Tame Secant Varieties and Group Actions

Let X be a complex projective variety embedded in a complex projective space. The dimensions of the secant varieties of X have an expected value, and it is important to know if they are equal or at least near to this expected value. Blomenhofer and Casarotti proved important results on the embeddings of G-varieties, G being an algebraic group, embedded in the projectivations of an irreducible G-representation, proving that no proper secant variety is a cone. In this paper, we give other conditions which assure that no proper secant varieties of X are a cone, e.g., that X is G-homogeneous. We consider the Segre product of two varieties with the product action and the case of toric varieties. We present conceptual tests for it, and discuss the information we obtained from certain linear projections of X. For the Segre–Veronese embeddings of ${\mathbb{P}}^n\times {\mathbb{P}}^n$ with respect to forms of bidegree $(1,d)$, our results are related to the simultaneous rank of degree d forms in $n+1$ variables.