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Components of the nested Hilbert scheme of few points

Michele Graffeo, Paolo Lella

TL;DR

The paper advances the understanding of the geometry of nested Hilbert schemes by (i) providing a new lower bound for the existence of non-smoothable nestings of fat points on smooth $n$-folds with $n\ge 4$, (ii) introducing a construction that yields generically non-reduced elementary components from reduced ones, and (iii) proving reducibility phenomena for Hilbert schemes of points on singular hypersurfaces in $\mathbb{A}^3$ when singularities have sufficiently high multiplicity. It leverages the Białynicki-Birula decomposition, Hilbert--Samual stratification, and the TNT criterion, together with an analysis of 2-step ideals, to derive explicit tangent-space and dimension computations that distinguish smoothable from non-smoothable or non-reduced components. The results illuminate how elementary components underpin the global structure of Hilb$^d X$ and provide concrete families and dimensional criteria to identify non-smoothable and non-reduced behavior, with implications for singularity theory and the geometry of zero-dimensional subschemes. Overall, the work offers a systematic framework to generate and recognize fundamental building blocks of Hilbert schemes of points in both smooth and singular settings, highlighting new thresholds and construction principles.

Abstract

We study the existence and the schematic structure of elementary components of the nested Hilbert scheme on a smooth quasi-projective variety. Precisely, we find a new lower bound for the existence of non-smoothable nestings of fat points on a smooth $n$-fold, for $n\geqslant 4$. Moreover, we implement a systematic method to build generically non-reduced elementary components. We also investigate the problem of irreducibility of the Hilbert scheme of points on a singular hypersurface of $\mathbb A^3$. Explicitly, we show that the Hilbert scheme of points on a hypersurface of $\mathbb{A}^3$ having a singularity of multiplicity at least 5 admits elementary components.

Components of the nested Hilbert scheme of few points

TL;DR

The paper advances the understanding of the geometry of nested Hilbert schemes by (i) providing a new lower bound for the existence of non-smoothable nestings of fat points on smooth -folds with , (ii) introducing a construction that yields generically non-reduced elementary components from reduced ones, and (iii) proving reducibility phenomena for Hilbert schemes of points on singular hypersurfaces in when singularities have sufficiently high multiplicity. It leverages the Białynicki-Birula decomposition, Hilbert--Samual stratification, and the TNT criterion, together with an analysis of 2-step ideals, to derive explicit tangent-space and dimension computations that distinguish smoothable from non-smoothable or non-reduced components. The results illuminate how elementary components underpin the global structure of Hilb and provide concrete families and dimensional criteria to identify non-smoothable and non-reduced behavior, with implications for singularity theory and the geometry of zero-dimensional subschemes. Overall, the work offers a systematic framework to generate and recognize fundamental building blocks of Hilbert schemes of points in both smooth and singular settings, highlighting new thresholds and construction principles.

Abstract

We study the existence and the schematic structure of elementary components of the nested Hilbert scheme on a smooth quasi-projective variety. Precisely, we find a new lower bound for the existence of non-smoothable nestings of fat points on a smooth -fold, for . Moreover, we implement a systematic method to build generically non-reduced elementary components. We also investigate the problem of irreducibility of the Hilbert scheme of points on a singular hypersurface of . Explicitly, we show that the Hilbert scheme of points on a hypersurface of having a singularity of multiplicity at least 5 admits elementary components.
Paper Structure (10 sections, 14 theorems, 56 equations, 1 figure)

This paper contains 10 sections, 14 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organisation of the content
  3. Preliminaries
  4. The Białynicki-Birula decomposition
  5. Hilbert functions and embedding dimension
  6. Trivial Negative tangents and elementary components
  7. The class of 2-step ideals
  8. Proof of Theorem A
  9. Proof of Theorem B
  10. Proof of Theorem C

Key Result

Theorem A

Let $R$ be the polynomial ring in $n\geqslant 4$ variables, and let $2\leqslant s \leqslant n-2$ be an integer. Consider where $H_2\subset R_2$ is a generic linear subspace of codimension $2$ of the space of quadratic forms. Then, the point $[I^{(1)}\supset I^{(2)}]\in\mathop{\mathrm{Hilb}}\nolimits^{1+s,n+3}{\mathbb{A}}^n$ is non-smoothable.

Figures (1)

  • Figure 1: Each box is associated to a pair $(n,s)$, $4\leqslant n \leqslant 15$ and $0\leqslant s\leqslant n$ and the corresponding Hilbert function $\underline{{\boldsymbol{h}}}=((1,s),(1,n,2))$. The label $a^b$ of a box means that: $a$ is the difference in \ref{['eq:gap']}; $b$ is $\mathsf t^{=-1}_{[\underline{I}]} \mathop{\mathrm{Hilb}}\nolimits\mathbb{A}^n$, for $[I]\in H_{\underline{{\boldsymbol{h}}}}^n$ generic.

Theorems & Definitions (44)

  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 1.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4: ELEMENTARY
  • Definition 2.6
  • Definition 2.8
  • Remark 2.9
  • ...and 34 more