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Nondefectivity of invariant secant varieties

Alexander Taveira Blomenhofer, Alex Casarotti

Abstract

We show that a large class of secant varieties is nondefective. In particular, we positively resolve most cases of the Baur-Draisma-de Graaf conjecture on Grassmannian secants in large dimensions. Our result improves the known bounds on nondefectivity for various other secant varieties, including Chow varieties, Segre-Veronese varieties and Gaussian moment varieties. We also give bounds for identifiability and the generic ranks.

Nondefectivity of invariant secant varieties

Abstract

We show that a large class of secant varieties is nondefective. In particular, we positively resolve most cases of the Baur-Draisma-de Graaf conjecture on Grassmannian secants in large dimensions. Our result improves the known bounds on nondefectivity for various other secant varieties, including Chow varieties, Segre-Veronese varieties and Gaussian moment varieties. We also give bounds for identifiability and the generic ranks.
Paper Structure (14 sections, 15 theorems, 47 equations, 2 figures)

This paper contains 14 sections, 15 theorems, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization
  4. Techniques and Acknowledgments
  5. Preliminaries
  6. The stationarity lemma
  7. Applications
  8. Grassmannian varieties
  9. Chow varieties
  10. Segre-Veronese varieties
  11. The Gaussian moment variety
  12. Low-rank Gaussians
  13. Other $G$-varieties
  14. Asymptotics and critical degree

Key Result

Theorem 1.1

Let $G$ be a group and $V$ an $N$-dimensional, irreducible affine cone in an irreducible $G$-module $\mathcal{L}$. If $V$ is closed under the $G$-action, then it holds that

Figures (2)

  • Figure 1: Comparison of the various bounds on secant nondefectivity of $\mathop{\mathrm{Gr}}\nolimits(d,n)$ when $d=5$.
  • Figure 2: Comparison of the various bounds on secant nondefectivity of $s\nu_{\underline{d}}(\underline{n})$ when $\underline{d}=(1,1,2)$ and $\underline{n}=(n,n,n)$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Definition 1
  • Lemma 2.1: Terracini, see Te12
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2: Stationarity Lemma
  • proof
  • ...and 26 more