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Linear preservers of secant varieties and other varieties of tensors

Fulvio Gesmundo, Young In Han, Benjamin Lovitz

TL;DR

The paper develops a unified framework for the linear preserver problem of algebraic varieties, focusing on secant varieties of Segre and related tensor varieties. It combines Lie algebra methods, prolongation of ideals, singular-locus analysis, and dual-geometry arguments to determine preservers in broad tensor settings, including many Secant varieties with the expected preserver $G_X=G(V_1,\dots,V_k)\rtimes \widetilde{\mathfrak{S}}$. A central result shows that $\sigma_r((\mathbb{P}^{n-1})^{\times k})$ has the expected preserver for all $r \le n^{\lfloor k/2 \rfloor}$, and the authors extend the analysis to subspace varieties, partition varieties, Waring-rank strata, and certain quantum-state classes such as biseparable and fully separable tensors. The work provides both structural theorems and computational techniques, with applications to quantum information and a range of tensor-geometry problems, while also offering geometric proofs of several known preservers.

Abstract

We study the problem of characterizing linear preserver subgroups of algebraic varieties, with a particular emphasis on secant varieties and other varieties of tensors. We introduce a number of techniques built on different geometric properties of the varieties of interest. Our main result is a simple characterization of the linear preservers of secant varieties of Segre varieties in many cases, including $σ_r((\mathbb{P}^{n-1})^{\times k})$ for all $r \leq n^{\lfloor k/2 \rfloor}$. We also characterize the linear preservers of several other sets of tensors, including subspace varieties, the variety of slice rank one tensors, symmetric tensors of bounded Waring rank, the variety of biseparable tensors, and hyperdeterminantal surfaces. Computational techniques and applications in quantum information theory are discussed. We provide geometric proofs for several previously known results on linear preservers.

Linear preservers of secant varieties and other varieties of tensors

TL;DR

The paper develops a unified framework for the linear preserver problem of algebraic varieties, focusing on secant varieties of Segre and related tensor varieties. It combines Lie algebra methods, prolongation of ideals, singular-locus analysis, and dual-geometry arguments to determine preservers in broad tensor settings, including many Secant varieties with the expected preserver . A central result shows that has the expected preserver for all , and the authors extend the analysis to subspace varieties, partition varieties, Waring-rank strata, and certain quantum-state classes such as biseparable and fully separable tensors. The work provides both structural theorems and computational techniques, with applications to quantum information and a range of tensor-geometry problems, while also offering geometric proofs of several known preservers.

Abstract

We study the problem of characterizing linear preserver subgroups of algebraic varieties, with a particular emphasis on secant varieties and other varieties of tensors. We introduce a number of techniques built on different geometric properties of the varieties of interest. Our main result is a simple characterization of the linear preservers of secant varieties of Segre varieties in many cases, including for all . We also characterize the linear preservers of several other sets of tensors, including subspace varieties, the variety of slice rank one tensors, symmetric tensors of bounded Waring rank, the variety of biseparable tensors, and hyperdeterminantal surfaces. Computational techniques and applications in quantum information theory are discussed. We provide geometric proofs for several previously known results on linear preservers.
Paper Structure (17 sections, 29 theorems, 55 equations)

This paper contains 17 sections, 29 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Techniques for computing the linear preserver
  4. Lie algebra approach
  5. Prolongation and partial derivatives
  6. Singular locus
  7. Dual varieties
  8. Preservers of secant varieties of Segre varieties
  9. Preservers of subspace varieties
  10. Preservers of partition varieties
  11. Waring rank preservers
  12. Linear preservers in quantum information theory
  13. Preservers of hyperdeterminantal surfaces and the "onion" of multipartite entanglement
  14. Biseparable and fully separable preservers
  15. Linear preservers computed directly
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\mathbf{n} = (n_1 , \dots , n_k)$ be a $k$-tuple of positive integers. Let $1\leq r \leq s(\mathbf{n})-1$ be an integer. Let $V_1 , \dots , V_k$ be vector spaces with $\dim V_i = n_i$, let $Y = \mathbb{P} V_1 \times \cdots \times \mathbb{P} V_k$ and $X = \sigma_r(Y)$. Then $G_{\sigma_r(Y)

Theorems & Definitions (54)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • ...and 44 more