Deformation of tropical Hirzebruch surfaces and enumerative geometry

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 Σ n \Sigma _n and Σ n + 2 \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.