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

Joachim Jelisiejew, Ritvik Ramkumar, Alessio Sammartano

TL;DR

This work studies the Hilbert scheme of $d$ points on smooth threefolds, introducing broken Gorenstein structures as a sharp criterion for smoothness and conjecturing their exhaustiveness on the smoothable component. It provides a complete monomial-case classification, linking smoothness to the absence of singularizing triples and licci-ness, and develops the bicanonical module as a central invariant guiding deformation and tangent-space analysis. The authors establish a Pfaffian structural framework for broken Gorenstein algebras without flips, prove Hu’s singularity conjectures in several key cases (monomial, tripod, and Borel-fixed), and connect smoothness to linkage and nested Hilbert schemes. The paper also uncovers a torus-equivariant rank-$d$ bundle on Hilb^d(A^2) arising from the bicanonical module and highlights how linkage preserves smoothable tangents, offering a nuanced picture of singularities in codimension three Hilbert schemes with substantial implications for combinatorics and algebraic geometry.

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.

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

TL;DR

This work studies the Hilbert scheme of points on smooth threefolds, introducing broken Gorenstein structures as a sharp criterion for smoothness and conjecturing their exhaustiveness on the smoothable component. It provides a complete monomial-case classification, linking smoothness to the absence of singularizing triples and licci-ness, and develops the bicanonical module as a central invariant guiding deformation and tangent-space analysis. The authors establish a Pfaffian structural framework for broken Gorenstein algebras without flips, prove Hu’s singularity conjectures in several key cases (monomial, tripod, and Borel-fixed), and connect smoothness to linkage and nested Hilbert schemes. The paper also uncovers a torus-equivariant rank- bundle on Hilb^d(A^2) arising from the bicanonical module and highlights how linkage preserves smoothable tangents, offering a nuanced picture of singularities in codimension three Hilbert schemes with substantial implications for combinatorics and algebraic geometry.

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.
Paper Structure (27 sections, 35 theorems, 95 equations, 1 figure)

This paper contains 27 sections, 35 theorems, 95 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Monomial ideals
  3. Broken Gorenstein structures
  4. Singular points
  5. Further directions
  6. The shape of the staircase:
  7. Generating functions
  8. Structure of licci ideals
  9. Preliminaries
  10. Zero-dimensional algebras
  11. Tangent spaces and abstract deformation functors
  12. Linkage
  13. Monomial ideals
  14. Macaulay's inverse systems
  15. Broken Gorenstein structures
  16. ...and 12 more sections

Key Result

Theorem 1.3

Let $I \subseteq S = {\Bbbk}[x, y, z]$ be a monomial ideal and $[S/I] \in \mathrm{Hilb}^d(\mathbb{A}^3)$ the corresponding point. The following conditions are equivalent

Figures (1)

  • Figure 3.1: Long exact sequence of $\mathrm{Ext}$-modules

Theorems & Definitions (89)

  • Definition 1.2
  • Theorem 1.3: \ref{['ThmSmoothMonomialClassification']}
  • Definition 1.4: Broken Gorenstein algebras
  • Theorem 1.5: \ref{['ref:smoothnessOfSmoothableBroken:cor']}
  • Example 1.6
  • Example 1.7: \ref{['exampleFlip']}
  • Conjecture 1.8
  • Theorem 1.9: Structure theorem for broken Gorenstein structures without flips
  • Proposition 1.10
  • Conjecture 1.11: H23
  • ...and 79 more