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Tropical curves in abelian surfaces III: pearl diagrams and multiple cover formulas

Thomas Blomme

Abstract

This paper is the third installment in a series of papers devoted to the computation of enumerative invariants of abelian surfaces through the tropical approach. We develop a pearl diagram algorithm similar to the floor diagram algorithm used in toric surfaces that concretely solves the tropical problem. These diagrams can be used to prove specific cases of Oberdieck's multiple cover formula that reduce the computation of invariants for non-primitive classes to the primitive case, getting rid of all diagram considerations and providing short explicit formulas. The latter can be used to prove the quasi-modularity of generating series of classical invariants, and the polynomiality of coefficients of fixed codegree in the refined invariants.

Tropical curves in abelian surfaces III: pearl diagrams and multiple cover formulas

Abstract

This paper is the third installment in a series of papers devoted to the computation of enumerative invariants of abelian surfaces through the tropical approach. We develop a pearl diagram algorithm similar to the floor diagram algorithm used in toric surfaces that concretely solves the tropical problem. These diagrams can be used to prove specific cases of Oberdieck's multiple cover formula that reduce the computation of invariants for non-primitive classes to the primitive case, getting rid of all diagram considerations and providing short explicit formulas. The latter can be used to prove the quasi-modularity of generating series of classical invariants, and the polynomiality of coefficients of fixed codegree in the refined invariants.
Paper Structure (67 sections, 34 theorems, 100 equations, 8 figures)

This paper contains 67 sections, 34 theorems, 100 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and problem setting
  3. Abelian surfaces.
  4. Curves in abelian surfaces.
  5. The tropical approach.
  6. Refined invariants.
  7. Explicit computations and floor diagrams.
  8. Results
  9. Pearl diagram algorithm.
  10. Multiple cover formulas.
  11. Relation to other Gromov-Witten invariants.
  12. Regularity results.
  13. Structure of the paper
  14. Little toolbox
  15. A small disgression on the arithmetic convolution
  16. ...and 52 more sections

Key Result

Theorem 1

theorem multiple cover formulas point We have the following multiple cover formulas

Figures (8)

  • Figure 1: Examples of pearl diagrams. The first is of respective genus $2$, $4$ and $7$.
  • Figure 2: Three examples of tropical curves in tropical tori of respective degrees $\left(1001\right)$, $\left(2003\right)$ and $\left(2101\right)$ .
  • Figure 3: A tropical curve realizing the class $(1,6)$ admitting a pearl decomposition in a "rectangular" abelian surface (a) and inside a small deformation of it (b).
  • Figure 4: A tropical torus associated to the matrix $\left(Lla/d_20l\right)$ and a small deformation.
  • Figure 5: Intersection between a genus $2$ curve and the lift of an edge of some tropical curve in Lemma \ref{['lem-finite-length']}.
  • ...and 3 more figures

Theorems & Definitions (80)

  • Theorem
  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem
  • Theorem
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 70 more