Moduli of stable maps in genus one and logarithmic geometry II

TL;DR

This paper develops a logarithmic Gromov--Witten theory for genus one curves, constructing a moduli space that facilitates enumerative geometry and tropical realizability in genus one.

Contribution

It constructs a logarithmically nonsingular moduli space of genus 1 stable maps to toric varieties, advancing the understanding of genus 1 enumerative geometry and tropical realizability.

Findings

Constructed a birational modification of existing moduli spaces.

Resolved the tropical realizability problem in genus 1.

Connected non-archimedean analytic skeleton to enumerative geometry.

Abstract

This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a logarithmically nonsingular moduli space of genus $1$ curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich--Chen--Gross--Siebert space of logarithmic stable maps and produces an enumerative genus $1$ curve counting theory. We describe the non-archimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus $1$.