Tropical Methods in Hurwitz-Brill-Noether Theory

TL;DR

This paper proves that splitting type loci in Hurwitz spaces have the expected dimension using tropical geometry, and provides algorithms and conjectures for their structure and enumeration.

Contribution

It offers a tropical geometric proof of a key theorem, describes the structure of splitting type loci, and proposes an algorithm for their enumeration.

Findings

Splitting type loci have the expected dimension for general Hurwitz space elements.

Tropical splitting type loci are connected in codimension one.

An algorithm for computing the cardinality of zero-dimensional loci is provided.

Abstract

Splitting type loci are the natural generalizations of Brill-Noether varieties for curves with a distinguished map to the projective line. We give a tropical proof of a theorem of H. Larson, showing that splitting type loci have the expected dimension for general elements of the Hurwitz space. Our proof uses an explicit description of splitting type loci on a certain family of tropical curves. We further show that these tropical splitting type loci are connected in codimension one, and describe an algorithm for computing their cardinality when they are zero-dimensional. We provide a conjecture for the numerical class of splitting type loci, which we confirm in a number of cases.