Moduli of relative stable maps to $\mathbb{P}^1$: cut-and-paste invariants

TL;DR

This paper investigates invariants of moduli spaces of genus zero stable maps to 1, revealing recursive formulas for Euler characteristics and invariance properties within certain parameter chambers, advancing understanding in enumerative geometry.

Contribution

It introduces new recursive formulas for Euler characteristics and demonstrates invariance of classes in the Grothendieck ring within chambers, providing novel insights into the structure of these moduli spaces.

Findings

Recursive differential equation for Euler characteristics.

Invariance of the class in the Grothendieck ring within chambers.

Conjectures on asymptotic behavior and chamber structure for Chern numbers.

Abstract

We study constructible invariants of the moduli space $\overline{\mathcal{M}}(\boldsymbol{x})$ of stable maps from genus zero curves to $\mathbb{P}^1$, relative to $0$ and $\infty$, with ramification profiles specified by ${\boldsymbol{x}\in \mathbb{Z}^n}$. These spaces are central to the enumerative geometry of $\mathbb{P}^1$, and provide a large family of birational models of the Deligne--Mumford--Knudsen moduli space $\overline{\mathcal{M}}_{0,n}$. For the sequence of vectors $\boldsymbol{x}$ corresponding to maps which are maximally ramified over $0$ and unramified over $\infty$, we prove that a generating function for the topological Euler characteristics of these spaces satisfies a differential equation which allows for its recursive calculation. We also show that the class of the moduli space in the Grothendieck ring of varieties is constant as $\boldsymbol{x}$ varies within a fixed chamber in the resonance decomposition of $\mathbb{Z}^n$. We conclude by suggesting several further directions in the study of these spaces, giving conjectures on (1) the asymptotic behavior of the Euler characteristic and (2) a potential chamber structure for the Chern numbers.