Bronowski's conjecture and the identifiability of projective varieties

TL;DR

This paper investigates Bronowski's conjecture on the identifiability of projective varieties, provides counterexamples, and proposes an amended conjecture linking identifiability to secant defectiveness.

Contribution

It offers counterexamples to Bronowski's conjecture and proves an amended version for a broad class of varieties, connecting identifiability to secant defectiveness.

Findings

Counterexamples to Bronowski's conjecture.

Amended conjecture valid for many varieties.

Reduces identifiability problem to secant defectiveness.

Abstract

Let $X\subset\mathbb{P}^{hn+h-1}$ be an irreducible and non-degenerate variety of dimension $n$. The Bronowski's conjecture predicts that $X$ is $h$-identifiable if and only if the general $(h-1)$-tangential projection $\tau_{h-1}^X:X\dashrightarrow\mathbb{P}^n$ is birational. In this paper we provide counterexamples to this conjecture. Building on the ideas that led to the counterexamples we manage to prove an amended version of the Bronowski's conjecture for a wide class of varieties and to reduce the identifiability problem for projective varieties to their secant defectiveness.