Hilbert-Burch matrices and explicit torus-stable families of finite subschemes of ${\mathbb{A}}^2$

Abstract

Using Hilbert-Burch matrices, we give an explicit description of the Białynicki-Birula cells on the Hilbert scheme of points on ${\mathbb{A}}^2$ with isolated fixed points. If the fixed point locus is positive dimensional we obtain an étale rational map to the cell. We prove Conjecture 4.2 from arXiv:2309.06871 which we realize as a special case of our construction. We also show examples when the construction provides a rational étale map to the Hilbert scheme which is not contained in any Białynicki-Birula cell. Finally, we give an explicit description of the formal deformations of any ideal in the Hilbert scheme of points on the plane.

accepted manuscript for The Quarterly Journal Of Mathematics

References in corpus (1)

• On the Cobordism Class of the Hilbert Scheme of a Surface