Seminar in Real and Complex Geometry

Monday, April 6, 2020, 16:00-18:00, online (via zoom)

Xiang He (The Hebrew University)

On the Severi problem with point constraints


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.