Tropical curves in abelian surfaces I: enumeration of curves passing through 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-Göttsche to get tropical refined invariants.

Tropical Mirror Symmetry in Dimension One

“…Since the first version of this paper appeared as preprint, other researchers have continued working on related topics. We would like to point out in particular a series of papers by Blomme who studies the enumerative geometry of line bundles over elliptic curves and generalized further to the enumerative geometry of abelian surfaces [7,8], and the papers on enumerative geometry of elliptic fibrations by Oberdieck and Pixton [34,35].…”

“…Leaky tropical covers show up as floor diagrams representing counts of tropical curves in toric surfaces resp. in P 1 -bundles over E (see, e.g., [2,7,8,10,18]). We introduce them here, since they can be treated in terms of Feynman integrals analogously to their balanced versions.…”