Prym-Brill-Noether Theory for General Covers

Abstract

We bound the dimension of the Prym-Brill-Noether variety for an open subset of the moduli space of étale double covers of k-elliptic curves. We also obtain new bounds on the dimension of the Prym-Brill-Noether variety for general étale double covers of k-gonal curves, disproving a conjecture of Creech, Len, Ritter, and Wu, and provide a new conjecture for its dimension. To do this, we completely describe the Prym-Brill-Noether variety of a double cover of a certain tropical curve known as the loop of loops. We use the combinatorics of Coxeter groups to establish several topological properties of these tropical Prym-Brill-Noether varieties, and prove a lifting result when the edge lengths are generic.