Identifiability and singular locus of secant varieties to Grassmannians

TL;DR

This paper investigates the geometric properties of secant varieties to Grassmannians, focusing on their singular loci, identifiability, and tangential-identifiability, advancing understanding in tensor decomposition and algebraic geometry.

Contribution

It determines the singular locus of secant varieties to Grassmannians and addresses key problems of identifiability and tangential-identifiability, using the structure of SL(V)-varieties.

Findings

Identified the singular locus of secant varieties to Grassmannians.

Resolved the problems of identifiability and tangential-identifiability.

Determined the second Terracini locus for 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.