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.

Keywords

• 14H10