Andrew P. Staal (University of Waterloo)
Date
Monday January 17, 20224:30 pm - 5:30 pm
Location
Online via ZoomAlgebra & Geometry Seminar
Monday, January 17th, 2022
Time: 4:30 p.m. Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)
Speaker: Andrew P. Staal (University of Waterloo)
Title: Small Elementary Components of Hilbert Schemes of Points
Abstract: Hilbert schemes of points are moduli spaces of fundamental importance in algebraic geometry, commutative algebra, and algebraic combinatorics. Since their construction by Grothendieck, they have seen broad-ranging applications from the McKay correspondence to Haiman's proof of the Macdonald positivity conjecture.\par I will present some recent progress in the study of Hilbert schemes Hilbd(An) of d points in affine space, and the related (local) punctual Hilbert schemes Hilbd(OAn,p) at fixed p∈An. Specifically, I will discuss some results on \emph{elementary} components of Hilbert schemes of points and tie these to a question posed by Iarrobino in the 80's: does there exist an irreducible component of the punctual Hilbert scheme Hilbd(OAn,p) of dimension less than (n−1)(d−1)? I will answer this question by describing a new infinite family of irreducible components satisfying this bound, when n=4. A secondary family of elementary components also arises, providing further new examples of elementary components of Hilbert schemes of points, and improving our knowledge surrounding a folklore question on the existence of certain Gorenstein local Artinian rings.\par This is joint work with Matt Satriano (U Waterloo).
Website details here: https://mast.queensu.ca/~georep/