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Bronowski's conjecture and the identifiability of projective varieties

Alex Massarenti, Massimiliano Mella

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 $τ_{h-1}^X:X\dashrightarrow\mathbb{P}^n$ 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.

Bronowski's conjecture and the identifiability of projective varieties

Abstract

Let be an irreducible and non-degenerate variety of dimension . The Bronowski's conjecture predicts that is -identifiable if and only if the general -tangential projection 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.
Paper Structure (9 sections, 19 theorems, 41 equations)

This paper contains 9 sections, 19 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Counterexamples to the Bronowski's conjecture
  3. Identifiability and the Bronowski's conjecture
  4. Applications to toric varieties and varieties in tensor spaces
  5. Toric and Segre-Veronese varieties
  6. Grassmannians and Flag varieties
  7. Lagrangian Grassmannians and Spinor varieties
  8. Gaussian moment varieties
  9. Varieties of powers of forms

Key Result

Theorem 1.3

Let $\Gamma\subset\mathbb{P}^N$ be a rational normal curve of degree $N\geq 7$ and assume $h = \frac{N+1}{2r}$ to be an integer. Then $\mathbb{S}ec_r(\Gamma)$ gives a counterexample to Conjecture CBroC and Conjecture CBro.

Theorems & Definitions (52)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 42 more