Tropical M\"obius strips and ruled surfaces

TL;DR

This paper explores the enumeration of tropical curves in M"obius strips and their relation to rational ruled surfaces, establishing regularity properties and modularity of associated generating series.

Contribution

It introduces a novel correspondence between tropical curves in M"obius strips and curves in rational ruled surfaces, proving new regularity and modularity results.

Findings

Piecewise quasi-polynomiality of relative invariants

Quasi-modularity of generating series

Relation between tropical curves and algebraic surfaces

Abstract

We consider the enumeration of tropical curves in M\"obius strips for two different lattice structures and relate them to the enumeration of curves in two rational ruled surfaces over a complex elliptic curve. Using this correspondence, we prove regularity results such as the piecewise quasi-polynomiality of relative invariants and the quasi-modularity of their generating series.