Virtual Hodge numbers of $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$: stability and calculations
Siddarth Kannan, Terry Dekun Song
TL;DR
The paper develops a framework to compute $\mathbb{S}_n$-equivariant motivic invariants, notably Serre characteristics and Hodge--Deligne polynomials, for moduli spaces of degree-$d$ maps from $n$-pointed genus-$g$ curves to $\mathbb{P}^r$. It introduces generating series in symmetric-function language, proves the rationality of a transformed generating function, and establishes a stability phenomenon for weight-graded Euler characteristics as $d\to\infty$. By reducing to $\mathcal{M}_{g,n}$ in genus $1$ and $2$, it yields explicit formulas for $\mathsf{e}^{\mathbb{S}_n}(\mathcal{M}_{1,n}(\mathbb{P}^r,d))$ for all $n,r,d$ (building on Getzler's results) and analogous statements for $\mathcal{M}_{2,n}(\mathbb{P}^r,d)$ under certain degree bounds. The work integrates quasimap stratifications, relative Picard stacks, and skewing operators on symmetric functions to generalize genus-zero computations and to offer concrete, computable expressions for higher-genus mapping spaces with potential homological-stability implications.
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. Since Getzler has calculated the Serre characteristic of $\mathcal{M}_{1, n}$, our formula in particular determines the Serre characteristic of $\mathcal{M}_{1, n}(\mathbb{P}^r, d)$ for arbitrary $n$, $r$, and $d$. Our formula also calculates the Serre characteristic of $\mathcal{M}_{2, n}(\mathbb{P}^r, d)$ whenever those of $\mathcal{M}_{2, n +k}$ are known for $k \leq d$.