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arXiv · 2409.17009

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

Joachim Jelisiejew·Ritvik Ramkumar·Alessio Sammartano
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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.

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Joachim Jelisiejew, Ritvik Ramkumar, Alessio Sammartano. 2024-09-25. The Hilbert scheme of points on a threefold: broken Gorenstein structures and linkage. https://doi.org/10.1515/crelle-2026-0008

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