Abstract: Tropical curves are piecewise linear objects arising as degenerations of algebraic curves. The close connection between algebraic curves and their tropical limits persists when considering moduli. This exhibits certain spaces of tropical curves as the tropicalizations of the moduli spaces of stable curves. It is, however, still unclear which properties of the algebraic moduli spaces of curves are reflected in their tropical counterparts.
In work with Renzo Cavalieri and Hannah Markwig we defined, in a purely tropical way, tropical psi classes in arbitrary genus. They are operational cocycles on a stack of tropical curves, which enjoy several properties that we know from their algebraic ancestors. We also computed two examples in genus one and gave a tropical explanation for the psi class on the moduli space of 1-marked stable genus-1 curves to be 1/24 times a point.
In my talk, I will report on joint work in progress with Renzo Cavalieri, where we explore the missing piece in the story: the link to algebraic geometry. I will explain how to obtain, if we are lucky, a family of tropical curves from a family of algebraic curves. Naturally, there also is a correspondence-type theorem that equates algebraic and tropical intersection products with psi classes, thus showing that the tropical computations done with Cavalieri and Markwig faithfully reflect the algebraic world.