Tropical Geometric Counting Problems

Abstract

A degeneration of algebraic counting problems to tropical geometry produces new concepts and computational approaches. The challenge is to find methods to lift the results in the degeneration back to the original problem. This thesis contributes towards the fundamental research in this area by investigating two geometric counting problems via tropical geometry.

The first part of this thesis closes a research gap on the recovery of the classical count of 4, 8, 16 or 28 real bitangents to smooth quartic curves by Plücker and Zeuthen from tropical geometry. Building on previous research results, we develop methods to produce an understanding of the global lifting of tropical bitangents over certain real closed fields. Moreover, we establish a computational tool in polymake for the investigation of tropical bitangents and their lifting behavior, and present results of analyses using this tool.

The second part of this thesis explores the question of tropically counting binodal surfaces. We exploit tropical floor plans, a recent enumerative tool from tropical geometry. We prove that tropical floor plans recover the algebraic counts of plane curves, while for surfaces the current technique is not sufficient to asymptotically recover the second order term. To improve the approach for surfaces, we provide a classification of the smallest examples of polytopes that can appear as Newton polytopes of a binodal surface together with instructions how to use them for tropical counting. This investigation is of computational nature and is aided by functions written for the computeralgebra system OSCAR. Furthermore, we extend the definition of tropical floor plans that contains the newly found cases. Additionally, we show that these smallest cases contribute to the third order term of the asymptotic count.