Equivariant Hodge polynomials of heavy/light moduli spaces

TL;DR

This paper provides a formula to compute equivariant Hodge-Deligne polynomials of certain moduli spaces of weighted pointed stable curves, linking them to known polynomials of classical moduli spaces.

Contribution

It introduces a change-of-variables formula that expresses the generating functions for equivariant Hodge-Deligne polynomials of Hassett's moduli spaces in terms of those of standard moduli spaces.

Findings

Derived a change-of-variables formula for generating functions

Expressed equivariant Hodge-Deligne polynomials in terms of known cases

Connected heavy/light moduli spaces to classical moduli space invariants

Abstract

Let $\bar{\mathcal{M}}_{g, m|n}$ denote Hassett's moduli space of weighted pointed stable curves of genus $g$ for the heavy/light weight data $\left(1^{(m)}, 1/n^{(n)}\right)$, and let $\mathcal{M}_{g, m|n} \subset \bar{\mathcal{M}}_{g, m|n}$ be the locus parameterizing smooth, not necessarily distinctly marked curves. We give a change-of-variables formula which computes the generating function for $(S_m\times S_n)$-equivariant Hodge-Deligne polynomials of these spaces in terms of the generating functions for $S_{n}$-equivariant Hodge-Deligne polynomials of $\bar{\mathcal{M}}_{g,n}$ and $\mathcal{M}_{g,n}$.