Tropical curves in abelian surfaces I: enumeration of curves passing through points

TL;DR

This paper introduces a tropical approach to counting genus g curves of fixed degree passing through g points in abelian surfaces, computing multiplicities and refining them to obtain tropical refined invariants.

Contribution

It develops a method to compute and refine enumerative invariants of abelian surfaces using tropical geometry, extending previous correspondence theorems.

Findings

Computed multiplicities for tropical curves using Nishinou's correspondence theorem

Refined multiplicities to define tropical refined invariants

Established enumeration results for genus g curves passing through g points

Abstract

This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus $g$ curves of fixed degree passing through $g$ points. We compute the multiplicity provided by a correspondence theorem due to T. Nishinou and show that it is possible to refine this multiplicity in the style of Block-G\"ottsche to get tropical refined invariants.