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

TL;DR

This paper advances the computation of enumerative invariants of abelian surfaces using pearl diagrams, enabling proofs of multiple cover formulas, and establishing properties like quasi-modularity and polynomiality of generating series.

Contribution

It introduces a pearl diagram algorithm for tropical invariants, simplifying calculations and proving multiple cover formulas for non-primitive classes.

Findings

Proves specific cases of Oberdieck's multiple cover formula.

Establishes quasi-modularity of generating series.

Shows polynomiality of coefficients in refined invariants.

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.