2013/04/11 by Alexey Petukhov, Petukhov, A., Valdemar V. Tsanov +1
Mathematics · Medicine · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #FOS: Mathematics #Phytoestrogen effects and research #Representation Theory (math.RT) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1304.3322
openalex publication_date 2013/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbb X⊂\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 Xr and its projective image by \mathbb Xr. The r-th secant variety of X is defined σr(\mathbb X):=\sqcups≤ r\mathbb Xs; it is called tame if σr(\mathbb X)=\sqcups≤ r \mathbb Xs and wild if the closure contains elements of higher rank. In this paper, we classify all equivariantly embedded homogeneous projective varieties \mathbb X⊂\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.