The Cohomological Excess of Certain Moduli Spaces of Curves of Genus $g$

TL;DR

This paper investigates the cohomological properties of certain moduli spaces of genus g curves, confirming a conjectured upper bound on cohomological excess for hyperelliptic loci when k=0.

Contribution

It proves that the conjectured upper bound on cohomological excess is sharp for the hyperelliptic locus in the moduli space of genus g curves.

Findings

Constructible sheaf with non-vanishing cohomology in degree 3g-2

Confirmation that the upper bound g-1+k is sharp for k=0

Insights into the cohomological structure of hyperelliptic loci

Abstract

The open subvariety $\overline{M}_g^{\leq k}$ of $\overline{M}_g$ parametrizes stable curves of genus $g$ having at most $k$ rational components. By the work of Looijenga, one expects that the cohomological excess of $\overline{M}_g^{\leq k}$ is at most $g-1+k$. In this paper we show that when $k=0$, the conjectured upper bound is sharp by showing that there is a constructible sheaf on $\overline{H}_g^{\leq k}$ (the hyperelliptic locus) which has non-vanishing cohomology in degree $3g-2$.