Graduate Student Colloquium
Xudong Zheng
UIC
Hilbert scheme of points and the $n!$ conjecture
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.
Thursday February 27, 2014 at 4:00 PM in SEO 636