Alexander Stathis : The Hilbert Scheme of Points on a Surface II

Posted by Alex Stathis , part of the Graduate Algebraic Geometry Seminar.

At

Oct. 17, 2013, 3:30 p.m.

In

SEO 712

Abstract

Let $X$ be a smooth projective surface, and fix a positive integer $n$. In this talk, we will follow the work of Fogarty in "Algebraic Families on an Algebraic Surface" to show that ${Hilb}^n(X) := X^{[n]}$, the Hilbert scheme parameterizing zero dimensional subschemes $Z \subset X$ of length $n$, is connected and smooth, and that the Hilbert-Chow morphism $h : X^{[n]} \to X^{(n)}$, where $X^{(n)}$ is the $n$-th symmetric product of $X$, is birational. We will begin this discussion where we left off last time.