# The embedding theorem in Hurwitz--Brill--Noether theory

The embedding theorem in Hurwitz--Brill--Noether theory

Brill--Noether theory studies the maps of general curves to projective spaces. The embedding theorem of Eisenbud and Harris states that a general degree $d$ map $C \rightarrow \mathbb{P}^r$ is an embedding when $r \geq 3$. Hurwitz--Brill--Noether theory starts with a curve $C$ already equipped with a fixed map $C \rightarrow \mathbb{P}^1$ (which often forces $C$ to be special) and studies the maps of $C$ to other projective spaces. In this setting, the appropriate analogue of the invariants $d$ and $r$ is a finer invariant called the splitting type. Our embedding theorem determines the splitting types $\vec{e}$ such that a general map of splitting type $\vec{e}$ is an embedding. This is joint work with Kaelin Cook--Powel, Dave Jensen, Eric Larson, and Isabel Vogt.