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Identifiability and singular locus of secant varieties to spinor varieties

Vincenzo Galgano

TL;DR

This work analyzes the secant variety of lines to Spinor varieties 𝕊_N^+ under the Spin$_{2N}$-action, solving identifiability and tangential-identifiability questions via Clifford apolarity. It provides a complete orbit stratification into secant orbits Σ_{l,N} and tangent orbits Θ_{l,N}, and derives explicit dimensional, identifiability, and Terracini-locus results. The second Terracini locus is shown to coincide with the distance-2 orbit closure, and the authors establish bounds for the singular locus of σ_2(𝕊_N^+), conjecturing equality with this orbit closure. The approach blends representation theory, nonabelian apolarity, and geometric analysis, contributing to the broader understanding of secant varieties for cominuscule varieties.

Abstract

In this work we analyze the $Spin(V)$-structure of the secant variety of lines $σ_{2}(\mathbb{S})$ to a Spinor variety $\mathbb{S}$ minimally embedded in its spin representation. In particular, we determine the poset of the $Spin(V)$-orbits and their dimensions. We use it for solving the problems of identifiability and tangential-identifiability in $σ_2(\mathbb S)$, and for determining the second Terracini locus of $\mathbb{S}$. Finally, we show that the singular locus $Sing(σ_{2}(\mathbb{S}))$ contains the two $Spin(V)$-orbits of lowest dimensions and it lies in the tangential variety $τ(\mathbb{S})$: we also conjecture what it set-theoretically is.

Identifiability and singular locus of secant varieties to spinor varieties

TL;DR

This work analyzes the secant variety of lines to Spinor varieties 𝕊_N^+ under the Spin-action, solving identifiability and tangential-identifiability questions via Clifford apolarity. It provides a complete orbit stratification into secant orbits Σ_{l,N} and tangent orbits Θ_{l,N}, and derives explicit dimensional, identifiability, and Terracini-locus results. The second Terracini locus is shown to coincide with the distance-2 orbit closure, and the authors establish bounds for the singular locus of σ_2(𝕊_N^+), conjecturing equality with this orbit closure. The approach blends representation theory, nonabelian apolarity, and geometric analysis, contributing to the broader understanding of secant varieties for cominuscule varieties.

Abstract

In this work we analyze the -structure of the secant variety of lines to a Spinor variety minimally embedded in its spin representation. In particular, we determine the poset of the -orbits and their dimensions. We use it for solving the problems of identifiability and tangential-identifiability in , and for determining the second Terracini locus of . Finally, we show that the singular locus contains the two -orbits of lowest dimensions and it lies in the tangential variety : we also conjecture what it set-theoretically is.
Paper Structure (19 sections, 32 theorems, 97 equations, 2 figures, 2 tables)

This paper contains 19 sections, 32 theorems, 97 equations, 2 figures, 2 tables.

Table of Contents

  1. Preliminaries
  2. Clifford algebras and spin groups.
  3. Spin representations.
  4. Spinor varieties.
  5. Maximal orthogonal Grassmannians.
  6. Hamming distance of varieties.
  7. Secant varieties.
  8. Identifiability.
  9. Nonabelian apolarity.
  10. Clifford apolarity
  11. The poset of $\mathop{\mathrm{Spin}}\nolimits_{2N}$--orbits in $\sigma_2(\mathbb S_N^+)$
  12. Secant orbits in $\sigma_2^\circ(\mathbb S_{N}^+)$
  13. Tangent orbits in $\tau(\mathbb S_{N}^+)$
  14. Inclusions among closures of $\mathop{\mathrm{Spin}}\nolimits_{2N}$--orbits
  15. Identifiability in $\sigma_2(\mathbb S_{N}^+)$
  16. ...and 4 more sections

Key Result

Theorem 1

The second Terracini locus $\mathop{\mathrm{Terr}}\nolimits_2(\mathbb S_N^+)$ of the Spinor variety $\mathbb S_N^+$ corresponds to the orbit closure $\overline{\Sigma_{2,N}}=\mathbb S_N^+ \sqcup \Sigma_{2,N}$.

Figures (2)

  • Figure 1: Poset graph of $\mathop{\mathrm{Spin}}\nolimits_{2N}$--orbits in $\sigma_2(\mathbb S_{N}^+)$.
  • Figure 2: Poset graph of the $\mathop{\mathrm{Spin}}\nolimits_{16}$--orbits in $\sigma_2(\mathbb S_{8}^+)$, and their dimensions.

Theorems & Definitions (76)

  • Theorem : Theorem \ref{['thm:terracini locus spinor']}
  • Theorem : Corollary \ref{['cor:sing locus spinor']}
  • Remark
  • Remark 1.1: Setting
  • Proposition 1.2
  • Remark 1.3
  • Proposition 1.4
  • Definition 1.5
  • Definition 1.6
  • Proposition 1.7
  • ...and 66 more