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Singularities and syzygies of secant varieties of smooth projective varieties

Doyoung Choi, Justin Lacini, Jinhyung Park, John Sheridan

Abstract

We study the higher secant varieties of a smooth projective variety embedded in projective space. We prove that when the variety is a surface and the embedding line bundle is sufficiently positive, these varieties are normal with Du Bois singularities and the syzygies of their defining ideals are linear to the expected order. We show that the cohomology of the structure sheaf of the surface completely determines whether the singularities of its secant varieties are Cohen--Macaulay or rational. We also prove analogous results when the dimension of the original variety is higher and the secant order is low, and by contrast we prove a result that strongly implies these statements do not generalize to higher dimensional varieties when the secant order is high. Finally, we deduce a complementary result characterizing the ideal of secant varieties of a surface in terms of the symbolic powers of the ideal of the surface itself, and we include a theorem concerning the weight one syzygies of an embedded surface -- analogous to the gonality conjecture for curves -- which we discovered as a natural application of our techniques.

Singularities and syzygies of secant varieties of smooth projective varieties

Abstract

We study the higher secant varieties of a smooth projective variety embedded in projective space. We prove that when the variety is a surface and the embedding line bundle is sufficiently positive, these varieties are normal with Du Bois singularities and the syzygies of their defining ideals are linear to the expected order. We show that the cohomology of the structure sheaf of the surface completely determines whether the singularities of its secant varieties are Cohen--Macaulay or rational. We also prove analogous results when the dimension of the original variety is higher and the secant order is low, and by contrast we prove a result that strongly implies these statements do not generalize to higher dimensional varieties when the secant order is high. Finally, we deduce a complementary result characterizing the ideal of secant varieties of a surface in terms of the symbolic powers of the ideal of the surface itself, and we include a theorem concerning the weight one syzygies of an embedded surface -- analogous to the gonality conjecture for curves -- which we discovered as a natural application of our techniques.

Paper Structure

This paper contains 56 sections, 100 theorems, 154 equations.

Table of Contents

  1. Preliminaries on Hilbert schemes
  2. Hilbert scheme of points
  3. Distinguished subschemes
  4. The Hilbert--Chow morphism
  5. The symmetric product
  6. The isospectral Hilbert scheme
  7. The discriminant and its root
  8. Smoothness and irreducibility
  9. The canonical bundle
  10. The tangent and cotangent sheaves
  11. The nested Hilbert scheme of points
  12. The cotangent sheaves
  13. Smoothness
  14. The residual morphism
  15. Properties of the universal families
  16. ...and 41 more sections

Key Result

Theorem A

(Singularity Theorem) Fix a positive integer $k$ and assume that either $n \leq 2$ or $k \leq 2$ and that $L$ is sufficiently positive. Then $\Sigma_k$ has normal Du Bois singularities which are Moreover, $\Sigma_k\subseteq \mathbb{P}^r$ is projectively normal (in fact it is arithmetically Cohen--Macaulay if $\Sigma_k$ itself is Cohen--Macaulay) with regularity $\operatorname{reg}(\mathcal{O}_{\S

Theorems & Definitions (124)

  • Theorem A
  • Theorem B
  • Corollary C
  • Theorem D
  • Corollary E
  • Theorem F
  • Theorem G
  • Theorem H
  • Lemma 1.1
  • Theorem 1.2
  • ...and 114 more