Xudong Zheng : Hilbert scheme of points and the $n!$ conjecture

Posted by Xudong Zheng , part of the Graduate Student Colloquium.

At

Feb. 27, 2014, 4 p.m.

In

SEO 636

Abstract

Given an algebraic variety $X$, the Hilbert scheme of points $Hilb^n(X)$ is the parameter space of all possible collection of $n$ points on $X$. If $X$ is a smooth curve, $Hilb^n(X)$ is isomorphic to the $n$-fold product of $C$ modulo the action of permuting factors by the symmetric group $S_n$. A second extensively studied case is when $X$ is a smooth surface. In particular, M. Haiman utilized geometric properties of the Hilbert scheme of points to study questions in combinatorics. Suppose $k[x_1, \dots, x_n]$ is the polynomial ring in $n$ variables. It is a free module of rank $n!$ over the subring $k[x_1, \dots, x_n]^{S_n}$, consisting of permutation invariant polynomials. The most naïve form of the $n!$ conjecture concerns the analogy for the polynomial ring $k[x_1, \dots, x_n; y_1, \dots, y_n]$ with the symmetric group $S_n$ simultaneously permuting the variables $(x_1, \dots, x_n)$ and $(y_1, \dots, y_n)$. I will explicitly give a system of coordinates on the Hilbert scheme and relate the Hilbert scheme to the $n!$ conjecture. If time permits, I will indicate some generalizations to the case where the underlying variety $X$ has some singularity.

There will be refreshment after the talk.