Classification and degenerations of small minimal border rank tensors via modules

We give a self-contained classification of $1_{\ast }$-generic minimal border rank tensors in ${\mathbb{C}}^m\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^m$ for $m\le 5$. 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 $m\le 5$: 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 $m\le 4$.