Secant varieties of generalised Grassmannians

TL;DR

This paper studies the geometric properties of secant varieties of generalized Grassmannians, providing a complete description for cominuscule cases and analyzing differences in non-cominuscule cases, with implications for representation theory.

Contribution

It offers a uniform description of the identifiable and singular loci of secant lines to cominuscule varieties and introduces a refined Terracini locus concept.

Findings

Complete description of loci for cominuscule varieties.

Determination of the 2nd strong-Terracini locus for these varieties.

Analysis of differences in non-cominuscule isotropic Grassmannians.

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.