DOI: https://doi.org/10.15368/theses.2019.129
Available at: https://digitalcommons.calpoly.edu/theses/2534
Date of Award
5-2019
Degree Name
MS in Mathematics
Department/Program
Mathematics
Advisor
Robert W. Easton
Abstract
A fundamental result in the classification of nonsingular projective curves is the canonical embedding. Our goal is to investigate how this canonical embedding interacts with the process of tropicalization. We first give an overview of classical algebraic geometry, beginning with affine varieties. We highlight several important tools used for the construction of maps from algebraic varieties to projective space, namely divisors and differentials, extracting the main properties necessary for the canonical embedding. For illustration, we explicitly describe the image of a certain genus six curve under the canonical embedding. We then transition to tropical geometry, with a brief overview of the tropical setting, including the competing notions of bend locus and congruence variety. Finally, we analyze how tropicalization interacts with the canonical map. We are particularly interested in whether the tropicalization of the canonical model agrees with a tropical canonical model for the tropicalization. For our analysis, we will focus on our explicit genus six curve as a test case