yv ) q−bεi (3.1) and where ∆(q) the weight 12 cusp form defined as ∆(q) = q ∏∞ k=1(1− q k)24 which satisfies q ∆(q) = 1 + 24q + 324q2 + 3200q3 + · · · Proof. Let (An, Ln) be a polarized abelian surface with polarization type (1, n) and with Kummer surface Km(An). As stated in Definition 3.4, the (restricted) Gromov-Witten potential function for this is Fn = ∑ m ∑ w∈K 〈 γmSS γ mF F ∏ v∈A[2] γmvv 〉 βn+w ymSS mS ! ymFF mF ! ∏ v∈A[2] ymvv mv! qβn+w Since on Km(An), as stated above, we only need to consider divisor classes the divisor equation simplifies this to Fn = ∑ m ∑ w∈K 〈 〉βn+w ( ∫ βn+w γS)mSy mS S mS ! ( ∫ βn+w γF )mF y mF F mF ! · ∏ v∈A[2] ( ∫ βn+w γv)mvymvv mv! qβn+w = ∑ w∈K 〈 〉βn+w exp ( ( ∫ βn+w γS)ys ) exp ( ( ∫ βn+w γF )yF ) · ∏ v∈A[2] exp ( ( ∫ βn+w γv)yv ) qβn+w. 29Since we are summing over K, by proposition 3.1 we can replace βn +w with 12Hn + w − ε̂i (with i ≡ n (mod 2)), and so noting that (see remark 3.5) ∫ 1 2Hn+w−bεi γS = 12 ∫ 1 2Hn+w−bεi γF = n2 ∫ 1 2Hn+w−bεi γv = ∫ w−bεi γv we can write this as F2k = exp(12yS) ∑ w∈K 〈 〉 1 2H2k+w−bε0 exp(12yF ) 2k ∏ v∈A[2] exp ( ( ∫ w−bε0 γv)yv ) q 1 2H2k+w−bε0 (and similarly for F2k+1). Let Cd be a primitive curve class in aK3 surfaceX satisfying C2d = 2d−2, and let Nd = 〈 〉XCd be the (reduced) genus 0 Gromov-Witten invariant. Then the Yau-Zaslow theorem [22, 7] states that these numbers satisfy ∞∑ d=0 Ndq d = q ∆(q) where ∆(q) is defined above to be ∏∞ k=1(1 − q k)24. Since Hn ∈ Λ⊥, and H2n = 4n, we have that ( 1 2H2k + w − ε̂0 )2 = 2k − 2 + 〈w,w − 2ε̂0〉 30(and similarly for the odd case). Thus we find 〈 〉 1 2H2k+w−bε0 = N k+12 〈w,w−2bε0〉 〈 〉 1 2H2k+1+w−bε0 = N k+12 〈w,w−2bε1〉 As mentioned in Remark 3.5, let F = ∑∞ n=0 Fn. We have F = ∞∑ k=0 F2k + ∞∑ k=0 F2k+1 = exp(12yS) ∞∑ k=0 ∑ w∈K N k+12 〈w,w−2bε0〉 exp(12yF ) 2k ∏ v∈A[2] exp ( ( ∫ w−bε0 γv)yv ) q 1 2H2k+w−bε0+ exp(12yS) ∞∑ k=0 ∑ w∈K N k+12 〈w,w−2bε1〉 exp(12yF ) 2k+1 ∏ v∈A[2] exp ( ( ∫ w−bε1 γv)yv ) q 1 2H2k+1+w−bε1 . If we then perform the substitutions given in equation (3.6), this simplifies to F = λ0 ∞∑ k=0 ∑ w∈K N k+12 〈w,w−2bε0〉 u2k ∏ v∈A[2] exp ( ( ∫ w γv)yv ) qw λ1 ∞∑ k=0 ∑ w∈K N k+12 〈w,w−2bε1〉 u2k+1 ∏ v∈A[2] exp ( ( ∫ w γv)yv ) qw To simplify this we perform the substitution n = k+ 12〈w,w− ε̂i〉 which 31yields F = λ0 ∞∑ n=0 ∑ w∈K Nnu 2nu−〈w,w−2bε0〉 ∏ v∈A[2] exp ( ( ∫ w γv)yv ) qw + λ0 ∞∑ n=0 ∑ w∈K Nnu 2nu−〈w,w−2bε1〉+1 ∏ v∈A[2] exp ( ( ∫ w γv)yv ) qw = λ0 u2 ∆(u2) ∑ w∈K u−〈w,w−2bε0〉 ∏ v∈A[2] exp ( ( ∫ w(γv)yv ) qw + λ1 u2 ∆(u2) ∑ w∈K u−〈w,w−2bε1〉+1 ∏ v∈A[2] exp ( ( ∫ w(γv)yv ) qw as claimed. 3.2 Application of the crepant resolution conjecture We now use apply the crepant resolution conjecture of [5] to compute the potential function of the orbifold [A/±1]. This conjecture is given as follows. Conjecture 3.7 ([5, Conjecture 1.2]). Given an orbifold X satisfying the hard Lefschetz condition and admitting a crepant resolution Y , there exists a graded linear isomorphism L : H∗(Y ) → H∗orb(X ) and roots of unity c1, . . . , cr such that the following conditions hold. 1. The inverse of L extends the map pi∗ : H∗(X ) → H∗(Y ). 322. Regarding the potential function F Y as a power series in y0, . . . , ya, and in q1, . . . , qs, the coefficients admit analytic continuations from (qs+1, . . . , qr) = (0, . . . , 0) to (qs+1, . . . , qr) = (cs+1, . . . cr). 3. The potential functions FX and F Y are equal after the substitution yi = ∑ j Ljixj qi =    ci for i > s citi for i ≤ s It should be noted that the crepant resolution conjecture, as stated, only applies to ordinary Gromov-Witten invariants and not to reduced Gromov- Witten invariants as we use in our case. The short explanation is simply that it appears to work. The longer explanation is that our situation (dealing with fibre-wise Gromov-Witten invariants of a non-Ka¨hler 3-fold) is similar enough to the CY3 case that it should work. Moreover, the local picture around the singular points is the same as that of the resolution T ∗P1 → [C2/ ± 1], a case where the equivariant version of the crepant resolution conjecture has been proven [5]. The only issue is whether or not we can extant this globally to the orbifold [A/± 1], which appears to be the case. As is usually the case, the change-of-variables is forced upon us by know- ing a few of the invariants and choosing data to match those. In our case this comes from two sources. As stated above, we have a good understanding of the local picture around a singular point, so we can use that knowledge. Moreover, we already know the number of genus 2 curves in an Abelian surface due to Go¨ttsche [11]. These two facts allow us to derive the full change-of-variables for the restricted potential function, which is given as 33follows. The function L : H∗ ( Km(A) ) → H∗orb([A/ ± 1]) used in the crepant resolution conjectures is the following. Let yv denote the formal cohomo- logical variable corresponding to the dual of an exceptional divisor Ev, and let zv denote the variable corresponding to the class in the twisted sector of [A/± 1]. Similarly, let yS and yF denote the formal variables corresponding to the classes S and F (with zS and zF downstairs), respectively. Then the map L on the formal variables is given by L(yv) = izv L(yS) = zS L(yF ) = zF while the roots of unity are given by applying, for w = ∑ v∈A[2] av 2 Ev qw mapsto→ (−1)w := (−1) P v∈A[2] av 2 and the substitutions qS/2 mapsto→ tS/2 qF/2 mapsto→ −tF/2 As in the statement of Theorem 3.6, we will continue with the equivalent substitution of u = ( exp(zF )t F )1/2. which yields that u mapsto→ −u under this substitution. All of this together yields the following. Proposition 3.8. Assume the crepant resolution conjecture holds for the 34resolutions Km(An) → [An/ ± 1]. Then the restricted genus 0 orbifold Gromov-Witten potential function for [A/ ± 1], when summed over all po- larizations, is given by F [A/±1] = ( exp(zS)t S)1/2 u 2 ∆(u2) · ( ∏ v∈ε0 e−izv/2 ∑ w∈K u−〈w,w−2bε0〉 ∏ v∈A[2] exp ( i( ∫ w γv)zv ) (−1)w + ∏ v∈ε1 e−izv/2 ∑ w∈K u−〈w,w−2bε1〉+1 ∏ v∈A[2] exp ( i( ∫ w γv)zv ) (−1)w ) . where (−1)w := (−1) P v∈A[2] av 2 for w = ∑ v∈A[2] av 2 Ev. Proof. If we combine all of the transformations given above, we find that the remaining composite terms in the potential transform as λ0 mapsto→ ( exp(zS)t S)1/2 ∏ v∈ε0 eizv/2 λ1 mapsto→ − ( exp(zS)t S)1/2 ∏ v∈ε1 eizv/2 This yields the result as claimed. Remark 3.9. We should remark that for simplicity, we will omit the term ( exp(zS)tS )1/2 occurring at the beginning of the expression, which con- tributes no further enumerative information. To simplify this, we recall that from corollary 2.5 we can write any element w¯ ∈ K as w¯ = w + η̂ for w ∈ Λ and η ∈ Π3. This allows us to replace the summation over K by a double summation over Λ and over Π3. More precisely, let η ∈ Π3, and (noting that (−1)bη = 1 by the definition 35above) define F [A/±1]η = u2 ∆(u2) ( ∏ v∈ε0 e−izv/2 ∑ w∈Λ u−〈w+bη,w+bη−2bε0〉 · ∏ v∈A[2] exp ( i( ∫ w+bη γv)zv ) (−1)w + ∏ v∈ε1 e−izv/2 ∑ w∈Λ u−〈w+bη,w+bη−2bε1〉+1 · ∏ v∈A[2] exp ( i( ∫ w+bη γv)zv ) (−1)w ) . Then we have that F [A/±1] = ∑ η∈Π3 F [A/±1] η . Since the intersection form restricted to Λ is diagonal, the functions F [A/±1]η can now be computed. For simplicity, the superscript [A/± 1] will now be omitted. Lemma 3.10. The function Fη can be written as Fη = u2 ∆(u2) ( u 1 2 |η+ε0|−2 ∏ v∈η+ε0 h(zv, u) ∏ v/∈η+ε0 g(zv, u) + u 1 2 |η+ε1|−2 ∏ v∈η+ε1 h(zv, u) ∏ v/∈η+ε1 g(zv, u) ) where h(z, u) = 2 ∞∑ k=0 (−1)k sin ( (2k + 1) z 2 ) u2k 2+2k g(z, u) = 1 + 2 ∞∑ k=1 (−1)k cos(kz)u2k 2 . Proof. Throughout this proof we will only work with the terms involving ε0; 36all of the proofs for ε1 are nearly identical. We begin by noting that 〈w + η̂, w + η̂ − 2ε̂0〉 = 〈w,w + 2η̂ − 2ε̂0〉+ 〈η̂, η̂〉 − 2〈η̂, ε̂0〉. Since |ε0| = 4, and for any two subsets η1, η2 ∈ Π0, we have 〈η̂1, η̂2〉 = −12 |η1 ∩ η2| (see remark 2.3), it follows that 〈η̂, η̂〉 − 2〈η̂, ε̂0〉 = 〈η̂, η̂〉 − 2〈η̂, ε̂0〉+ 〈ε̂0, ε̂0〉+ 2 = − 1 2 ( |η| − 2|η ∩ ε0|+ |ε0| ) + 2 = − 1 2 |η + ε0|+ 2 where we recall that the summation is done in Π0 (and so is the symmetric difference of the two sets). We work now to simplify the expression ∏ v∈ε0 e−izv/2 ∑ w∈Λ u−〈w,w+2bη−2bε0〉 ∏ v∈A[2] exp ( i( ∫ w+bη γv)zv ) (−1)w. To begin with, we note that exp ( i( ∫ bη γv)zv ) = exp(izv/2) if v ∈ η (and is 1, otherwise), and so this term is ∏ v∈ε0\η e−izv/2 ∏ v∈η\ε0 eizv/2 ∑ w∈Λ u−〈w,w+2bη−2bε0〉 ∏ v∈A[2] exp ( i( ∫ w γv)zv ) (−1)w Because of the fact that the intersection form restricted to Λ is diagonal, we 37can write the latter sum as a product ∑ w∈Λ u−〈w,w+2bη−2bε0〉 ∏ v∈A[2] exp ( i( ∫ w γv)zv ) (−1)w = ∏ v∈η\ε0 ∞∑ k=−∞ (−1)ku2k 2+2keikzv · ∏ v∈ε0\η ∞∑ k=−∞ (−1)ku2k 2−2keikzv · ∏ v/∈η+ε0 ∞∑ k=−∞ (−1)ku2k 2 eikzv . By rearranging the summation index on the middle product it can be written as ∏ v∈ε0\η ∞∑ k=−∞ (−1)ku2k 2−2keikzv = ∏ ε0\η ( − eizv ) ∞∑ k=−∞ (−1)ku2k 2+2keikzv and so the whole term becomes ∏ v∈ε0\η e−izv/2 ∏ v∈η\ε0 eizv/2 ∑ w∈Λ u−〈w,w+2bη−2bε0〉 ∏ v∈A[2] exp ( i( ∫ w γv)zv ) (−1)w = ∏ v∈η+ε0 eizv/2 ∞∑ k=−∞ (−1)ku2k 2+2keikzv ∏ v/∈η+ε0 ∞∑ k=−∞ (−1)ku2k 2 eikzv Since we have that ∞∑ k=−∞ (−1)ku2k 2 eikz = 1 + 2 ∞∑ k=1 (−1)k cos(kz)u2k 2 and also that eiz/2 ∞∑ k=−∞ (−1)ku2k 2+2keikz = 2 ∞∑ k=0 sin ( (2k + 1)z/2 ) u2k 2+2k 38the conclusion follows. 3.3 Proof of Theorem 1.3 To prove Theorem 1.3 it remains to study the functions h and g given above. As they are written in terms of trigonometric functions, we begin by writing them in terms of Chebyshev polynomials. Recall that Chebyshev polynomials Tn(x) are given by the relationship Tn(cos θ) = cos(nθ). They can equivalently be defined recursively via T0(x) = 1 T1(x) = x Tn(x) = 2xTn−1(x)− Tn−2(x). (3.2) They also satisfy the relationships T2n+1(sin θ) = (−1) n sin ( (2n+ 1)θ ) and Tn(1− 2x 2) = (−1)nT2n(x). With these last two in mind, we have the following fact. 39Proposition 3.11. The functions h(x, u) and g(x, u) can be written as h(z, u) = 2 ∞∑ n=0 T2n+1 ( sin(z/2) ) u2n 2+2n g(z, u) = 1 + 2 ∞∑ n=1 T2n ( sin(z/2) ) u2n 2 Proposition 3.11 and Lemma 3.10 give us an explicit description for the (relevant portion of the) potential function of the orbifold [A/± 1]. We now perform the substitution xv = 2 sin(zv/2) (See Theorem 2.13), and use the following result from Chapter 6. Theorem 3.12 (Theorem 6.1). Let H(x, u) = 2 ∑∞ n=0 T2n+1(x/2)q n2+n and G(x, u) = 1 + 2 ∑∞ n=1 T2n(x/2)q n2. Then H(x, u) = (q2; q2)3∞ ∞∑ k=0 Ak(q 2)x2k+1 G(x, u) = (q; q)∞ (−q; q)∞ ∞∑ k=0 Ck(q)x 2k where (a; q)∞ = ∏∞ k=0(1− aq k) is a q-Pockhammer symbol, and Ak, Ck are MacMahon’s generalized sum-of-divisors functions (See [16]). We are now in a position to prove theorem 1.3. Proof of theorem 1.3. From Lemma 3.10, Proposition 3.11, and Theorem 403.12, we can write Fη = u2 ∆(u2) ( u 1 2 |η+ε0|−2 ∏ v∈η+ε0 (u4;u4)3∞ ∞∑ k=0 Ak(u 4)x2k+1v · ∏ v/∈η+ε0 (u2;u2)∞ (−u2;u2)∞ ∞∑ k=0 Ck(u 2)x2kv + u 1 2 |η+ε1|−2 ∏ v∈η+ε1 (u4;u4)3∞ ∞∑ k=0 Ak(u 4)x2k+1v · ∏ v/∈η+ε1 (u2;u2)∞ (−u2;u2)∞ ∞∑ k=0 Ck(u 2)x2kv ) Collecting all the Pockhammer symbols (and the extra powers of u) we find (for the ε0 term) u 1 2 |η+ε0|−2 u2 ∆(u2) ( (u4;u4)3∞ )|η+ε0| ( (u2;u2)∞ (−u2;u2)∞ )16−|η+ε0| . Using the fact that u 2 ∆(u2) = (u 2;u2)−24∞ and that (q; q)∞(−q; q)∞ = (q 2; q2)∞ this simplifies to ( u ( (u2;u2)∞(−u 2;u2)2∞ )4 )1 2 |η+ε0|−2 Using a theorem of Legendre [15] which states that ( (q; q)∞(−q; q)2∞ )4 = ∑∞ k=0 σ1(2k + 1)q k, it follows that this term is in fact E(u) 1 2 |η+ε0|−2 where E(u) = ∞∑ k=0 σ1(2k + 1)u 2k+1 = 1 16 ϑ2(u) 4 and where ϑ2 is the Jacobi theta function ϑ2(q) = ∑∞ k=0 q (k+1/2)2 . An 41identical computation yields that the terms in front of the ε1 term simplify to E(u) 1 2 |η+ε1|−2. We can now rewrite Fη as Fη = E(u) 1 2 |η+ε0|−2 ∏ v∈η+ε0 ∞∑ k=0 Ak(u 4)x2k+1v ∏ v/∈η+ε0 ∞∑ k=0 Ck(u 2)x2kv + E(u) 1 2 |η+ε1|−2 ∏ v∈η+ε1 ∞∑ k=0 Ak(u 4)x2k+1v ∏ v/∈η+ε1 ∞∑ k=0 Ck(u 2)x2kv . This is the generating function for counting hyperelliptic curves f : Σ → An with the property that f(w) ∈ A[2] for all Weirstrass points w where we include the data of how many Weirstrass points lie on a given 2-torsion point. Specifically, the coefficient of the monomial ∏ v∈A[2] x k(v) v gives the number of such curves with k(v) Weirstrass points having as image the point v ∈ A[2]. From this description, the conclusion follows. Corollary 3.13. The coefficient of a monomial ∏ v∈A[2] x k(v) v in the gener- ating function F lies in the ring of quasimodular forms. Proof. This follows directly from Theorem 1.3 and from Corollary 6.4. 3.4 A simple consequence One consequence of this formula is a prediction for the lower bound on the arithmetic genus of a hyperelliptic curve in an Abelian surface A, which we prove. Note that the lowest degree terms of Ak(q) and Ck(q), respectively, are 42(k+1 2 ) and k2. Thus the lowest degree term of E(u) 1 2 |S|−2 ∏ v∈S Ak(v)−1 2 (u4) ∏ v/∈S Ck(v) 2 (u2) (see equation (1.1)) is given by −2 + ∑ v∈S 1 2 + ∑ v∈S 4 (1 2(k(v)− 1) + 1 2 ) + ∑ v/∈S 2 (k(v) 2 )2 = −2 + ∑ v∈A[2] 1 2k(v) 2. It follows that this formula predicts that the minimal arithmetic genus of a hyperelliptic curves in A (with discrete data k) is −1 + ∑ v∈A[2] 1 2k(v) 2 by the definition of Fg,k(u). We now show without assuming the crepant resolution conjecture that this is true. Theorem 3.14. Fix k, and let g be such that 2g+2 = |k|, and suppose that A has Picard number 1. Then the minimal arithmetic genus of a hyperelliptic curve in A with discrete data k is ∑ v∈A[2] 1 2k(v) 2 − 1. Proof. Given such discrete data, the geometric genus is given by g = 12 |k|−1. For each v ∈ A[2] with k(v) > 1, we note that any curve which produces such data must be nodal, since more than one Weirstrass point will have the same image. More specifically, we introduce at least (k(v) 2 ) nodes for each 43k(v) > 1. As such, the total arithmetic genus is given by 1 2 |k| − 1 + ∑ v∈A[2] ( k(v) 2 ) = 12 ∑ v∈A[2] k(v)− 1 + ∑ v∈A[2] 1 2 ( k(v)2 − k(v) ) = −1 + ∑ v∈A[2] 1 2k(v) 2 as claimed. Corollary 3.15. There are no smooth hyperelliptic curves in A of genus greater than 5. Proof. We first note that if any k(v) > 1, then there must be at least one node in the image curve, as we have two Weirstrass points with the same image, v. Next, note that due to the requirement that P , the set of those v with k(v) odd must be congruent to εi mod K (or more accurately, mod Π3) yields that no more than twelve of the sixteen 2-torsion points can have k(v) = 1 (this maximal case is when P is the complement of ε0). In this case, we have that 2g + 2 = 12, or g = 5. 44Chapter 4 Proofs for Low Genera 4.1 Introduction We will prove the formula (1.1) for the case of genus 1 and genus 2 curves. Recall that a hyperelliptic curve of genus g yields a P1 with 2g + 2 stacky points, and so to enumerate genus g curves we need only consider those monomials of total degree 2g + 2. For genus 1, this is degree 4 (we will explain below what a genus one curve in A, which should generically have no elliptic curves, means), and for genus 2, this is of degree 6. We assume as in Section 2.3 that the Abelian surface is a product A ∼= S × F of non-isogenous elliptic curves. Recall that there is then an elliptic fibration A/± 1 → S/± 1 with general fibre F , and with four special fibres F/ ± 1 over the points v ∈ S[2]. We will further (by abuse of notation) denote by S and by F the classes in A/ ± 1 of a section S/ ± 1 and fibre F/± 1, respectively. In the case that the Picard number of A is 1 and that the curve class is primitive, the source curves are combs with collapsing teeth. This allows us to perform the substitution given in in Theorem 2.13. When the Picard number of A is greater than 1—as is the case when A ∼= S × F—or when the class is not primitive, it is possible that there are collapsing components 450 × × × × n1 n2 Figure 4.1: A collapsing component joining two non-collapsing ones 46which join two components mapped into A with non-zero degree (see figure 4.1). These do not contribute to the Gromov-Witten invariant (see Proposi- tion 5.9). From these two considerations, we can consider only those maps which do not collapse any components. Let f : Σ → [A/ ± 1] be a rational curve. As we saw, the source is a tree of rational curves Σ = Σ0 ∪ Σ1 ∪ · · · ∪ Σk, and such that f∗[Σ0] = S and f∗[Σi] = niF for i ≥ 1 with ∑ ni = n (and ni > 0, as discussed above). Moreover, the leaves of the tree of curves must have at least 3 stacky points on them (plus a node which connects them to the tree). For convenience, we now need to label the 2-torsion points of A. We label them as 2-torson from F d47d47 E3d63d62 d61d60 d56d57 d58d59 E7 E11 E15 E2 E6 E10 E14 Translates of Sd111d111 d118d118d109d109 d109d109 d109d109 d109d109 d109 d104d104d81d81d81d81d81d81d81d81d81 E1 E5 E9 E13 E0d55d54 d53d52d48d49 d50d51E4 E8 E12 2-torsion from Sd111d111 (4.1) with monomials labeled similarly. With this labelling, we have the explicit descriptions ε0 = 12(E0+E4+E8+E12) and ε1 = 1 2(E1+E2+E3)+ 1 2(E4+ E8 + E12). 474.2 Genus 1 The genus 1 case is somewhat of an aberation; there are of course no elliptic curves in a generic Abelian surface. It turns out that the genus 1 case corresponds to a ‘polarization’ of type (1, 0). By Poincare´’s Reducibility Theorem [14, Chapter 5, Theorem 3.5], such an A is isogenous to a product of elliptic curves. Since the source curve is genus 1, on the orbifold side we have exactly 4 marked stacky points. From the description above, if the source curve were to consist of more than one component, it would have at least 6 marked points, and so the source curve must be irreducible with f mapping it iso- morphically onto a section S/±1. There is only one such map (as S, F were chosen to be generic). The only term with monomials of degree 4 from all of the functions Fη are x0x4x8x12, x1x5x9x13, x2x6x10x14, and x3x7x11x15. If we look at the formula (1.1), the prediction for each of these is also 1, and so the conjecture is verified. It should be noted that in terms of computing the actual number of such curves, these latter three monomials yield curves that are simply translations of the first, and so the number of genus 1 curves in A in the class S is 1, as we would expect. 4.3 Genus 2 The component which maps isomorphically onto a section curve S/±1 must have three of the six marked points, and so our source curve must consist 48of exactly two components, Σ = Σ0 ∪ Σ1, and where f∗[Σ1] = nF . Thus the component Σ1 is an n-fold cover of F/± 1 (with certain data about the images of the stacky points in Σ1), and so we are reduced to computing the number of such covers. Analogous to the genus 1 case, since we only care up to translation in A, we need only focus on certain monomials. These are x1x2x3x4x8x12 x 3 0x4x8x12 x0x4x8x12x 2 1 x0x4x8x12x 2 2 x0x4x8x12x 2 3 and the corresponding predictions are E(u) A1(u 4) C1(u 2) C1(u 2) C1(u 2). The first comes simply from the term ε1 and the others all come from ε0. In each case, we are counting maps Σ1 → [F/ ± 1] of a curve with four Z/2-points to the orbifold [F/± 1]. In this case, we can lift to the cover E d47d47 d15d15 F d15d15 Σ1 d47d47 F/± 1 and so we can compute this by counting covers of F by elliptic curves E satisfying certain conditions based on the image of the 2-torsion of E. These are the following. 1. The first monomial corresponds to those maps E → F such that the 2-torsion of E surjects onto the 2-torsion of F . 492. The second monomial corresponds to those maps such that all 2- torsion of E maps to 0 ∈ F . 3. The last three monomials all correspond to the three possible cases where E[2] surjects onto a subgroup 〈ei〉 for ei a non-zero 2-torsion point in F . We first need a classically known fact. Proposition 4.1. Let E be an elliptic curve. Then the number of degree n isogenies F → E is given by σ1(n) = ∑ d|n d. Proof. As maps of Abelian varieties are determined by their lattices, count- ing degree n isogenies F → E is the same as counting index n sublattices of a fixed rank 2 lattice. Up to change of basis of the respective lattices, this is the same as counting matrices of the form    a b 0 d    with ad = n and 0 ≤ b < d. This is clearly equal to σ1(n) as claimed. All that is now required are the following lemmata. We will provide a proof of Lemma 4.4, the rest of them having a similar flavour. Lemma 4.2. Suppose that g : E → F is a map of elliptic curves. Then the degree of g is odd if and only if the 2-torsion surjects. In this case the generating function is simply the odd-degree part of A1(u), i.e. E(u). 50Lemma 4.3. Suppose that g : E → F is a map of elliptic curves such that g(E[2]) = 0. Then g factors through the degree 4 map E g d47d47 d35d35 F F/F [2] d59d59 d120 d120 d120 d120 d120 d120 d120 d120 d120 In this case, we are actually counting all covers of the curve F/F[2]—but this is of course the same as counting covers of degree n/4 of an arbitrary curve. That is, we obtain the function A1(u4) as desired. Lemma 4.4. Suppose that g : E → F is a map of elliptic curves such that g(E[2]) surjects onto the subgroup 〈v〉 for some non-zero v ∈ F [2]. Then g factors through the degree 2 map E g d47d47 d34d34 F F/〈v〉 d60d60 d121 d121 d121 d121 d121 d121 d121 d121 Proof. As before, we consider maps of elliptic curves via the maps on their underlying lattices. That is, we consider the map g˜ : ΛE ↪→ ΛF . Choose a basis λ1, λ2 of ΛF , and µ1, µ2 of ΛE such that 12λ1 represents the 2-torsion point v. We will show that its image lies in the sublattice 〈λ1, 2λ2〉, which proves the lemma. The condition above yields (up to labelling) that g ( 1 2µ1 ) ≡ 12λ1 (mod ΛF ) and g ( 1 2µ2 ) ∈ ΛF 51or equivalently that g(µ1)− λ1 ∈ 2ΛF and g(µ2) ∈ 2ΛF . It follows that g(ΛE) ⊂ 〈λ1, 2ΛF 〉 = 〈λ1, 2λ2〉 as claimed. In this final case, we are counting those covers of degree n/2 of F/〈ei〉 (i.e. A1(u2)) less those whose 2-torsion is all mapped to zero (i.e. A1(u4), from before). That is, the generating function is A1(u2)−A1(u4) = C1(u2). These three lemmas together prove the conjecture for the genus 2 case. 4.4 Proof of Go¨ttsche’s genus 2 formula If we consider all the previous cases, it follows that the number of genus 2 curves in A up to translation in A is given by the sum of all of the given terms above. That is, if we let F2(u) denote the number of genus 2 curves in A up to translation, then we have F2(u) = E(u) +A1(u 4) + 3 ( A1(u 2)−A1(u 4) ) = E(u) + 3A1(u 2)− 2A1(u 4). (4.2) It is clear that the odd powers of u in the right-hand side are the same as those in A1(u). The fact that the even powers match those of A1(u) will follow from the following lemmata. We once more prove only the second, the first being similar. 52Lemma 4.5. Let n > 0 be congruent to 2 (mod 4). Then σ1(n) = 3σ1(n/2). Lemma 4.6. Let n > 0 be congruent to 0 (mod 4). Then σ1(n) = 3σ1(n/2)− 2σ1(n/4). Proof. Write n = 2km with 2 notbar m and with k ≥ 2. Then as σ1 is a multi- plicative function, we have that σ1(n) = σ1(2 k)σ1(m) = (2k+1 − 1)σ1(m) = ( 3(2k − 1)− 2(2k−1 − 1) ) σ1(m) = 3σ1(n/2)− 2σ1(n/4) as claimed. In [11], the following is proven. Theorem 4.7 (Go¨ttsche, Theorem 3.2). Let (A,L) be a polarized Abelian surface with polarization of type (1, n). Then the generating function for the number of genus 2 curves in the linear system |L|, summed over all polarization types, is given by F˜2(u) = ∞∑ n=1 n2σ1(n)u n = D2A1(u) 53To see that these are equivalent, we note that the difference between the two counts—that is, curves in a fixed linear system vs. curves up to translation—comes from translating by elements in the kernel of the isogeny A → Â = Pic0(A) given by a mapsto→ L ⊗ t∗aL −1. If the polarization is of type (1, n), then the map on lattices H1(A) → H1(Â) can be represented by the matrix (2.2) (see Section 2.1). The map is thus of degree n2, and so the kernel consists of exactly n2 elements. This yields the claim. 54Chapter 5 Structure of the Moduli Space In this Chapter we gather a few facts about the structure of the moduli space M ([A/± 1];k;n), and in particular about its reduced virtual fundamental class. All throughout we assume that A is a generic Abelian surface and has Picard number 1. As stated before, the Gromov-Witten theory of an Abelian or K3 surface, strictly speaking is trivial, as any of these can be deformed into a non- algebraic surface. To account for that, we look at a reduced theory for these surfaces. For more detail, see [7, 6, 17]. To construct the reduced class on M ([A/± 1]; 2g + 2;n), we use the following approach. Let A be a fixed polarized Abelian surface with polar- ization of type (1, n), and let B be the family of Ka¨hler metrics arising from the hyperka¨hler structure. Note that B ∼= S2, the real 2-sphere. Let A pi −→ B be the family of Abelian surfaces over B given by this family of metrics. That is, Ab = pi−1(b) is A with the Ka¨hler structure given by b. We can take the fiber-wise quotient to obtain the family [A / ± 1] pi −→ B which we use to construct our reduced class. 55Remark 5.1. The family A is not an algebraic family. In fact, to work with this we must leave the algebraic setting and move into the complex analytic category. However, while it is not algebraic, it is fiber-wise Ka¨hler, and so we are still able to work with Gromov-Witten invariants of this family. It is worth noting that the construction of the reduced class for families of K3 surfaces has been done in [17] purely in the algebraic category. It seems likely that their methods would work similarly to obtain an algebraic reduced class for the moduli space of maps into an Abelian surface, and that we should similarly be able to do this for the orbifold [A/ ± 1]. We do not however pursue this approach in this work. In the end we use the notion of the Twistor family as it is a well- understood and concrete approach. This concrete approach suits us well, as it permits us to define our invariants with as little pain as possible. Note that we have an inclusion ι : [A/ ± 1] → [A / ± 1] as one of the fibres. For brevity’s sake, define M = M ([A/± 1]; 2g + 2;β) and M ′ = M ([A /± 1]; 2g + 2; ι∗β) . We have the following lemma. Lemma 5.2. Let [A/ ± 1] be as above, and suppose that C ⊂ [A/ ± 1] is a holomorphic curve. Then the only Ka¨hler structure in B that has a holomorphic curve in the class [C] is the original Ka¨hler structure for which C is holomorphic. Proof. Suppose that there are two complex structures b, b′ for which there are curves in the class [C] which are holomorphic. We can then lift these 56to the cover A → [A/ ± 1] to obtain two differing complex structures on A which support curves in a fixed homology class; this contradicts [6, Lemma 3.4]. From this we obtain the following. Proposition 5.3. The moduli spaces M and M ′ are isomorphic in the cat- egory of complex analytic stacks. Proof. There is an obvious mapM →M ′ induced by the inclusion ι. Specif- ically, for a family C˜ d47d47 d15d15 A d15d15 C d47d47 d15d15 [A/± 1] T si d71d71 we can compose with the inclusion ι to obtain C˜ d47d47 d15d15 A d15d15 d47d47 A d15d15 C d47d47 d15d15 [A/± 1] ι d47d47 [A /± 1] T si d71d71 For the reverse direction, note that all holomorphic curves in [A /± 1] land 57in a fixed fibre [A/± 1]b. This given a diagram C˜ d47d47 d15d15 A d15d15 C f d47d47 d15d15 [A /± 1] T si d71d71 we note that the map f factors through this fixed fibre [A/±1]b. This yields the inverse map j : M ′ →M . We compute the virtual dimension of M ′ to be ∫ ι∗β c1 ( [A /± 1] ) + (1− g)(dim[A /± 1] ︸ ︷︷ ︸ =3 −3) + (2g + 2)− 2g+2∑ i=1 age(pi) ︸ ︷︷ ︸ =1 = 0 and so we have a virtual fundamental class [M ′]vir in degree zero. Definition 5.4 We define the reduced virtual fundamental class on M to be [M ]red = j∗[M ′]vir. We next investigate the structure of the spaceMk,n = M ([A/± 1];k;n). Recall that, for k : A[2] → Z≥0, this is the moduli space of genus 0 twisted stable maps into [A/± 1] such that k(v) stacky points have as image k(v). Denote by λ turnstileleft k a ‘multipartition’ of k. That is, a collection of partitions λv turnstileleft k(v) with parts (λv1, . . . , λ v rv), indexed by v ∈ A[2]. We say that a 58twisted stable map has partition type λ if we can write the source curve as Σ = Σ0 ∪ ⋃ v∈A[2] rv⋃ i=1 Σvi with each Σvi a (potentially nodal) genus 0 curve with λ v i marked Z/2-points, and where Σvi is attached to the curve Σ0 at some point, which is will be stacky depending on the parity of λvi . The map then collapses each Σ v i to the stacky point in [A/± 1] corresponding to v ∈ A[2]. Lastly, denote byM λ,n the closed substack consisting of those maps with partition type λ. The main result is the following. Proposition 5.5. If the Picard number of A is 1, then Mk,n = ∐ λturnstileleftk M λ,n. Proof. As stated before, each M λ,n is a closed substack, and so we must show that they are also open in Mk,n. We claim that any deformation of the nodes connected a tooth to the handle cannot be smoothed. We first note that the collection of rational curves (excluding collapsing components) in [A/ ± 1] is 0-dimensional. Indeed, if it were not then by looking at the proper transform we would obtain a positive dimensional family of rational curves in Km(A), which cannot exist (See, e.g., [8]). Now, since the Picard number is 1 and the curve class is primitive, the source curve must be the normalization of its image. We will assume for simplicity that the number of teeth on the curve is 1, and that this tooth is itself irreducible. Thus if we were to smooth the node joining this tooth, 59the resulting source curve must be irreducible. Consider now a flat family of twisted stable maps into [A/ ± 1] over a punctured base T ′ = T \ {p}. C d47d47 d15d15 [A/± 1] T ′ Since each fibre is the normalization of the image curve in [A/ ± 1], this family must be constant. As the moduli spaces Mk,n are separated and proper, there is a unique way to fill in the central fibre, which in this case must also be a constant family. In particular, the resulting central fibre is also the normalization of the image, and so must have no teeth. It follows that no nodes joining the teeth to the handle can be smoothed. The main result of this decomposition is the following. Recall (See Proposition 2.11) that all curves in M ([A/± 1]; 2g + 2;n) are comb curves. We claimed that the components corresponding to even parts contribute zero to the Gromov-Witten invariant—this is equivalent to saying that if a comb curve has a tooth which is joined to the handle at a non-orbifold point, then it contributes zero to the total invariant. Lemma 5.6. Let v ∈ A[2] be fixed, and let M 0;2g+2|2k,v denote the compo- nent consisting of those twisted stable maps into [A/ ± 1] which collapse a component to v with 2k marked stacky points, and with 2g+2 marked stacky 60points elsewhere. Then ∫ [M0;2g+2|2k,v ]red 1 = 0 Proof. We will assume without loss of generality (by induction on the num- ber of collapsing components) that the source curve consists of two compo- nents, Σ1 and Σ2 joined at a non-stacky point P , and with Σ1 being the handle. We begin with a little notation. Define M1 = M ◦ ([A/± 1]; 2g + 2, 1;n) to be the moduli space of twisted stable maps into [A/±1] with no collapsing components and with one ordinary (i.e. non-stacky) marked point. This has (reduced) virtual dimension 1. Similarly, define M2 = M (BZ/2; 2k, 1; 0) to be the moduli space of twisted stable maps into BZ/2 = [C2/± 1], which we think of as the local model for one of the stacky points in [A/± 1]. This also has virtual dimension 1. Now, unlike the case where the node is stacky, we do not have an iso- morphism M0;2g+2|2k,v ∼= M1 ×M2 due to the fact that the curves are joined at a non-stacky point. What we 61do have, however, is a morphism ι : M0;2g+2|2k,v →M1 ×M2 which fits into the gluing diagram M0;2g+2|2k,v d15d15 ι d47d47 M1 ×M2 ev×ev d15d15 I[A/± 1] ∆ d47d47 I[A/± 1]× I[A/± 1] where the evaluation maps on the right are from the non-stacky points. From this we obtain (see [2, Proposition 5.3.1]) that [M0;2g+2|2k,v] red = ∆! ( [M1] red × [M2] red). Now, since the evaluations are at non-stacky points, they in fact lie in the non-twisted sector, which is [A/±1]. Since [A/±1] satisfies Poincare´ duality (rationally, at least), we can choose a basis (γi) of the cohomology of A/±1 so that this is given by ∆! ( [M1] red × [M2] red) = ∑ i ∫ ev∗[M1]red γi ∫ ev∗[M2]red γi. Since [M1]red and [M2]red are classes in H2, we see that the only cohomol- ogy classes which may contribute are those in dimension 2. However, since the map ev : M2 → [A/ ± 1] is constant (recall that this is a collapsing component), it doesn’t intersect any classes in H2, and so each of the in- tegrals ∫ ev∗[M2]red γi are zero. It follows then that [M0;2g+2|2k,v] red = 0 as claimed. We will next provide the proofs of several facts concerning the non- 62enumerative nature of the Gromov-Witten invariants stated in section 2.2. First, recall that we have the decomposition of moduli spaces Uλ,n = M ◦ ([A/± 1];kλ;n)× ∏ v∈A[2] rv∏ i=1 M (BZ/2;λvi + 1; 0) together with a projection map pλ : Uλ,n →M◦ ([A/± 1];kλ;n). Theorem 5.7 (Theorem 2.12). Let λ = (λv)v∈A[2] be a collection of parti- tions of k, all of which consist of odd parts. Then the virtual degree of pλ is ( − 14 ) 1 2 (|k|−|kλ|). That is, (pλ)∗[Uλ,n] red = ( − 1 4 ) 1 2 (|k|−|kλ|)[ M◦ ([A/± 1];kλ;n) ]red . The proof of this is virtually identical to that in [20, Section 3.6]. Let pi : C → Uλ,n denote the universal curve, and let Σ = Σ0 ∪ Σ1 ∪ · · · ∪ Σk denote a comb curve. Since all deformations of the nodes which join the teeth to the handle are obstructed, we have the exact sequence 0 → ⊕ Pi pi∗(TPiΣ0 ⊗ TPiΣi) → Obs(f) → Obs(Σ, f) → 0. (5.1) We will compute Obs(f), and use this exact sequence to compute the ob- struction bundle Obs(Σ, f). Lemma 5.8. Over a point [f : Σ → A /± 1], the bundle Obs(f) is isomor- phic to Obs(f) ∼= H1 ( Σ0, f ∗T [A /± 1]|Σ0 ) ⊕ k⊕ i=1 H1(Σi, ρ1 ⊕ ρ1) 63where ρ1 is the non-trivial representation of Z/2. Proof. As above, let f : Σ → [A / ± 1] be a comb curve with teeth Σi for 1 ≤ i ≤ k, and let T = f∗T [A /± 1]. Recall that over such a point that Obs(f) ∼= H1(Σ, T ). To compute this, we look at the normalization sequence H0(Σ0, T )⊕ k⊕ i=1 H0(Σi, T |Σi) → k⊕ i=1 H0(Pi, T |Pi) → → H1(Σ, T ) → k⊕ i=0 H1(Σi, T |Σi) → 0 where Pi is the node joining Σi to Σ0. As f is representable, the image of Pi must lie in the twisted sector, and so we see that T |Σi ∼= T |Pi ∼= ρ1⊕ρ1⊕ρ0 where ρ1 and ρ0 denote the non-trivial and trivial representations of Z/2, respectively. Since Σi has stacky points, and since Pi ∼= BZ/2, we have that H0(Σ0, ρ1) ∼= 0 (and similarly for Pi). Moreover, it is clear that H0(Σi, ρ0) → H0(Pi, ρ0) is surjective. It follows then that Obs(f) ∼= H1(Σ, T ) ∼= H1(Σ0, T |Σ0)⊕ k⊕ i=1 H1(Σi, T |Σi). Since T |Σi ∼= ρ1 ⊕ ρ1 ⊕ ρ0 and ρ0, being the trivial representation, has no higher cohomology, the lemma follows. Proof of Theorem 2.12. Since M◦0;kλ is zero-dimensional, on components of Uλ,n the summand of Obs(f) coming from H1 ( Σ0, f∗T [A / ± 1]|Σ0 ) is a 64fixed vector space, and so it corresponds to a trivial summand. Moreover, as discussed in [21], the terms H1(Σi, ρ1⊕ρ1) contributed a dual Hodge bundle summand; that is, Obs(f) ∼= O⊕d⊕ ⊕k i=1 E ∨ i ⊕E ∨ i . Our exact sequence (5.1) thus reads 0 → ⊕ Pi pi∗(TPiΣ0 ⊗ TPiΣi) → O ⊕d ⊕ k⊕ i=1 E∨i ⊕ E ∨ i → Obs(Σ, f) → 0. from which we compute that the total Chern class of Obs(Σ, f) over Uλ,n is given by c(O⊕d) ︸ ︷︷ ︸ =1 k∏ i=1 c(E∨i ) 2 c ( pi∗(TPiΣ0 ⊗ TPiΣi) ) . We need to integrate this obstruction class over the fibres of the projection map pλ : Uλ,n → M◦ ([A/± 1];kλ;n). Specifically, we need to compute the integrals (using the notation of [10]) ∫ M(BZ/2;2gi+2;0) c(Ei)2 1− 12ψ1 (5.2) which are computed in [21, 10] to be (−14) gi . Since the fibre is the product ∏ v∈A[2] rv∏ i=1 M (BZ/2;λvi + 1; 0) 65it follows that the degree of the pushforward is ∏ v∈A[2] rv∏ i=1 ( − 1 4 )1 2 (λ v i−1) = ∏ v∈A[2] ( − 1 4 )1 2k(v)− 1 2 rv = ( − 1 4 )1 2 (|k|−|kλ|) as claimed. Consider now case that A ∼= S ×F . The same reasoning as above yields that on any component of M ([A/± 1]; 2g + 2;n) which collapses a compo- nent between two non-collapsing components (see figure 4.1), the virtual class is obtained by computing the integral ∫ M(BZ/2;2g+2;0) c(E)2 (1− 12ψ1)(1− 1 2ψ2) which arises due to the two nodes whose smoothings are obstructed. Proposition 5.9. Let g > 0. Then the integral ∫ M(BZ/2;2g+2;0) c(E)2 (1− 12ψ1)(1− 1 2ψ2) is zero. Proof. We follow a method similar to the one given in [10] to prove that (5.2) is equal to ( −14 )g . More specifically, we assemble the Gromov-Witten invariants into a generating function, which we will see must be equal to zero. Let X = [C2/ ± 1]. Let 1 and v, respectively, denote the generators 66of the untwisted and twisted sectors of H∗(X ), and let 〈· · · 〉 denote the integral ∫ M · · · where the integral is over the appropriate moduli space of genus 0 twisted stable maps into X . Let g > 1, and let a, b be non-negative integers. The topological recursion relations in this case yield 〈v2g−1, τa+1v, τbv, v〉 = 2 g−1∑ i=1 ( 2g − 1 2i ) 〈v2i, τav, v〉〈v 2(g−i)−1, τbv, v, v〉 + 2 g∑ i=1 ( 2g − 1 2i− 1 ) 〈v2i−1, τav, 1〉〈v 2(g−i), τbv, v, 1〉 = 2 g−1∑ i=1 ( 2g − 1 2i ) 〈v2i+1, τav〉〈v 2(g−i)+1, τbv〉 + 2 g∑ i=1 ( 2g − 1 2i− 1 ) 〈v2i−1, τa−1v〉〈v 2(g−i)+1, τb−1v〉 where the second equality is given by the string equation. For g = 1, this reads 〈v2, τa+1v, τbv〉 = 2〈v, τav, 1〉〈v, τbv, 1〉 due to the requirement that each component of the curve have an even number of stacky points. Multiplying both sides of this by 2−a−b−1 and summing a and b from 0 67to ∞ yields 〈 v2g, v 1− 12ψ1 , v 1− 12ψ2 〉 − 〈 v2g+1, v 1− 12ψ2 〉 = g−1∑ i=1 ( 2g − 1 2i )〈 v2i+1, v 1− 12ψ1 〉〈 v2(g−i)+1, v 1− 12ψ2 〉 + 1 4 g∑ i=1 ( 2g − 1 2i− 1 )〈 v2i−1, v 1− 12ψ1 〉〈 v2(g−i)+1, v 1− 12ψ2 〉 (5.3) We now assemble these into a generating series. Let H(q) and h(q) be given by H(q) = ∞∑ g=1 〈 v2g, v 1− 12ψ1 , v 1− 12ψ2 〉 q2g−1 (2g − 1)! h(q) = q + ∞∑ g=1 〈 v2g+1, v 1− 12ψ 〉 q2g+1 (2g + 1)! From [10], we see that h(q) = 2 sin(q/2). Moreover, if we multiply (5.3) by q2g−1 (2g−1)! and sum from g = 1 to ∞, we obtain H(q)− h′′(q) = ∞∑ g=1 g−1∑ i=1 〈 v2i+1, v 1− 12ψ1 〉 q2i (2i)! 〈 v2(g−i)+1, v 1− 12ψ2 〉 q2(g−i)−1 (2g − 2i− 1)! + 1 4 ∞∑ g=1 g∑ i=1 〈 v2i−1, v 1− 12ψ1 〉 q2i−1 (2i− 1)! 〈 v2(g−i)+1, v 1− 12ψ2 〉 q2(g−i) (2g − 2i)! (5.4) A somewhat tedious computation yields that the right-hand side of (5.4) is 68equal to ( h′(q)− 1 ) h′′(q) + 1 4 h(q)h′(q) = h′(q)h′′(q)− h′′(q) + 1 4 h(q)h′(q). Since h′′(q) = −14h(q), this is simply equal to −h ′′(q). It follows then that H(q) = 0. Since 〈 v2g, v 1− 12ψ1 , v 1− 12ψ2 〉 = ∫ M(BZ/2;2g+2;0) c(E)2 (1− 12ψ1)(1− 1 2ψ2) the claim follows. Next, recall that we defined the generating functions Fn(zv) = ∑ k:A[2]→Z≥0 GWk,n ∏ v∈A[2] zk(v)v k(v)! F ◦n(xv) = ∑ k:A[2]→Z≥0 GW ◦k,n ∏ v∈A[2] xk(v)v k(v)! . We prove now the following. Theorem 5.10 (Theorem 2.13). The two generating functions Fn and F ◦n are equal after the substitution xv = 2 sin(zv/2). Proof. We prove this by computing F ◦n ( 2 sin(zv/2) ) , and showing that this is equal to Fn(zv). For simplicity we use the notation A∨ = Map(A[2],Z≥0). Note that 2 sin(zv/2) = ∞∑ lscript=0 ( − 1 4 )lscript z2lscript+1v (2lscript+ 1)! . 69Substituting this into the definition for F ◦n we obtain F ◦n ( 2 sin(zv/2) ) = ∑ k∈A∨ GW ◦k,n ∏ v∈A[2] 1 k(v)! ( ∞∑ lscript=0 ( − 1 4 )lscript z2lscript+1v (2lscript+ 1)! )k(v) = ∑ k∈A∨ GW ◦k,n ∏ v∈A[2] ∞∑ lscript=0 ( − 1 4 )lscript z2lscript+k(v)v (2lscript+ k)(v)! s ( 2lscript+ k(v),k(v) ) where s(k, lscript) = 1 lscript! ∑ a1+···+alscript=k ai odd ( k a1, . . . , alscript ) . Exchanging the order of the summation over lscript and the product over v in this expression we find that F ◦n ( 2 sin(zv/2) ) = ∑ k∈A∨ GW ◦k,n ∑ lscript∈A∨ ( − 1 4 )|lscript| ∏ v∈A z(2lscript+k)(v)v (2lscript+ k)(v)! s ( (2lscript+ k)(v),k(v) ) = ∑ k,lscript∈A∨ GW ◦k,n ( − 1 4 )|lscript| ∏ v∈A z(2lscript+k)(v)v (2lscript+ k)(v)! s ( (2lscript+ k)(v),k(v) ) If we then re-index the summation by letting k′ = 2lscript+k (and for simplicity of notation omitting the ′), we find F ◦n ( 2 sin(zv/2) ) = ∑ k,lscript∈A∨ GW ◦k−2lscript,n ( − 1 4 )|lscript| ∏ v∈A zk(v)v k(v)! s ( k(v), (k− 2lscript)(v) ) The claim then that these two generating functions are equal is equivalent then to the claim that GWk,n = ∑ lscript∈A∨ GW ◦k−2lscript,n ( − 1 4 )|lscript| ∏ v∈A s ( k(v), (k− 2lscript)(v) ) 70This follows from theorem 2.12 and from the following interpretation of the numbers s(k, lscript). The number s(k, lscript) gives the count of all possible ways of partitioning k marked points into lscript (unordered) odd-sized collections of points. In our case, by summing over all possible functions lscript : A[2] → Z≥0, the numbers s ( k(v), (k − 2lscript)(v) ) yield the count of all possible ways of partitioning the k(v) points mapping to a given 2-torsion point v by bubbling off collapsing components (all of which must have odd numbers of marked points). Order does not matter as they all map to the same point. For each such possibility, the Gromov-Witten invariant is then GW ◦k−2lscript (the invariant coming from the non-collapsing component) times ( −14 )|lscript| (the virtual degree of the map which forgets the collapsing components), as discussed above. This proves the theorem. 71Chapter 6 MacMahon’s Generalized Sum-of-Divisors Functions 6.1 Statement of results In this chapter we prove some relations satisfied by the generalized sum-of- divisors functions introduced by P. A. MacMahon in [16, pp. 303, 309]. In particular, as stated in Theorem 1.3, we prove that they are in the ring of quasi-modular forms. Recall that the sum-of-divisors function σk(n) is defined to be σk(n) = ∑ d|n dk. For k = 1, this has as a generating function A1(q) = ∞∑ k=1 σ1(n)q n = ∞∑ k=1 qk (1− qk)2 . 72Generalizing this notion, MacMahon introduces the generating functions Ak = ∑ 0<m1<···<mk qm1+···+mk (1− qm1)2 · · · (1− qmk)2 Ck = ∑ 0<m1<···<mk q2m1+···+2mk−k (1− q2m1−1)2 · · · (1− q2mk−1)2 . (With A0 and C0 defined to be 1) These provide generalizations in the following sense. Fix a positive integer k. We define an,k to be the sum an,k = ∑ s1 · · · sk where the sum is taken over all possible ways of writing n = s1m1+· · ·+skmk with 0 < m1 < · · · < mk, and si > 0. Note that for k = 1 this is nothing but σ1(n), the usual sum-of-divisors function. It can then be shown that we have Ak(q) = ∞∑ n=1 an,kq n. Similarly, we define cn,k to be cn,k = ∑ s1 · · · sk where the sum is over all partitions of n into n = s1(2m1 − 1) + · · ·+ sk(2mk − 1) with, as before 0 < m1 < · · · < mk. For k = 1 this is the sum over all 73divisors whose conjugate is an odd number. As for the case of an,k, we have Ck(q) = ∞∑ n=1 cn,kq n. We recall also that Chebyshev polynomials are defined via the relation Tn(cos θ) = cos(nθ). With these we form the following generating functions. F (x, q) := 2 ∞∑ n=0 T2n+1(12x)q n2+n G(x, q) := 1 + 2 ∞∑ n=1 T2n(12x)q n2 . The results of this Chapter are the following. Theorem 6.1. We have the following equalities: F (x, q) = (q2; q2)3∞ ∞∑ k=0 Ak(q 2)x2k+1 G(x, q) = (q; q)∞ (−q; q)∞ ∞∑ k=0 Ck(q)x 2k where (a; q)∞ = ∏∞ k=0(1− aq k). 74Corollary 6.2. The functions Ak(q) and Ck(q) can be written as Ak(q) = (−1)k (2k + 1)!(q; q)3∞ ∞∑ n=0 (−1)n(2n+ 1) (n+ k)! (n− k)! q 1 2n(n+1) Ck(q) = (−1)k(−q; q)∞ (2k)!(q; q)∞ ∞∑ n=1 (−1)n2n (n+ k − 1)! (n− k)! qn 2 . Corollary 6.3. The functions Ak and Ck satisfy the recurrence relations Ak(q) = 1 (2k + 1)2k (( 6A1(q) + k(k − 1) ) Ak−1(q)− 2q d dq Ak−1(q) ) Ck(q) = 1 2k(2k − 1) (( 2C1(q) + (k − 1) 2)Ck−1(q)− q d dq Ck−1(q) ) . The final result of this chapter settles a long-standing conjecture of P. A. MacMahon. In his paper [16, p. 328] he makes the claim The function Ak = ∑ an,kqn has apparently the property that the coefficient an,k is expressible as a linear function of the sum of the uneven powers of the divisors of n. I have not succeeded in reaching the general theory... What we prove is the following. Corollary 6.4. The functions Ak(q) are in the ring of quasi-modular forms. 756.2 Proofs Proof of theorem 6.1. Beginning with the series F (x, q), and letting x = 2 cos θ we find F (x, q) = 2 ∞∑ n=0 T2n+1(cos θ)q n2+n = 2 ∞∑ n=0 cos ( (2n+ 1)θ ) qn 2+n = ∞∑ n=0 ( ei(2n+1)θ + e−i(2n+1)θ ) qn 2+n = ∞∑ n=0 ei(2n+1)θqn 2+n + ∞∑ n=0 e−i(2n+1)θqn 2+n where in the latter sum, letting n mapsto→ −n− 1 we obtain F (x, q) = eiθ ∞∑ n=−∞ e2niθqn 2+n. Using the Jacobi triple product we see that this is equal to eiθ ∞∑ n=−∞ e2niθqn 2+n = eiθ(−e−2iθ; q2)∞(−q 2e2iθ; q2)∞(q 2; q2)∞ = (eiθ + e−iθ) ︸ ︷︷ ︸ x (q2; q2)∞ ∞∏ m=1 ( 1 + 2 cos(2θ) ︸ ︷︷ ︸ x2−2 q2m + q4m ) = x(q2; q2)∞ ∞∏ m=1 ( (1− q2m)2 + x2q2m ) = x(q2; q2)3∞ ∞∏ m=1 ( 1 + x2 q2m (1− q2m)2 ) = (q2; q2)3∞ ∞∑ k=0 Ak(q 2)x2k+1 76and thus comparing coefficients of x2k+1 yeilds the result. We ply a similar trick for G(x, k). In that case we have G(x, q) = 1 + 2 ∑ n>0 T2n(cos θ)q n2 = 1 + 2 ∑ n>0 cos(2nθ)qn 2 = ∞∑ n=−∞ e2niθqn 2 which, again, by the Jacobi triple product yields ∞∑ n=−∞ e2niθqn 2 = (q2; q2)∞(−qe 2iθ; q2)∞(−qe −2iθ; q2)∞ = (q2; q2)∞ ∞∏ m=1 ( 1 + 2 cos(2θ) ︸ ︷︷ ︸ x2−2 q2m−1 + q4m−2 ) = (q2; q2)∞ ∞∏ m=1 ( (1− q2m−1)2 + x2q2m−1 ) = (q2; q2)∞(q; q 2)2∞︸ ︷︷ ︸ (q;q)∞ (−q;q)∞ ∞∏ m=1 ( 1 + x2 q2m−1 (1− q2m−1)2 ) = (q; q)∞ (−q; q)∞ ∞∑ k=0 Ck(q)x 2k which completes the theorem. To deduce Corollary 6.2, we begin by expanding the series F (x, q) (and 77similarly, G(x, q)) in powers of x, i.e. F (x, q) = xf0(q) + x 3f1(q) + x 5f2(q) + · · ·+ x 2k+1fk(q) + · · · G(x, q) = g0(q) + x 2g1(q) + x 4g2(q) + · · ·+ x 2kgk(q) + · · · . Now, the coefficients of x2k in 2T2n(12x) and of x 2k+1 in 2T2n+1(12x) are respectively given by 2n(−1)n−k (n+ k − 1)! (n− k)!(2k)! (−1)n−k(2n+ 1) (n+ k)! (n− k)!(2k + 1)! and thus we have fk(q) = (−1)k (2k + 1)! ∞∑ n=0 (−1)n(2n+ 1) (n+ k)! (n− k)! qn 2+n gk(q) = (−1)k (2k)! 2 ∞∑ n=1 (−1)nn (n+ k − 1)! (n− k)! qn 2 . with g0 defined to be 1 + 2 ∑∞ n=1(−1) nqn 2 . As Theorem 6.1 implies that fk(q) = (q2; q2)3∞Ak(q 2) and gk(q) = (q;q)∞ (−q;q)∞ Ck(q), we see that Corollary 6.2 follows. Next, letting f0(q) = ∞∑ n=0 (−1)n(2n+ 1)qn 2+n = (q2; q2)3∞ g0(q) = 1 + 2 ∞∑ n=1 (−1)nqn 2 = (q; q)∞ (−q; q)∞ and defining the operators Dlscript = q ddq − lscript(lscript− 1) and D ′ lscript = q d dq − (lscript− 1) 2, we 78then have that fk(q) = (−1)k (2k + 1)! Dk · · ·D1f0(q) gk(q) = (−1)k (2k)! D′k · · ·D ′ 1g0(q). From these formulae we note that the functions fk, gk satisfy the recur- sion relations fk(q) = −1 (2k + 1)2k ( q d dq − k(k − 1) ) fk−1(q) gk(q) = −1 2k(2k − 1) ( q d dq − (k − 1)2 ) gk−1(q). Noting again that fk(q) = (q2; q2)3∞Ak(q 2) and gk(q) = (q;q)∞ (−q;q)∞ Ck(q), together with the relations q d dq (q2; q2)3∞ = −6(q 2; q2)3∞A1(q 2) q d dq (q; q)∞ (−q; q)∞ = −2 (q; q)∞ (−q; q)∞ C1(q), we now obtain the recurrence relation of Corollary 6.3 between the functions Ak(q) and Ck(q). Our final result requires a bit of explanation. It is well known that the ring of modular forms for the full modular group Γ = PSL2(Z) is the polynomial ring in the generators E4, E6, where E2k(q) = 1 2 ζ(1− 2k) + ∞∑ n=1 σ2k−1(n)q n 79are the Eisenstein series of weights 2k. There are no modular forms of weight 2 for Γ, but E2 = − 124 + ∑∞ n=1 σ1(n)q n is a quasi-modular form (See [13]). The relevant fact for this paper is that the ring of all such objects (which contains the ring of modular forms as a subring) is the ring generated either by E2, E4, and E6, or by q ddq and by E2. Noting then thatA1(q) = E2(q)+ 1 24 , the recurrence relation from Corollary 6.3 implies that each Ak(q) lies in this ring, and hence the conclusion follows. 80Chapter 7 Conclusion 7.1 Summary The main result of this thesis is the complete solution (assuming the crepant resolution conjecture for the resolution Km(A) → A/ ± 1) of the problem stated in Question 1.1 regarding the number of hyperelliptic curves in a generic polarized Abelian surface. In face we provide a stronger result, in that we refine this count based on the discrete data of the image of the Weirstrass points in A[2]. Moreover, we show that there are strong restrictions on what possible patterns of these images are allowed, which is not a priori obvious. The formula obtained from this thesis is also of interest. It has a certain elegance to it, and it provides further connections between geometry, number theory, and combinatorics. For example, it is by no means obvious that counting certain restricted partitions of h − 1 should provide information about the number of hyperelliptic curves of arithmetic genus h in a generic abelian surface A. This also proves once more the importance of the study of modular forms in geometry. Lastly, this work offered as a nice side-effect the proof of a more-than- 90-year old conjecture of P. A. MacMahon’s [16] regarding the patterns that 81arise when studying the functions Ak(q) and Ck(q). While MacMahon did not have available the language of (quasi-) modular forms to describe the generating functions he constructed to count partitions, it is in some ways exactly the result of Corollary 6.4 that he predicted. 7.2 Further research The most obvious direction to guide further research is to remove the reliance of this proof on the crepant resolution conjecture. That is, we would like to prove that the generation functions given hold using only the methods of orbifold Gromov-Witten theory. The first stages and the general method are outlined in Chapter 4 where the case of genus 1 and genus 2 are proven, but for higher genera more care is required. This is first seen in the genus 3 case. As stated in Section 2.3, the moduli space M ([A/± 1]; 2g + 2;n) in the case that A ∼= S × F will be a product of moduli spaces of Hurwitz covers of the rational curve F/ ± 1. In the case of genus 3, it can be shown that this space is one dimensional, although its virtual dimension is zero. Thus we need to take care with virtual fundamental classes and obstruction bundles to obtain our desired results, as well as a careful study of certain moduli spaces of Hurwitz covers. The use of Gromov-Witten theory to study Hurwitz covers has proven to be a fruitful method of study (see [19]), but as yet is limited to some strong restrictions on allowable monodromy. As the monodromy which results from computing the number of higher genus hyperelliptic curves in A ∼= S × F falls out of this class, it would be worthwhile to work to extend these known 82results. A further approach that is of interest is to prove Theorem 1.3 is to follow along the ideas for proving the genus 2 case given in [11]. In that paper, the count of genus 2 curves is given by a purely lattice-theoretic approach. The idea is that a map C → A for an Abelian variety A factors as C d47d47 d39d39d79 d79 d79 d79 d79 d79 d79 d79 d79 d79 d79 d79 d79 d79 J(C) d15d15 A where J(C) is the Jacobian variety of the curve C. Since J(C) is itself an Abelian variety, we can study the map C → A up to translation in A by instead looking at the map H1 ( J(C) ) ∼= H1(C) → H1(A) using this method. As the Jacobian of hyperelliptic curves has some special structure (see [18]), it follows that we should be able to derive these results simply by studying the maps of lattices. Finally, one other direction of interest is inspired by Section 4.4 and Equation (4.2): A1(u) = E(u) + 3A1(u 2)− 2A1(u 4). We see in this case that a certain linear combination of quasi-modular forms of higher level combine to produce one of lower level. An analysis of the genus 3 case yields a similar phenomena for the generation functions A2(u) and for C2(u), and so the question arises under which conditions this can occur. This leads into the Galois theory of modular functions, in an attempt 83to understand the particulars of how linear combinations of non-invariant objects become invariant under the larger group PSL2(Z). 84Bibliography [1] Dan Abramovich, Tom Graber, and Angelo Vistoli. Algebraic orbifold quantum products. In Orbifolds in mathematics and physics (Madison, WI, 2001), volume 310 of Contemp. Math., pages 1–24. Amer. Math. Soc., Providence, RI, 2002. [2] Dan Abramovich, Tom Graber, and Angelo Vistoli. Gromov-Witten theory of Deligne-Mumford stacks. Amer. J. Math., 130(5):1337–1398, 2008. [3] George E. Andrews and Simon C.F. Rose. Macmahon’s sum-of-divisors functions, Chebyshev polynomials, and quasi-modular forms. J. Reine Angew. Math., 2010. To appear. [4] W. Barth, C. Peters, and A. Van de Ven. Compact Complex Surfaces, chapter VIII. Springer-Verlag, 1984. [5] Jim Bryan and Tom Graber. The crepant resolution conjecture. In Algebraic geometry—Seattle 2005. Part 1, volume 80 of Proc. Sympos. Pure Math., pages 23–42. Amer. Math. Soc., Providence, RI, 2009. 85[6] Jim Bryan and Naichung Conan Leung. Generating functions for the number of curves on abelian surfaces. Duke Math. J., 99(2):311–328, 1999. [7] Jim Bryan and Naichung Conan Leung. The enumerative geometry of K3 surfaces and modular forms. J. Amer. Math. Soc., 13(2):371–410 (electronic), 2000. [8] X. Chen. Rational curves on K3 surfaces. J. Algebraic Geom., 8:245– 278, 1999. [9] William Gillam. Hyperelliptic Gromov-Witten Theory. PhD thesis, Columbia University, 2008. [10] William Gillam. Letter to J. wise, 2008. [11] Lothar Go¨ttsche. A conjectural generating function for numbers of curves on surfaces. Comm. Math. Phys., 196(3):523–533, 1998. [12] Tom Graber. Enumerative geometry of hyperelliptic plane curves. J. Algebraic Geom., 10(4):725–755, 2001. [13] Masanobu Kaneko and Don Zagier. A generalized Jacobi theta function and quasimodular forms. In The moduli space of curves (Texel Island, 1994), volume 129 of Progr. Math., pages 165–172. Birkha¨user Boston, Boston, MA, 1995. [14] H. Lange and Ch. Birkenhake. Complex Abelian Varieties, chapter 5. Springer-Verlag, 1992. 86[15] A. M. Legendre. Traite´ des Fonctions Elliptiques, pages 133–134. Im- primerie de Huzard-Courcier, 1828. [16] P. A. MacMahon. Divisors of numbers and their continuations in the theory of partitions. In G. Andrews, editor, Reprinted: Percy A. MacMahon Collected Papers, pages 305–341. MIT Press, Cambridge, 1986. [17] D. Maulik, R. Pandharipande, and R. P. Thomas. Curves on K3 sur- faces and modular forms. J. Topol., 3(4):937–996, 2010. With an ap- pendix by A. Pixton. [18] D. Mumford. Tata Lectures on Theta: Jacobian theta functions and differential equations. Progress in mathematics. Birkha¨user, 1984. [19] A. Okounkov and R. Pandharipande. Gromov-witten theory, hurwitz theory, and completed cycles. Anal. Math., 163(2):517–560, 2006. [20] Jonathan Wise. The genus zero Gromov-Witten invariants of [Sym2P2] and the enumerative geometry of hyperelliptic curves in P2. ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–Brown University. [21] Jonathan Wise. A hyperelliptic hodge integral, 2008. [22] Shing-Tung Yau and Eric Zaslow. BPS states, string duality, and nodal curves on K3. Nuclear Phys. B, 471(3):503–512, 1996. 87

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