Lines in Tropical Quadrics

• Main Author: O'Neill, Kevin
• Format: Others
• Published: Scholarship @ Claremont 2013
• Subjects: 14T05 Tropical geometry; Physical Sciences and Mathematics
• Online Access: http://scholarship.claremont.edu/hmc_theses/43; http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1053&context=hmc_theses

Classical algebraic geometry is the study of curves, surfaces, and other varieties defined as the zero set of polynomial equations. Tropical geometry is a branch of algebraic geometry based on the tropical semiring with operations minimization and addition. We introduce the notions of projective space and tropical projective space, which are well-suited for answering enumerative questions, like ours. We attempt to describe the set of tropical lines contained in a tropical quadric surface in $\mathbb{TP}^3$. Analogies with the classical problem and computational techniques based on the idea of a tropical parameterization suggest that the answer is the union of two disjoint conics in $\mathbb{TP}^5$.