A refined Brill-Noether theory over Hurwitz spaces

Abstract

Let $f:C\to {\mathbb{P}}^1$ be a degree $k$ genus $g$ cover. The stratification of line bundles $L\in {\mathrm{Pic}}^d(C)$ by the splitting type of $f_{\ast }L$ is a refinement of the stratification by Brill-Noether loci $W_d^r(C)$. We prove that for general degree $k$ covers, these strata are smooth of the expected dimension. In particular, this determines the dimensions of all irreducible components of $W_d^r(C)$ for a general $k$-gonal curve (there are often components of different dimensions), extending results of Pflueger and Jensen-Ranganathan. The results here apply over any algebraically closed field.

removed characteristic hypotheses