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

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.