# Project Tropikalisierungen von Modulraeumen von Kurven und Ueberlagerungen

## Basic data

Title:

Tropikalisierungen von Modulraeumen von Kurven und Ueberlagerungen

Duration:

01/03/2016 to 28/02/2018

Abstract / short description:

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.

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.

Keywords:

algebraic geometry

Algebraische Geometrie

geometry

Geometrie

## Involved staff

### Managers

Department of Mathematics

Faculty of Science

Faculty of Science

## Local organizational units

Department of Mathematics

Faculty of Science

University of Tübingen

University of Tübingen

## Funders

Bonn, Nordrhein-Westfalen, Germany