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 $d \geq 4$. We describe the stratification of the boundary of $\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underline{\mu})$ by lower-dimensional $\overline{\mathcal{H}}_{\underline{d'},\underline{g'}}(\underline{\mu'})$, and set up an inductive framework which may be used for future arguments involving the odd cohomology of $\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underline{\mu})$.