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.

中文翻译：

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迈向Strassen渐近秩猜想的几何方法

我们对\（\ mathbb {C} ^ m \ otimes \ mathbb {C} ^ m \ otimes \ mathbb {C} ^ m \）（对于每个m）中的三个变体进行首次几何研究，包括Zariski封闭紧张量的集合，这些张量具有连续的规则对称性。我们的动机是为Strassen的渐近秩猜想开发一个几何框架，该构想使任何紧张量的渐近秩最小。特别地，我们确定紧张量集的尺寸。我们证明此维等于斜张量集的维，这是Strassen引入的限制较小的类。