Genus 0 logarithmic and tropical fixed-domain counts for Hirzebruch surfaces

TL;DR

This paper establishes genus 0 correspondence theorems for logarithmic stable maps to toric varieties and derives explicit tropical formulas for counting such maps on Hirzebruch surfaces.

Contribution

It proves genus 0 correspondence theorems for toric varieties and provides explicit tropical formulas for counts on Hirzebruch surfaces.

Findings

Genus 0 correspondence theorem for logarithmic stable maps.

Explicit tropical formulas for Hirzebruch surface counts.

Complete enumeration of contributing tropical curves.

Abstract

For a non-singular projective toric variety $X$, the virtual logarithmic Tevelev degrees are defined as the virtual degree of the morphism from the moduli stack of logarithmic stable maps $\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)$ to the product $\overline{\mathcal{M}}_{g,n} \times X^n$. In this paper, after proving the genus $0$ correspondence theorem in this setting, we use tropical methods to provide closed formulas for the case in which $X$ is a Hirzebruch surface. In order to do so, we explicitly list all the tropical curves contributing to the count.