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Secant varieties of generalised Grassmannians

Vincenzo Galgano

TL;DR

This work develops a uniform, representation-theoretic description of secant varieties of lines to cominuscule generalized Grassmannians, exploiting G–actions to reduce analysis to parabolic orbit representatives in G/P_k and in the tangent space 𝔤/𝔭_k. It delivers a complete orbit poset for σ_2(G/P_k) in the cominuscule setting, identifies the identifiable and singular loci, and introduces the 2–nd strong-Terracini locus, providing explicit dimensions of decomposition and singular loci. The paper also extends the comparison to non-cominuscule cases by detailing IG(k,2N): a non-cominuscule example with a richer, non-totally ordered tangent-orbit structure, including explicit orbit invariants (r,h,t) and dimensions, and analyzes the tangential variety τ(IG(k,2N)). Overall, the results unify and extend prior works on Grassmannians and Spinor varieties, highlighting when and how the cominuscule structure governs secant geometry and when non-cominuscule isotropic Grassmannians deviate from this pattern.

Abstract

Secant varieties of a homogeneously embedded generalised Grassmannian $G/P$ inherit the natural group action, and one can reduce the study of their local geometric properties to $G$-orbit representatives. The case of secant varieties of lines is particularly elegant as their $G$-orbits are induced by $P$-orbits in both $G/P$ and $\mathfrak{g}/\mathfrak{p}$. Parabolic orbits are a classical problem in Representation Theory, well understood when $G/P$ is cominuscule. Exploiting them, we provide a complete and uniform description of both the identifiable and singular loci of the secant variety of lines to any cominuscule variety. We also introduce a finer version of the $2$-nd Terracini locus, called $2$-nd strong-Terracini locus, and we determine it for cominuscule varieties. Finally, we analyse the non-cominuscule case of isotropic Grassmannians for comparison, and we highlight a few differences.

Secant varieties of generalised Grassmannians

TL;DR

This work develops a uniform, representation-theoretic description of secant varieties of lines to cominuscule generalized Grassmannians, exploiting G–actions to reduce analysis to parabolic orbit representatives in G/P_k and in the tangent space 𝔤/𝔭_k. It delivers a complete orbit poset for σ_2(G/P_k) in the cominuscule setting, identifies the identifiable and singular loci, and introduces the 2–nd strong-Terracini locus, providing explicit dimensions of decomposition and singular loci. The paper also extends the comparison to non-cominuscule cases by detailing IG(k,2N): a non-cominuscule example with a richer, non-totally ordered tangent-orbit structure, including explicit orbit invariants (r,h,t) and dimensions, and analyzes the tangential variety τ(IG(k,2N)). Overall, the results unify and extend prior works on Grassmannians and Spinor varieties, highlighting when and how the cominuscule structure governs secant geometry and when non-cominuscule isotropic Grassmannians deviate from this pattern.

Abstract

Secant varieties of a homogeneously embedded generalised Grassmannian inherit the natural group action, and one can reduce the study of their local geometric properties to -orbit representatives. The case of secant varieties of lines is particularly elegant as their -orbits are induced by -orbits in both and . Parabolic orbits are a classical problem in Representation Theory, well understood when is cominuscule. Exploiting them, we provide a complete and uniform description of both the identifiable and singular loci of the secant variety of lines to any cominuscule variety. We also introduce a finer version of the -nd Terracini locus, called -nd strong-Terracini locus, and we determine it for cominuscule varieties. Finally, we analyse the non-cominuscule case of isotropic Grassmannians for comparison, and we highlight a few differences.

Paper Structure

This paper contains 19 sections, 19 theorems, 95 equations, 5 figures, 4 tables.

Table of Contents

  1. Preliminaries
  2. Miscellanea on Lie groups and Lie algebras
  3. Generalised Grassmannians
  4. Secant varieties, identifiability and Terracini loci
  5. Hamming distance on projective varieties
  6. Secants of lines to generalised Grassmannians
  7. Orbit poset in $\sigma_2(G/P_k)$ in the cominuscule case
  8. Orbits in $\sigma_2^\circ(G/P_k)$
  9. Orbits in $\tau(G/P_k)$
  10. Poset in $\sigma_2(G/P_k)$
  11. Identifiability, singular locus and Terracini locus in the cominuscule case
  12. Identifiability
  13. Singular locus
  14. Strong-Terracini locus
  15. A non-cominuscule example: $\mathop{\mathrm{IG}}\nolimits(k,2N)$
  16. ...and 4 more sections

Key Result

Proposition 2.2

There are $d_{k}^{_G}+1$ many $P_k$--orbits in a cominuscule variety $G/P_k$, for $d_{k}^{_G}$ as in table:secants to cominuscule. More precisely, given $(\beta_1,\ldots, \beta_{d_k^{_G}})$ a maximal orthogonal sequence of long roots in $\Phi_k^+$, the $P_k$--orbits in $G/P_k$ are where $w_j:= s_{\beta_1}\cdots s_{\beta_j}$ for any $j \in [d_k^{_G}]$. Moreover, such orbits are totally ordered, na

Figures (5)

  • Figure 2.1: Hasse diagrams of Schubert cells (\ref{['figure:Bruhat G(2,4).a']}) and of $P_2$--orbits (\ref{['figure:Bruhat G(2,4).b']}) in $\mathop{\mathrm{Gr}}\nolimits(2,4)$.
  • Figure 2.2: Poset graph of $G$--orbits in $\sigma_2(G/P_k)$ for $G/P_k$ cominuscule.
  • Figure 4.1: The orbit inclusions are described in the above lattice cube, centered at the red point $(0,0,0)$ corresponding to a fixed triplet of invariants $(r,h,t)$. Any other point $(a,b,c)$ in the lattice corresponds to the triplet (not necessarily of invariants) $(r+a,h+b,t+c)$. The gray points are triplets which are not of invariants (constraints \ref{['constraints on invariants']} are not satisfied). \ref{['fig:lattice orbit inclusions.a']} shows the orbits in the lattice degenerating to $\cal O_{(r,h,t)}$: the cyan arrows denote the minimal degenerations. \ref{['fig:lattice orbit inclusions.b']} shows the orbits in the lattice to which $\cal O_{(r,h,t)}$ degenerates: the green arrows denote the minimal degenerations.
  • Figure 4.2: Poset graph of $P_3$--orbits in $\mathfrak{sp}_8/\mathfrak{p}_3$, and their dimensions.
  • Figure 4.3: Poset graph of $\mathop{\mathrm{Sp}}\nolimits_8$--orbits in $\tau(\mathop{\mathrm{IG}}\nolimits(3,8))$.

Theorems & Definitions (31)

  • Proposition 2.2: richardson
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Theorem 2.9
  • proof
  • Proposition 3.1
  • ...and 21 more