TGiF-Seminar: Tropical geometry in Frankfurt (First meeting Summer Semester 2023)

Lou-Jean Cobigo (Universität Tübingen)

Abstract: Classical Hurwitz numbers count the number of branched covers of a fixed target curve that exhibit a certain ramification behaviour. It is an enumerative problem deeply rooted in mathematical history.
A modern twist: Spin Hurwitz numbers were introduced by Eskin-Okounkov-Pandharipande for certain computations in the moduli space of differentials on a Riemann surface.
Similarly to Hurwitz numbers they are defined as a weighted count of branched coverings of a smooth algebraic curve with fixed degree and branching profile. In addition, they include information about the lift of a theta characteristic of fixed parity on the base curve.

In this talk we investigate them from a tropical point of view. We start by defining tropical spin Hurwitz numbers as result of an algebraic degeneration procedure, but soon notice that these have a natural place in the tropical world as tropical covers with tropical theta characteristics on source and target curve.
Our main results are two correspondence theorems stating the equality of the tropical spin Hurwitz number with the classical one.