Date Available

8-20-2023

Year of Publication

2023

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department/School/Program

Mathematics

First Advisor

Dr. Christopher Manon

Abstract

A projective, normal variety is called a Mori dream space when its Cox ring is finitely generated. These spaces are desirable to have, as they behave nicely under the Minimal Model Program, but no complete classification of them yet exists. Some early work identified that all toric varieties are examples of Mori dream spaces, as their Cox rings are polynomial rings. Therefore, a natural next step is to investigate projectivized toric vector bundles. These spaces still carry much of the combinatorial data as toric varieties, but have more variable behavior that means that they aren't as straightforward as Mori dream spaces. Expanding on Gonzalez's 2012 result that all rank 2 projectivized toric vector bundles are Mori dream spaces, we give a combinatorial sufficient condition for when a rank r bundle is Mori dream, using Kaveh and Manon's description of a toric vector bundle by a linear ideal and an integral matrix. We then address the question: if a toric vector bundle projectivizes to a Mori dream space, when is the projectivization of the direct sum of that bundle with itself a Mori dream space? Expanding on the nice families of bundles found, we compute the positivity-related cones for these bundles and provide a description of additional classes of toric vector bundles that uphold the Fujita conjectures. Finally, we conclude with the subduction and KM algorithms, two Macaulay2-implemented algorithms that allow us to produce finite presentations of Cox rings of projectivized toric vector bundles, provided they exist, allowing for future work in the study of these bundles as Mori dream spaces.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2023.398

Download

Share

COinS