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Media type:
Doctoral Thesis;
E-Book;
Electronic Thesis
Title:
Algorithmic aspects of tropical intersection theory
Contributor:
Hampe, Simon
[Author]
imprint:
KLUEDO - Publication Server of University of Kaiserslautern-Landau (RPTU), 2014
Language:
English
Origination:
Footnote:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Description:
In the first part of this thesis we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus is the study of moduli spaces, where the underlying combinatorics of the varieties involved allow a much more efficient way of computing certain tropical cycles. The algorithms discussed here have been implemented in an extension for polymake, a software for polyhedral computations. In the second part we apply the algorithmic toolkit developed in the first part to the study of tropical double Hurwitz cycles. Hurwitz cycles are a higher-dimensional generalization of Hurwitz numbers, which count covers of \(\mathbb{P}^1\) by smooth curves of a given genus with a certain fixed ramification behaviour. Double Hurwitz numbers provide a strong connection between various mathematical disciplines, including algebraic geometry, representation theory and combinatorics. The tropical cycles have a rather complex combinatorial nature, so it is very difficult to study them purely "by hand". Being able to compute examples has been very helpful in coming up with theoretical results. Our main result states that all marked and unmarked Hurwitz cycles are connected in codimension one and that for a generic choice of simple ramification points the marked cycle is a multiple of an irreducible cycle. In addition we provide computational examples to show that this is the strongest possible statement. ; Im ersten Teil dieser Arbeit betrachten wir algorithmische Aspekte tropischer Schnitttheorie. Wir analysieren, wie Divisoren und Schnittprodukte tropischer Zykel mithilfe polyedrischer Geometrie berechnet werden können. Der Fokus liegt hierbei auf der Betrachtung von Modulräumen, deren zugrunde liegende Kombinatorik eine deutlich effizientere Berechnung bestimmter tropischer Zykel erlaubt. Die Algorithmen, die hier vorgestellt werden, wurden in einer Erweiterung für polymake implementiert, einer Software für ...