From non-defectivity to identifiability

Abstract

A point $p$ in a projective space is $h$-identifiable via a variety $X$ if there is a unique way to write $p$ as a linear combination of $h$ points of $X$. Identifiability is important both in algebraic geometry and in applications. In this paper we propose an entirely new approach to study identifiability, connecting it to the notion of secant defect for any smooth projective variety. In this way we are able to improve the known bounds on identifiability and produce new identifiability statements. In particular, we give optimal bounds for some Segre and Segre–Veronese varieties and provide the first identifiability statements for Grassmann varieties.