Ian Cavey (UIUC): Hilbert schemes of points on toric surfaces

The Hilbert scheme of n points on a smooth surface X is a natural compactification of the space of n unordered points on X. In this talk, I will discuss several results on these Hilbert schemes for X a toric surface. First, sections of line bundles on the Hilbert scheme can be identified with certain collections of diagonally symmetric/alternating polynomials satisfying a polytopal support condition. Linear bases of these global sections can be indexed by certain n-tuples of integer points in the polygon(s) corresponding to X, which I will describe explicitly in some cases. Time permitting, I will discuss how this concrete representation of sections can be applied to the minimal model program for these spaces. Some of these results are based on joint work with Eugene Gorsky, Alexei Oblomkov, and Joshua Turner.