Bronowski's conjecture and the identifiability of projective varieties

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 ${\tau }_{h-1}^X:X⇢{\mathbb{P}}^n$ 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.