# A refined Brill-Noether theory over Hurwitz spaces

## TL;DR

This paper refines Brill-Noether theory over Hurwitz spaces by analyzing stratifications of line bundles via pushforward splitting types, proving smoothness and expected dimensions for general covers, and extending known results to all algebraically closed fields.

## Contribution

It introduces a refined stratification of line bundles over Hurwitz spaces and proves their smoothness and dimension properties for general covers, extending prior work to all algebraically closed fields.

## Key findings

- Strata are smooth of expected dimension for general degree k covers.
- Determines dimensions of all irreducible components of W^r_d(C) for general k-gonal curves.
- Extends Brill-Noether theory results to arbitrary algebraically closed fields.

## Abstract

Let $f\colon C \rightarrow \mathbb{P}^1$ be a degree $k$ genus $g$ cover. The stratification of line bundles $L \in \mathrm{Pic}^d(C)$ by the splitting type of $f_*L$ is a refinement of the stratification by Brill-Noether loci $W^r_d(C)$. 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 $W^r_d(C)$ 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.