Seminario Nacional de Geometría Algebraica en línea

Resumen:

Counting plane algebraic curves of fixed degree d and genus g through 3d+g-1 generic points is a classical enumerative problem. Although these numbers are finite and independent of the chosen point configuration, determining them is highly nontrivial. In a groundbreaking series of papers, Caporaso and Harris established a powerful degeneration technique that produces a recursive formula for these curve counts, now known as the Caporaso--Harris recursion.

Mikhalkin's correspondence theorem later showed that plane curve counts agree with counts of tropical plane curves, which can be described combinatorially as weighted graphs in the plane. This tropical viewpoint dramatically simplifies several aspects of the Caporaso--Harris argument, as first observed by Gathmann and Markwig.

In this talk, I will introduce the Caporaso-Harris recursion, explain how its proof becomes more transparent in the tropical world, and outline the key ideas involved. I will conclude by discussing ongoing joint work with Andrés Jaramillo Puentes, Hannah Markwig, and Felix Röhrle, in which we investigate how to extend these methods to plane curve counts over arbitrary fields.