Tropikalisierungen von Modulraeumen von Kurven und Ueberlagerungen

01.03.2016 bis 28.02.2018

In tropical geometry, algebraic varieties are degenerated to polyhedral complexes called tropical varieties. We refer to this degeneration process as tropicalization.
Tropical geometry has been particularly succesfully applied to questions in enumerative geometry.
Many enumerative numbers can be expressed in terms of Chow cycles on a suitable moduli space parametrizing the objects to count. This holds true both in algebraic and tropical geometry. A connection at the level of moduli spaces still remains to be understood in general. In this proposal, we plan to study several moduli spaces which are important in enumerative geometry, namely compactifications of spaces of stable curves and Hurwitz schemes. We need to study compactifications in order to perform the intersection theory necessary to produce enumerative numbers. In the ideal case, a moduli space and its tropical counterpart are related by a tropical compactification - a compactification dictated by the tropicalization. When dealing with curves of higher genus, we have to take toroidal structure and Berkovich analytification into account.