Geometry & Topology Seminar

Andrew Staal, Department of Pure Mathematics, University of Waterloo

"Ubiquity of Smooth Hilbert Schemes"

What are the geometric properties of a typical Hilbert scheme? In this talk, we consider Hilbert schemes parametrizing closed subschemes in some projective space with a fixed Hilbert polynomial. We first explain how the collection of all such Hilbert schemes forms a collection of trees and a discrete probability space.  Exploiting the underlying combinatorial structure, we show why a random Hilbert scheme is smooth and irreducible with probability greater than 0.5 and how to classify Hilbert schemes with unique Borel-fixed points.

MC 5403