Deformation of tropical Hirzebruch surfaces and enumerative geometry

TL;DR

This paper uses tropical geometry to relate enumerative invariants of Hirzebruch surfaces, generalizing existing formulas by employing tropical deformations and enumerative techniques in three-dimensional tropical surfaces.

Contribution

It introduces a tropical deformation approach to connect enumerative invariants of different Hirzebruch surfaces, extending prior formulas by Abramovich, Bertram, and Vakil.

Findings

Established relations between invariants of $ obreak ext{Hirzebruch surfaces}$ using tropical methods.

Developed a tropical deformation framework for Hirzebruch surfaces.

Applied tropical enumerative geometry to derive new relations.

Abstract

We illustrate the use of tropical methods by generalizing a formula due to Abramovich and Bertram, extended later by Vakil. Namely, we exhibit relations between enumerative invariants of the Hirzebruch surfaces $\Sigma_n$ and $\Sigma_{n+2}$, obtained by deforming the first surface to the latter. Our strategy involves a tropical counterpart of deformations of Hirzebruch surfaces, and tropical enumerative geometry on a tropical surface in three-space.