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d-elliptic loci in genus 2 and 3

Carl Lian

TL;DR

The answers exhibit quasimodularity properties similar to those in the Gromov–Witten theory of a fixed genus 1 curve; it is conjecture that the quasIModularity persists in higher genus and a number of possible variants are indicated.

Abstract

We consider the loci of curves of genus 2 and 3 admitting a $d$-to-1 map to a genus 1 curve. After compactifying these loci via admissible covers, we obtain formulas for their Chow classes, recovering results of Faber-Pagani and van Zelm when $d=2$. The answers exhibit quasimodularity properties similar to those in the Gromov-Witten theory of a fixed genus 1 curve; we conjecture that the quasimodularity persists in higher genus, and indicate a number of possible variants.

d-elliptic loci in genus 2 and 3

TL;DR

The answers exhibit quasimodularity properties similar to those in the Gromov–Witten theory of a fixed genus 1 curve; it is conjecture that the quasIModularity persists in higher genus and a number of possible variants are indicated.

Abstract

We consider the loci of curves of genus 2 and 3 admitting a -to-1 map to a genus 1 curve. After compactifying these loci via admissible covers, we obtain formulas for their Chow classes, recovering results of Faber-Pagani and van Zelm when . The answers exhibit quasimodularity properties similar to those in the Gromov-Witten theory of a fixed genus 1 curve; we conjecture that the quasimodularity persists in higher genus, and indicate a number of possible variants.

Paper Structure

This paper contains 42 sections, 42 theorems, 64 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Conventions
  5. Intersection numbers on moduli spaces of curves
  6. $\overline{\mathcal{M}}_{1,2}$
  7. $\overline{\mathcal{M}}_{1,3}$
  8. $\overline{\mathcal{M}}_2$
  9. $\overline{\mathcal{M}}_{2,1}$
  10. $\overline{\mathcal{M}}_{3}$
  11. Admissible covers
  12. Quasimodular forms
  13. Auxiliary computations
  14. Counting isogenies
  15. The 2-pointed $d$-elliptic locus on $\overline{\mathcal{M}}_{1,2}$
  16. ...and 27 more sections

Key Result

Theorem 1.1

For $g\ge2$, the generating series is a quasimodular form of weight $6g-6$.

Figures (19)

  • Figure 1: Boundary classes in $A^1(\overline{\mathcal{M}}_{1,2})$
  • Figure 2: Some boundary classes in $A^1(\overline{\mathcal{M}}_{1,3})$
  • Figure 3: Some boundary classes in $A^2(\overline{\mathcal{M}}_{1,3})$
  • Figure 4: Boundary classes in $A^1(\overline{\mathcal{M}}_2)$
  • Figure 5: Boundary classes in $A^2(\overline{\mathcal{M}}_2)$
  • ...and 14 more figures

Theorems & Definitions (78)

  • Theorem 1.1: dijkgraaf
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4: hm
  • ...and 68 more