Homogeneous projective varieties with semi-continuous rank function

Let $\mathbb X\subset\mathbb P(V)$ be a projective variety, which is not contained in a hyperplane. Then every vector $v$ in $V$ can be written as a sum of vectors from the affine cone $X$ over $\mathbb X$. The minimal number of summands in such a sum is called the rank of $v$. The set of vectors of rank $r$ is denoted by $X_r$ and its projective image by $\mathbb X_r$. The r-th secant variety of $X$ is defined $σ_r(\mathbb X):=\bar{\sqcup_{s\le r}\mathbb X_s}$; it is called tame if $σ_r(\mathbb X)=\sqcup_{s\le r} \mathbb X_s$ and wild if the closure contains elements of higher rank. In this paper, we classify all equivariantly embedded homogeneous projective varieties $\mathbb X\subset\mathbb P(V)$ with tame secant varieties. Classical examples are: the variety of rank one matrices (Segre variety with two factors) and the variety of rank one quadratic forms (quadratic Veronese variety). In the general setting, $\mathbb X$ is the orbit in $\mathbb P(V)$ of a highest weight line in an irreducible representation $V$ of a reductive algebraic group $G$. Thus, our result is a list of all irreducible representations of reductive groups, where the resulting $\mathbb X$ has tame secant varieties.

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