The Hilbert scheme of points on a threefold: broken Gorenstein structures and linkage

TL;DR

This paper studies the smoothness of the Hilbert scheme of points on a threefold, introducing broken Gorenstein structures that guarantee smoothness and exploring their prevalence.

Contribution

It introduces the concept of broken Gorenstein structures for finite schemes and conjectures their exhaustiveness in characterizing smooth points.

Findings

Broken Gorenstein structures guarantee smoothness on the Hilbert scheme.

Explicit characterization of smooth points for monomial ideals in A^3.

Several conjectures by Hu are proved regarding singular points.

Abstract

We investigate the Hilbert scheme of points on a smooth threefold. We introduce a notion of broken Gorenstein structure for finite schemes, and show that its existence guarantees smoothness on the Hilbert scheme. Moreover, we conjecture that it is exhaustive: every smooth point admits a broken Gorenstein structure. We give an explicit characterization of the smooth points on the Hilbert scheme of A^3 corresponding to monomial ideals. We investigate the nature of the singular points, and prove several conjectures by Hu. Along the way, we obtain a number of additional results, related to linkage classes, nested Hilbert schemes, and a bundle on the Hilbert scheme of a surface.