Vanishing $H^1$ for Hurwitz spaces of fully-marked admissible covers of degree 3

Abstract

We show that the first cohomology group of the Hurwitz space of fully-marked admissible covers $H^1({\overline{\mathcal{H}}}_{\underline{d},\underline{g}}(\underline{\mu }))$ vanishes for covers of degree $d=3$ and deduce the same result for the classical Hurwitz space of simply-branched covers. In degree 4, we compute examples where $H^1({\overline{\mathcal{H}}}_{\underline{4},\underline{g}}(\underline{\mu }))$ is nonzero, which implies that $H^1({\overline{\mathcal{H}}}_{\underline{d},\underline{g}}(\underline{\mu }))$ is nonvanishing for $d\ge 4$. We describe the stratification of the boundary of ${\overline{\mathcal{H}}}_{\underline{d},\underline{g}}(\underline{\mu })$ by lower-dimensional ${\overline{\mathcal{H}}}_{\underline{d^{\prime }},\underline{g^{\prime }}}(\underline{{\mu }^{\prime }})$, and set up an inductive framework which may be used for future arguments involving the odd cohomology of ${\overline{\mathcal{H}}}_{\underline{d},\underline{g}}(\underline{\mu })$.