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Collineation varieties of tensors

Fulvio Gesmundo, Hanieh Keneshlou

TL;DR

The paper introduces the $k$-th collineation variety of a third-order tensor as the closure of the image of a minors-map on a first flattening, and develops a comprehensive framework to study these varieties via projections of Veronese varieties. It provides complete classifications for pencils ($\dim V_1=2$) and for small nets (notably $\mathbb{C}^3\otimes\mathbb{C}^2\otimes\mathbb{C}^{n_3}$ and $\mathbb{C}^3\otimes\mathbb{C}^3\otimes\mathbb{C}^3$), linking collineation varieties to invariant theory (Kronecker, Nur, DitDeGrMar) and to stratifications of tensor spaces. The work shows that, in the studied cases, collineation varieties are rational and of minimal degree, and it connects these varieties to classical tensor invariants and orbit closures, thereby providing a finer invariant than rank or border rank. It also outlines computational approaches (e.g., Macaulay2) for explicit determination and paves directions for understanding higher-dimensional cases and boundaries of strata. Overall, this framework offers new geometric invariants and classification tools for distinguishing tensor orbit-closures via collineation geometry with potential applications in algebraic statistics, quantum information, and complexity theory.

Abstract

In this article, we introduce the $k$-th collineation variety of a third order tensor. This is the closure of the image of the rational map of size $k$ minors of a matrix of linear forms associated to the tensor. We classify such varieties in the case of pencils of matrices, and nets of matrices of small size. We discuss the natural stratification of tensor spaces induced by the invariants and the geometric type of the collineation varieties.

Collineation varieties of tensors

TL;DR

The paper introduces the -th collineation variety of a third-order tensor as the closure of the image of a minors-map on a first flattening, and develops a comprehensive framework to study these varieties via projections of Veronese varieties. It provides complete classifications for pencils () and for small nets (notably and ), linking collineation varieties to invariant theory (Kronecker, Nur, DitDeGrMar) and to stratifications of tensor spaces. The work shows that, in the studied cases, collineation varieties are rational and of minimal degree, and it connects these varieties to classical tensor invariants and orbit closures, thereby providing a finer invariant than rank or border rank. It also outlines computational approaches (e.g., Macaulay2) for explicit determination and paves directions for understanding higher-dimensional cases and boundaries of strata. Overall, this framework offers new geometric invariants and classification tools for distinguishing tensor orbit-closures via collineation geometry with potential applications in algebraic statistics, quantum information, and complexity theory.

Abstract

In this article, we introduce the -th collineation variety of a third order tensor. This is the closure of the image of the rational map of size minors of a matrix of linear forms associated to the tensor. We classify such varieties in the case of pencils of matrices, and nets of matrices of small size. We discuss the natural stratification of tensor spaces induced by the invariants and the geometric type of the collineation varieties.
Paper Structure (11 sections, 12 theorems, 40 equations)

This paper contains 11 sections, 12 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Projections of Veronese varieties
  4. Pencils
  5. Nets
  6. Nets in $\mathbb{C}^2 \otimes \mathbb{C}^{n_3}$
  7. Nets in $\mathbb{C}^3 \otimes \mathbb{C}^3$
  8. Stratification of tensor spaces via collineation varieties
  9. Stratification of spaces of pencils
  10. Stratification of spaces of small nets
  11. Future directions

Key Result

Theorem 3.1

Let $T \in V_1 \otimes V_2 \otimes V_3$ be a tensor. Let $n_i = \dim V_i$, with $n_1 = 2$ and $n_2 \leq n_3$ and let $1\leq k\leq n_2$ with the second inequality strict if $n_2=n_3$. Suppose $B_k^1(T)$ is $0$-dimensional and $p=\deg(B_k^1(T))$. If $p < k$, then $\mathscr{C}^1_k(T)$ is the rational n

Theorems & Definitions (23)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of \ref{['thm: pencils general']}
  • Corollary 3.3
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 13 more