On the weight zero compactly supported cohomology of $\mathcal{H}_{g, n}$

TL;DR

This paper explicitly computes the weight zero compactly supported cohomology of hyperelliptic moduli spaces using dual complexes and graph formulas, enabling calculations of Euler characteristics for various genera and covers.

Contribution

It introduces a dual complex and graph complex approach to compute weight zero cohomology of hyperelliptic moduli spaces, extending to abelian covers of genus zero curves.

Findings

Explicit dual complex for hyperelliptic moduli spaces.

A sum-over-graphs formula for Euler characteristics.

Computer-aided calculations for genera up to 7.

Abstract

For $g\ge 2$ and $n\ge 0$, let $\mathcal{H}_{g,n}\subset \mathcal{M}_{g,n}$ denote the complex moduli stack of $n$-marked smooth hyperelliptic curves of genus $g$. A normal crossings compactification of this space is provided by the theory of pointed admissible $\mathbb{Z}/2\mathbb{Z}$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of $\mathcal{H}_{g, n}$. Using this graph complex, we give a sum-over-graphs formula for the $S_n$-equivariant weight zero compactly supported Euler characteristic of $\mathcal{H}_{g, n}$. This formula allows for the computer-aided calculation, for each $g\le 7$, of the generating function $\mathsf{h}_g$ for these equivariant Euler characteristics for all $n$. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissible $G$-covers of genus zero curves, when $G$ is abelian, as a symmetric $\Delta$-complex. We use these complexes to generalize our formula for $\mathsf{h}_g$ to moduli spaces of $n$-pointed smooth abelian covers of genus zero curves.