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.
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