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.
Highlights
Introduction and resultsLet K be an algebraically closed field of characteristic 0; all varieties appearing here will be over K
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
Let G be a connected affine algebraic group, and let X be a projective variety on which G acts transitively
Summary
Let K be an algebraically closed field of characteristic 0; all varieties appearing here will be over K. The Segre–Veronese embedding of P1 × P1 of degree (d, e) with d ≥ e ≥ 1 is non-defective unless e = 2 and d is even, in which case the (d + 1)-st secant variety has codimension 1 rather than the expected 0. We shall prove our theorems using a polyhedral-combinatorial lower bound on higher secant dimensions introduced by the second author in [10]. This goes as follows: to a given X and V we associate a finite set B of points in Rdim X , which parameterises a certain basis in V.
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