The Hilbert scheme of points on a threefold: broken Gorenstein structures and linkage
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.
Cite
@article{arxiv.2409.17009,
title = {The Hilbert scheme of points on a threefold: broken Gorenstein structures and linkage},
author = {Joachim Jelisiejew and Ritvik Ramkumar and Alessio Sammartano},
journal= {arXiv preprint arXiv:2409.17009},
year = {2026}
}
Comments
Major revision. Final version, to appear in the Journal f\"ur die reine und angewandte Mathematik