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Identifiability and singular locus of secant varieties to Grassmannians

Vincenzo Galgano, Reynaldo Staffolani

Abstract

Secant varieties are among the main protagonists in tensor decomposition, whose study involves both pure and applied mathematical areas. Grassmannians are the building blocks for skewsymmetric tensors. Although they are ubiquitous in the literature, the geometry of their secant varieties is not completely understood. In this work we determine the singular locus of the secant variety of lines to a Grassmannian Gr(k,V) using its structure as SL(V)-variety. We solve the problems of identifiability and tangential-identifiability of points in the secant variety: as a consequence, we also determine the second Terracini locus to a Grassmannian.

Identifiability and singular locus of secant varieties to Grassmannians

Abstract

Secant varieties are among the main protagonists in tensor decomposition, whose study involves both pure and applied mathematical areas. Grassmannians are the building blocks for skewsymmetric tensors. Although they are ubiquitous in the literature, the geometry of their secant varieties is not completely understood. In this work we determine the singular locus of the secant variety of lines to a Grassmannian Gr(k,V) using its structure as SL(V)-variety. We solve the problems of identifiability and tangential-identifiability of points in the secant variety: as a consequence, we also determine the second Terracini locus to a Grassmannian.
Paper Structure (19 sections, 28 theorems, 89 equations)

This paper contains 19 sections, 28 theorems, 89 equations.

Table of Contents

  1. Preliminaries
  2. Ranks and secant varieties.
  3. Identifiability.
  4. Tangential-identifiability
  5. The $2$-nd secant variety of Grassmannians.
  6. The defective toy-case $\mathop{\mathrm{Gr}}\nolimits(2,\mathbb C^N) \subset \mathbb P(\bigwedge^2\mathbb C^N)$
  7. $\mathop{\mathrm{SL}}\nolimits(N)$-orbit structure of $\sigma_2(\mathop{\mathrm{Gr}}\nolimits(k,\mathbb C^N))$
  8. The secant branch
  9. The tangent branch
  10. Orbit closures: inclusions; dimensions of secant orbits
  11. Orbit inclusions.
  12. Orbit dimensions.
  13. Identifiability in $\sigma_2(\mathop{\mathrm{Gr}}\nolimits(k,\mathbb C^N))$
  14. Unidentifiability of $\Sigma_{2,\mathop{\mathrm{Gr}}\nolimits(k,\mathbb C^N)}$.
  15. Identifiability of $\Sigma_{l,\mathop{\mathrm{Gr}}\nolimits(k,\mathbb C^N)}$ for $l\geq 3$.
  16. ...and 4 more sections

Key Result

Theorem 1

The orbits $\Sigma_{l,\mathop{\mathrm{Gr}}\nolimits(k,V)}$ for $l\geq 3$ are the only ones in $\sigma_2(\mathop{\mathrm{Gr}}\nolimits(k,V))$ being identifiable. The orbits $\Theta_{l,\mathop{\mathrm{Gr}}\nolimits(k,V)}$ for $l\geq 3$ are the only ones in $\tau(\mathop{\mathrm{Gr}}\nolimits(k,V))$ be

Theorems & Definitions (63)

  • Theorem
  • Definition 1.0.1
  • Definition 1.0.2
  • Theorem 2.0.1
  • proof
  • Definition 3.0.1
  • Lemma 3.0.2
  • proof
  • Definition 3.0.3
  • Corollary 3.0.4
  • ...and 53 more