Secant dimensions of low-dimensional homogeneous varieties

Abstract

Abstract We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all Segre–Veronese embeddings of ℙ1 × ℙ1, ℙ1 × ℙ1 × ℙ1, and ℙ2 × ℙ1, as well as for the flag variety ℱ of incident point-line pairs in ℙ2. For ℙ2 × ℙ1 and ℱ the results are new, while the proofs for the other two varieties are more compact than existing proofs. Our main tool is the second author's tropical approach to secant dimensions.