# MATH Seminar

Title: A refined Brill-Noether theory over Hurwitz spaces
Seminar: Algebra
Speaker: Hannah Larson of Stanford Univeristy
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-10-29 at 4:00PM
Venue: MSC W303
Abstract:
The celebrated Brill-Noether theorem says that the space of degree $d$ maps of a general genus $g$ curve to $\mathbb{P}^r$ is irreducible. However, for special curves, this need not be the case. Indeed, for general $k$-gonal curves (degree $k$ covers of $\mathbb{P}^1$), this space of maps can have many components, of different dimensions (Coppens-Martens, Pflueger, Jensen-Ranganathan). In this talk, I will introduce a natural refinement of Brill-Noether loci for curves with a distinguished map $C \rightarrow \mathbb{P}^1$, using the splitting type of push forwards of line bundles to $\mathbb{P}^1$. In particular, studying this refinement determines the dimensions of all irreducible components of Brill-Noether loci of general $k$-gonal curves.