Towards a Geometric Approach to Strassens Asymptotic Rank Conjecture

We make a first geometric study of three varieties in $mathbb{C}^m otimes mathbb{C}^m otimes mathbb{C}^m$ (for each $m$), including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is to develop a geometric framework for Strassens Asymptotic Rank Conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we determine the dimension of the set of tight tensors. We prove that this dimension equals the dimension of the set of oblique tensors, a less restrictive class introduced by Strassen.