Virtual Hodge numbers of ${\mathcal{M}}_{g,n}({\mathbb{P}}^r,d)$: stability and calculations

We study ${\mathbb{S}}_n$-equivariant motivic invariants of the moduli space ${\mathcal{M}}_{g,n}({\mathbb{P}}^r,d)$ of degree-$d$ maps from $n$-pointed curves of genus $g$ to ${\mathbb{P}}^r$. In particular, we obtain formulas for the Serre characteristic, which specializes to the Hodge–Deligne polynomial. Fixing $g,r\ge 1$, we prove that an explicit invertible transform of the generating function for the Serre characteristics is rational. We use our formula to prove a stability result for the weight-graded compactly-supported Euler characteristics of ${\mathcal{M}}_{g,n}({\mathbb{P}}^r,d)$ as $d\to \mathrm{\infty }$. In genus one and two, we reduce the calculation of the Serre characteristic of ${\mathcal{M}}_{g,n}({\mathbb{P}}^r,d)$ to those of the moduli spaces ${\mathcal{M}}_{g,n}$ of $n$-pointed curves. Formulas for the latter follow from work of Getzler and Petersen, so our formula in particular determines the Serre characteristic of ${\mathcal{M}}_{g,n}({\mathbb{P}}^r,d)$ for arbitrary $n$, $r$, and $d$ when $g=1$ and $g=2$.