2024/10/10 by Vincenzo Galgano, Galgano, Vincenzo
Mathematics · Physics and Astronomy · #14L30 #14M15 #14N07 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2410.08158
openalex publication_date 2024/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathfrakg/\mathfrakp. 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.