Title: The ubiquity of smooth Hilbert schemes
Abstract:
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 to make the collection of all such Hilbert schemes
into 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.