  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 Strassen's
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.
