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Local Equations for Hilbert Schemes of Points

Nathan Ilten, Francesco Meazzini, Andrea Petracci

TL;DR

This work computes the local structure of the Hilbert scheme $\mathrm{Hilb}_{n+1}^{n}$ at the worst point $[I]$ with $I=\\langle x_1,\\dots,x_n\\rangle^2$, by presenting an explicit description of the completed local ring as $\\Bbbk\\llbracket t_{ij}^{k}\\rrbracket/\\mathfrak{J}$ where $\\gamma_{ijk}^{\ell}$ generate $\\mathfrak{J}$. It achieves this via two complementary approaches: a classical deformation-theoretic lifting of syzygies and a differential graded Lie algebra (DGLA) framework built on a Koszul-Tate resolution, with a third perspective through deformations of based algebras. The paper also describes the universal family in this formal neighborhood, identifies the miniversal base space, and constructs large linear subspaces of the Hilbert scheme near $[I]$, yielding new insights into component structure and irreducibility, including new results for certain cases such as $\mathrm{Hilb}_{5}^{4}$. Overall, the work clarifies the local geometry around the most singular point and demonstrates the compatibility and complementarity of classical and DGLA deformation theories in explicit, high-dimensional settings.

Abstract

We compute the completion of the local ring of the Hilbert scheme of degree $n+1$ subschemes of $\mathbb{A}^n$ at the point corresponding to the ideal $\langle x_1,\ldots,x_n\rangle^2$, and describe the completion of the universal family. For the purposes of comparison, we do this computation with both classical and DGLA methods. We use our explicit equations to produce high dimensional linear subspaces of the Hilbert scheme, and compare our equations with those coming from deformations of based algebras.

Local Equations for Hilbert Schemes of Points

TL;DR

This work computes the local structure of the Hilbert scheme at the worst point with , by presenting an explicit description of the completed local ring as where generate . It achieves this via two complementary approaches: a classical deformation-theoretic lifting of syzygies and a differential graded Lie algebra (DGLA) framework built on a Koszul-Tate resolution, with a third perspective through deformations of based algebras. The paper also describes the universal family in this formal neighborhood, identifies the miniversal base space, and constructs large linear subspaces of the Hilbert scheme near , yielding new insights into component structure and irreducibility, including new results for certain cases such as . Overall, the work clarifies the local geometry around the most singular point and demonstrates the compatibility and complementarity of classical and DGLA deformation theories in explicit, high-dimensional settings.

Abstract

We compute the completion of the local ring of the Hilbert scheme of degree subschemes of at the point corresponding to the ideal , and describe the completion of the universal family. For the purposes of comparison, we do this computation with both classical and DGLA methods. We use our explicit equations to produce high dimensional linear subspaces of the Hilbert scheme, and compare our equations with those coming from deformations of based algebras.

Paper Structure

This paper contains 22 sections, 17 theorems, 82 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The worst point of the Hilbert scheme
  4. Relations among $\gamma$
  5. The Taylor complex
  6. The tangent space
  7. Obstructions
  8. Classical approach
  9. First order deformations and lifted relations
  10. Second order
  11. Higher order and proof of Theorem \ref{['thm:main']}
  12. DG-Lie methods and abstract deformations
  13. The derivation Lie algebra
  14. A truncated Koszul-Tate resolution
  15. First order deformations
  16. ...and 7 more sections

Key Result

Theorem A

Let $n\geq 3$. The completion of the local ring of $\mathop{\mathrm{Hilb}}\nolimits_{n+1}^{n}$ at the point $[I]$ is isomorphic to the quotient of the power series ring $\Bbbk \llbracket t_{ij}^{k} \rrbracket$ by $\mathfrak{J}$. The universal family in a formal neighborhood of $[I]$ is given by the Furthermore, one obtains the miniversal base space and family of $\mathop{\mathrm{Spec}}\nolimits S

Theorems & Definitions (43)

  • Theorem A
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.8
  • proof
  • Proposition 2.9
  • ...and 33 more