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

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 ${\overline{M}}_{\Gamma }\left(X\right)$ to the product ${\overline{M}}_{g,n}\times X^n$. In this paper, after proving that Mikhalkin's correspondence theorem holds in genus 0 for logarithmic virtual Tevelev degrees, 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.