Download or read book Brill--Noether Theory Over the Hurwitz Space written by Hannah Kerner Larson and published by . This book was released on 2022 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Historically, algebraic curves were defined as the solutions to a collection of polynomial equations inside of some ambient space. In the 19th century, however, mathematicians defined the notion of an abstract curve. With this perspective, the same abstract curve may sit in an ambient (projective) space in more than one way. The foundational Brill--Noether theorem, proved in the 1970s and 1980s, bridges these two perspectives by describing the maps of "most" abstract curves to projective spaces. However, the theorem does not hold for all curves. In nature, we often encounter curves already in (or mapping to) a projective space, and the presence of such a map may force the curve to have unexpected maps to other projective spaces! The first case of this is a curve that already has a map to the projective line. From the 1990s through the late 2010s, several mathematicians investigated this first case. They found that the space of maps of such a curve to other projective spaces can have multiple components of varying dimensions and eventually determined the dimension of the largest component. In this thesis, I develop analogues of all the main theorems of Brill--Noether theory for curves that already have a map to the projective line. The moduli space of curves together with a map to the line is called the Hurwitz space, so we call this work Brill--Noether theory over the Hurwitz space.