SFU NTAG Seminar: Andrew Staal

I will present some recent progress in the study of Hilbert schemes ${\mathrm{Hilb}}^{}({\mathbb{A}}^n)$ of $d$ points in affine space. Specifically, I will describe some recent examples of elementary components of Hilbert schemes of points. One infinite family of these answers a question posed by Iarrobino in the 80's: does there exist an irreducible component of the (local) punctual Hilbert scheme ${\mathrm{Hilb}}^{}({\mathcal{O}}_{{\mathbb{A}}^n,p})$ of dimension less than $(n-1)(d-1)$? A different family of elementary components arises from the Galois closure operation introduced by Bhargava--Satriano. In both situations, secondary families of elementary components also arise, providing further new examples of elementary components of Hilbert schemes of points.

This is joint work with Matt Satriano (U Waterloo).