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

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.

Published version

Cited by in corpus (1)

• On the Motivic Class of the Moduli Stack of Twisted $G$-Covers