# ProjectArithmetic counts of bitangents to plane quartics via tropical geometry

## Basic data

Title:

Arithmetic counts of bitangents to plane quartics via tropical geometry

Duration:

01/08/2022 to 01/08/2025

Abstract / short description:

Tropical geometry has been particularly successful in the study of real algebraic geometry resp. in the study of connections of real and complex geometry: this is true because often, the tropical objects we use are the same no matter whether we model complex or real geometry - it is just the lifting properties for the tropical objects in use that differ. This project deals with the tropical geometry of an old and basic topic concerning embedded curves: the study of their tangents. Classically, the study of bitangents to plane quartics dates back to Pluecker who studied not only the complex but also the real version: A complex quartic has exactly 28 bitangents. A real quartic can have 4, 8, 16 or 28 real bitangents, depending on the topology of the real curve. The tropicalization of bitangent lines of a quartic plane curve was studied by several researchers from various perspectives before. A tropical quartic in the plane can have infinitely many bitangent lines, but by imposing a natural equivalence relation, we obtain exactly seven classes. The number of complex lifts for each representative of a class (counted with multiplicity) sum up to 4 for each class, leading to a tropical count of the 7x4=28 complex bitangents. In this project, we study the real lifting behaviour of tropical bitangent lines to quartics. In addition, there are two extensions of this research which shall be studied, leading to connections to the theory of tropical moduli spaces of curves and to the theory of tropical divisors, and to the application of the tropical method in the construction of real sextic space curves with prescribed tangencies.

## 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