Elementary components of Hilbert schemes

TL;DR

This paper extends the Bialynicki-Birula decomposition to singular schemes and identifies new elementary, smooth components of the Hilbert scheme of points on affine four-space, providing criteria for smoothness and smoothability.

Contribution

It introduces a generalized decomposition method for singular schemes and discovers new elementary components of the Hilbert scheme of points on affine four-space.

Findings

Identified infinite families of elementary components in the Hilbert scheme of points on affine four-space.

Provided verifiable conditions for smoothness of points on the Hilbert scheme.

Established a necessary condition for the smoothability of finite subschemes given by homogeneous ideals.

Abstract

We generalize the Bialynicki-Birula decomposition to singular schemes and apply it to the Hilbert scheme of points on an affine space. We find an infinite family of small, elementary and generically smooth components of the Hilbert scheme of points of the affine four-space. Our method gives easily verifiable sufficient conditions for proving that a point of the Hilbert scheme is smooth and lies on an elementary component. We also present a necessary condition for smoothability of a finite subscheme given by a homogeneous ideal.