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Homogeneous projective varieties with semi-continuous rank function

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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.

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Petukhov, A.V., Tsanov, V.V. Homogeneous projective varieties with semi-continuous rank function. manuscripta math. 147, 269–303 (2015). https://doi.org/10.1007/s00229-014-0723-5

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