Severi varieties parameterize reduced irreducible curves of given geometric genus in a given
linear system on an algebraic surface. The irreducibility of Severi varieties is established
firstly by Harris in 1986 for the projective plane. Few more irreduciblity results for other
surfaces have been obtained since then, and the arguments more or less involves an inductive
process on the genus. In this talk, I will review these approaches to the irreducibility
problem, and then present a proof of the irreducibility of Severi varieties on the projective
plane via tropical geometry. The proof does not involve induction and also shows the
irreducibility of the sublocus of the Severi variety consisting of curves passing through given
points. Time permitting, I will indicate generalizations of this idea to other toric surfaces.
This is joint work with Karl Christ and Ilya Tyomkin.
