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

TL;DR

This paper derives formulas for motivic invariants of moduli spaces of maps from pointed curves to projective space, proving stability and explicit calculations for genus one and two cases.

Contribution

It provides explicit formulas for the Serre characteristic of these moduli spaces and establishes stability results as the degree increases.

Findings

Formulas for Serre characteristics of moduli spaces of maps.

Proved rationality of a generating function transform.

Established stability of Euler characteristics as degree tends to infinity.

Abstract

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 $g, r \geq 1$, 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 $d \to \infty$. 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 $g = 2$.