Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cactus scheme, catalecticant minors, and scheme theoretic equations

Jarosław Buczyński, Hanieh Keneshlou

Abstract

The $r$-th cactus variety of a subvariety $X$ in a projective space generalizes the $r$-th secant variety of $X$ and it is defined using linear spans of finite subschemes of $X$ of degree $r$. One of its original purposes was to study the vanishing sets of catalecticant minors. In this article, we equip the cactus variety with a scheme structure, via ``relative linear spans'' of families of finite schemes over a potentially non-reduced base. In this way, we are able to study the vanishing scheme of the catalecticant minors. For a sufficiently high degree Veronese variety, we show that $r$-th cactus scheme and the zero scheme of appropriate catalecticant minors agree on a dense open subset which is the complement of the $(r-1)$-th cactus variety (or scheme). This article is the first part of a series. In the follow-up, as an application, we can describe the singular locus of (in particular) secant varieties to high degree Veronese varieties in terms of singularities of the Hilbert scheme. We will also generalize the result to high degree Veronese reembeddings of other varieties and schemes.

Cactus scheme, catalecticant minors, and scheme theoretic equations

Abstract

The -th cactus variety of a subvariety in a projective space generalizes the -th secant variety of and it is defined using linear spans of finite subschemes of of degree . One of its original purposes was to study the vanishing sets of catalecticant minors. In this article, we equip the cactus variety with a scheme structure, via ``relative linear spans'' of families of finite schemes over a potentially non-reduced base. In this way, we are able to study the vanishing scheme of the catalecticant minors. For a sufficiently high degree Veronese variety, we show that -th cactus scheme and the zero scheme of appropriate catalecticant minors agree on a dense open subset which is the complement of the -th cactus variety (or scheme). This article is the first part of a series. In the follow-up, as an application, we can describe the singular locus of (in particular) secant varieties to high degree Veronese varieties in terms of singularities of the Hilbert scheme. We will also generalize the result to high degree Veronese reembeddings of other varieties and schemes.

Paper Structure

This paper contains 25 sections, 54 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Cactus scheme and catalecticant minors
  3. Structure of Hilbert scheme of finite Gorenstein schemes and catalecticant minors
  4. Conjecture of Deery and Geramita
  5. Overview
  6. Delibarations about schemes, modules and ideals
  7. Scheme inclusion criterion
  8. Finite subschemes as tensors
  9. Gorenstein duality for submodules of free modules
  10. Relative Macaulay bound and Gotzmann's persistence
  11. Families of linear subspaces and linear span
  12. Examples, properties, and saturatedness conjecture
  13. Familiars and linear span of a family
  14. Linear spans of independent familiars and minors
  15. Minors and linear spans
  16. ...and 10 more sections

Key Result

Theorem 1.1

We have the following relations between the locally closed subschemes of ${\mathbb P}_{{\Bbbk}}(S^{(d)}V)$:

Theorems & Definitions (102)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3: Deery, Geramita
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • ...and 92 more