Combinatorics Seminar: Toric geometry and Hilbert schemes

Abstract: The study of toric varieties lives in the intersection of combinatorics and algebraic geometry. Each toric variety corresponds to a fan whose combinatorics perfectly encodes the geometry of toric variety. In this talk, I will discuss an extension of the toric combinatorics to certain spaces parametrizing finite collections of points in toric surfaces called Hilbert schemes. Pleasantly, in some (conjecturally all!) cases the “polytopes” for these Hilbert schemes parametrize finite collections of points in the polytopes of the underlying toric surface. We will see how this bridge to combinatorics can be used to answer fundamental geometric questions about the Hilbert scheme.

Host: Laura Escobar Vega