Homogeneous projective varieties with semi-continuous rank function

Abstract

Let $${\mathbb{X}\subset\mathbb{P}(V)}$$ be a projective variety, which is not contained in a hyperplane. Then every vector v in V may be written as a sum of vectors from the affine cone over $${\mathbb{X}}$$ . The minimal number of summands in such a sum is called the rank of v. In this paper, we classify all equivariantly embedded homogeneous projective varieties $${\mathbb{X}\subset\mathbb{P}(V)}$$ whose rank function is lower semi-continuous. 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, for which the corresponding rank function is lower semi-continuous.