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Secant variety and syzygies of Hilbert scheme of two points

Chiwon Yoon, Haesong Seo

Abstract

In this paper, we prove that the singular locus of $\mathrm{Sec} (X^{[2]})$ coincides with $X^{[2]}$ under the Grothendieck-Plücker embedding $X^{[2]} \hookrightarrow \mathbb{P}^N$ when $X$ is embedded by a $4$-very ample line bundle. We also prove that the embedding $X^{[2]} \hookrightarrow \mathbb{P}^N$ satisfies Green's condition $(N_p)$ when the embedding of $X$ is positive enough. As an application, we describe the geometry of a resolution of singularities from the secant bundle to $\mathrm{Sec}(X^{[2]})$ when $X$ is a surface.

Secant variety and syzygies of Hilbert scheme of two points

Abstract

In this paper, we prove that the singular locus of coincides with under the Grothendieck-Plücker embedding when is embedded by a -very ample line bundle. We also prove that the embedding satisfies Green's condition when the embedding of is positive enough. As an application, we describe the geometry of a resolution of singularities from the secant bundle to when is a surface.
Paper Structure (13 sections, 25 theorems, 107 equations)

This paper contains 13 sections, 25 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Identifiability of Grassmannians.
  4. Hilbert scheme of points.
  5. Syzygies and Koszul cohomologies.
  6. Identifiability of Hilbert scheme of points
  7. Technical theorems.
  8. Tangent lines to Hilbert scheme of points.
  9. Proof of Theorem \ref{['thm:identifiable-locus-for-Hilbert-scheme-of-points']}.
  10. Singularities of secant varieties
  11. Syzygies of Hilbert scheme of two points
  12. Application
  13. Open questions

Key Result

Theorem 1.1

Let $X$ be a smooth projective variety, and let $\mathcal{L}$ be a $4$-very ample line bundle on $X$. Under the embedding $\varphi_1:X^{[2]} \hookrightarrow \mathbb{P}^N$, the non-identifiable locus of $\mathrm{Sec} (X^{[2]})$ is exactly $X^{[2]}$. In particular, the singular locus of $\mathrm{Sec}

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 2.1: GS2022
  • Definition 2.2
  • Example 2.3
  • Theorem 2.4: CG1990
  • Lemma 2.5: EL1993
  • Lemma 2.6: EL1993
  • Theorem 3.1
  • ...and 33 more