Classification and degenerations of small minimal border rank tensors via modules

TL;DR

This paper classifies all minimal border rank tensors in three-way tensor spaces of dimension up to 5, describing their degenerations and proving the absence of certain degeneracies for smaller dimensions.

Contribution

It provides a complete classification of minimal border rank tensors in $ ext{C}^m imes ext{C}^m imes ext{C}^m$ for $m \u003d 5$, including degenerations and non-existence results.

Findings

107 isomorphism classes for $m=5$

37 classes up to permutation of factors

No 1-degenerate minimal border rank tensors for $m \u003d 4$

Abstract

We give a self-contained classification of $1_*$-generic minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$. Together with previous results, this gives a classification of all minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$: there are $107$ isomorphism classes (only $37$ up to permuting factors). We fully describe possible degenerations among the tensors. We prove that there are no $1$-degenerate minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m $ for $m \leq 4$.