Hilbert–Burch matrices and explicit torus-stable families of finite subscheme of 𝔸2

ABSTRACT We give an explicit description of the Białynicki-Birula cells on the Hilbert scheme of points on $\mathbb {A}^2$ with isolated fixed points, with the Hilbert–Burch matrices as a main tool. If the fixed point locus is positive dimensional, we obtain an étale rational map to the cell. We prove a conjectural description by R. Homs and A.-L. Winz of certain Gröbner cells. We realise the description 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.