Cut and paste invariants of moduli spaces of stable maps to toric surfaces

TL;DR

This paper investigates the structure of moduli spaces of logarithmic stable maps to toric surfaces, introducing a chamber decomposition based on tangency data that affects their classes in the Grothendieck ring.

Contribution

It defines a chamber decomposition on tangency conditions for stable maps to toric surfaces, generalizing resonance chambers, and explores their impact on moduli space classes.

Findings

The class of the moduli space remains constant within each chamber.

Chambers are characterized by cyclic orderings of tangency data and toric rays.

Open question on whether Gromov-Witten invariants are polynomial across chambers.

Abstract

We study moduli spaces of logarithmic stable maps to proper toric surfaces with prescribed tangency conditions to the toric boundary. Fixing a surface, we define a chamber decomposition on the space of all tangencies such that as a function of the tangency data, the class of the corresponding moduli space in the Grothendieck ring of varieties is constant. The tangency data defines a collection of lines through the origin which, along with the rays of the toric fan, are cyclically ordered. The chambers of the decomposition are regions for which this cyclic ordering is constant and generalise the well-known resonance chambers. We pose the open question of whether the Gromov-Witten invariants of the moduli spaces are polynomial on these chambers.