From non Defectivity to Identifiability

TL;DR

This paper introduces a novel approach linking secant defect to identifiability, leading to improved bounds and new results for various algebraic varieties, including Grassmann varieties.

Contribution

It proposes a new method connecting secant defect to identifiability, improving bounds and establishing first results for Grassmann varieties.

Findings

Optimal bounds for Segre and Segre-Veronese varieties.

First identifiability results for Grassmann varieties.

Enhanced understanding of secant defect's role in identifiability.

Abstract

A projective variety $X\subset\mathbb{P}^N$ is $h$-identifiable if the generic element in its $h$-secant variety uniquely determines $h$ points on $X$. In this paper we propose an entirely new approach to study identifiability, connecting it to the notion of secant defect. In this way we are able to improve all known bounds on identifiability. In particular we give optimal bounds for some Segre and Segre-Veronese varieties and provide the first identifiability statements for Grassmann varieties.