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Hilbert schemes of points on fold-like curves and their combinatorics

Ángel David Ríos Ortiz, Javier Sendra-Arranz

TL;DR

This work develops a detailed combinatorial and geometric understanding of the Hilbert scheme of $m$ points on curves with rational $n$-fold (fold-like) singularities. By analyzing the punctual Hilbert scheme at the singularity via a moment map to a hypersimplicial complex, the authors explicitly classify irreducible components, describe their intersections, and relate punctual data to the global Hilbert scheme for curves with a single fold-like singularity. They establish reducedness, distinguish smoothable and non-smoothable components, and provide precise normalizations: non-smoothable parts are smooth with affine-local structure, while smoothable parts are normal toric with toric singularities; their normalization is a product of symmetric powers and a Grassmannian factor. The paper culminates in a cohesive framework linking deformation theory, toric geometry, and combinatorics to illuminate the Hilbert schemes of these non-locally planar curves, with further directions toward Quot schemes and higher-dimensional configurations.

Abstract

We investigate the Hilbert scheme of points on curves with n-fold singularities, that is curves that look locally around their singular points as the axis in an affine space. We describe the structure and number of its irreducible components, and provide a detailed analysis of their singularities, revealing rich combinatorial patterns governing its geometry.

Hilbert schemes of points on fold-like curves and their combinatorics

TL;DR

This work develops a detailed combinatorial and geometric understanding of the Hilbert scheme of points on curves with rational -fold (fold-like) singularities. By analyzing the punctual Hilbert scheme at the singularity via a moment map to a hypersimplicial complex, the authors explicitly classify irreducible components, describe their intersections, and relate punctual data to the global Hilbert scheme for curves with a single fold-like singularity. They establish reducedness, distinguish smoothable and non-smoothable components, and provide precise normalizations: non-smoothable parts are smooth with affine-local structure, while smoothable parts are normal toric with toric singularities; their normalization is a product of symmetric powers and a Grassmannian factor. The paper culminates in a cohesive framework linking deformation theory, toric geometry, and combinatorics to illuminate the Hilbert schemes of these non-locally planar curves, with further directions toward Quot schemes and higher-dimensional configurations.

Abstract

We investigate the Hilbert scheme of points on curves with n-fold singularities, that is curves that look locally around their singular points as the axis in an affine space. We describe the structure and number of its irreducible components, and provide a detailed analysis of their singularities, revealing rich combinatorial patterns governing its geometry.

Paper Structure

This paper contains 13 sections, 53 theorems, 219 equations, 18 figures.

Table of Contents

  1. Irreducible components of the punctual Hilbert scheme
  2. The moment map and its combinatorics
  3. From the punctual to the global Hilbert scheme
  4. The local Hilbert scheme and its schematic structure
  5. Singularities
  6. Singular Locus
  7. Local picture of the singularities
  8. Smoothable and non-smoothable components
  9. Non-smoothable components
  10. Smoothable components
  11. Ongoing work and future questions
  12. The Hypersimplicial complex
  13. Some Commutative Algebra

Key Result

Theorem A

Let $C$ be an irreducible curve with a unique rational $n$--fold singularity and denote by $C_{\mathrm{sm}}$ its smooth locus. The irreducible components of $\mathop{\mathcal{Hilb}}\nolimits^m(C)$ are birational to In particular, the number of irreducible components of $\mathop{\mathcal{Hilb}}\nolimits^m(C)$ is $\mathrm{min}\{n-1,m\}$.

Figures (18)

  • Figure 1: Graph of the number of irreducible components of the Hilbert scheme of $m$ points on a curve with an $n$-fold singularity.
  • Figure 2: Planes spanned by each pairs of lines in $X_3$.
  • Figure 3: The $m$ rational irreducible components of $\mathop{\mathrm{Hilb}}\nolimits^m_{\mathbf{0}}(X_2)$.
  • Figure 4: Decomposition of $(m-1)\cdot\Delta_1$ according to the moment map $\mu_m$ for $m=2,3,4$ and $n=2$. It corresponds to the hypersimplicial complexes $\mathcal{K}^{[2]}_2$, $\mathcal{K}^{[3]}_2$, and $\mathcal{K}^{[4]}_2$.
  • Figure 5: Hypersimplicial complex $\mathcal{K}^{[m]}_3$ subdividing $(m-1)\Delta_2$ for $m=1,2$ and $3$. The blue triangles correspond to the image of the maps $\mu_{\mathbf u,2}$ whereas the purple triangles correspond to the image of the maps $\mu_{\mathbf u,1}$.
  • ...and 13 more figures

Theorems & Definitions (138)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Proposition 1.4
  • proof
  • Definition 1.5
  • ...and 128 more