Brill-Noether theory for totally ramified covers of the projective line

TL;DR

This paper proves Pflueger's conjectures, extending Brill-Noether theory to totally ramified covers of the projective line and analyzing associated transmission loci in the Picard scheme.

Contribution

The paper establishes the validity of Pflueger's conjectures, advancing the understanding of transmission loci for totally ramified covers in Brill-Noether theory.

Findings

Proved Pflueger's conjectures for transmission loci.

Connected transmission loci to the extended $k$-affine symmetric group.

Extended classical Brill-Noether results to ramified covers.

Abstract

Given a curve $C$ that is a degree $k$ cover $C \to \mathbb{P}^1$ totally ramified at two points $p$ and $q$, we can seek to understand the space of degree $d$ line bundles on $C$ with prescribed ramification at $p$ and $q$. The corresponding subschemes of $\text{Pic}^d(C)$ are called transmission loci and are parameterized via elements of the (extended) $k$-affine symmetric group $\widetilde{\Sigma}_k$. Transmission loci provide a refinement of the splitting loci that have recently been extensively studied for $k$-gonal curves. Pflueger has conjectured analogues of the classic Brill-Noether theorem should hold for transmission loci. In this paper we prove Pflueger's conjectures.