Year of Publication

2021

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department/School/Program

Mathematics

First Advisor

Dr. David Jensen

Abstract

The geometry of a curve can be analyzed in many ways. One way of doing this is to study the set of all divisors on a curve of prescribed rank and degree, known as a Brill-Noether variety. A sequence of results, starting in the 1980s, answered several fundamental questions about these varieties for general curves. However, many of these questions are still unanswered if we restrict to special families of curves. This dissertation has three main goals. First, we examine Brill-Noether varieties for these special families and provide combinatorial descriptions of their irreducible components. Second, we provide a natural generalization of Brill-Noether varieties, known as Splitting-Type varieties, that parameterize this decomposition. Lastly, we provide purely combinatorial descriptions of these Splitting-Type varieties and explore the geometric consequences of these descriptions. These results are based upon and extend tools and techniques from tropical geometry.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2021.231

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