ISBN 978-3-8439-2687-4

978-3-8439-2687-4, Reihe Mathematik

Anna Lena Birkmeyer
The Realizability of Tropical Hypersurfaces in Matroid Fans

119 Seiten, Dissertation Technische Universität Kaiserslautern (2016), Softcover, B5

Zusammenfassung / Abstract

An interesting problem in tropical geometry is the (relative) realization problem: Given a tropical variety, tropical geometer would like to know if it is the tropicalization of an algebraic variety. In this thesis, we study this problem in a relative case. Given a linear space W in the n-dimensional projective space over the Puiseux series K{{t}}, where K is any algebraically closed field, and a tropical variety contained in the tropicalization of W (of codimension 1), we develop an algorithm able to decide whether this tropical variety is the tropicalization of a (not necessarily irreducible) subvariety of W. This algorithm has been implemented in the computer algebra system Singular and is based on the following idea: The tropicalization of W can be written as the Bergman fan of the matroid associated to W and hence we can use matroid theory to split our relative realization problem into several (absolute) realization problems of tropical hypersurfaces in k-space which can be solved more easily.

In the first interesting case of tropical curves in the general tropical plane in 3-space, we use the idea of our algorithm to prove sufficient and necessary conditions to relative realizability including and generalizing known obstructions to relative realizability by Brugallé-Shaw and Bogart-Katz.

In this work, we moreover explore the structure of the realization space of a tropical hypersurface in a matroid fan and the space of all relatively realizable tropical hypersurfaces in a fixed matroid fan. This can be used to investigate related relative realization problems: Instead of finding a subvariety of W tropicalizing to a given tropical hypersurface H in the tropicalization of W, we can ask if there is an irreducible subvariety of W realizing H. In the case of tropical fan varieties, we are able to algorithmically solve this problem. For tropical curves in 2-dimensional matroid fans, we additionally give an algorithmic solution to the question if the realization space of a tropical curve C in the tropicalization of W intersects a given Severi variety. This is related to the geometric genus of subvarieties of W realizing C.