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

TL;DR

This paper uses Hilbert-Burch matrices to explicitly describe Bia{
42}ynicki-Birula cells on the Hilbert scheme of points in A2^2, providing new insights into fixed points, deformations, and rational maps.

Contribution

It provides an explicit construction of Bia{
42}ynicki-Birula cells using Hilbert-Burch matrices and proves a conjecture related to these structures.

Findings

Explicit description of Bia{
42}ynicki-Birula cells on the Hilbert scheme.

Construction of rational E9tale maps to the Hilbert scheme.

Characterization of formal deformations of ideals in the Hilbert scheme.

Abstract

Using Hilbert-Burch matrices, we give an explicit description of the Bia{\l}ynicki-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 \'etale 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 \'etale map to the Hilbert scheme which is not contained in any Bia{\l}ynicki-Birula cell. Finally, we give an explicit description of the formal deformations of any ideal in the Hilbert scheme of points on the plane.