Abstract
We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.
Highlights
We get a notion of symmetric-rank and of ∧-rank for which one can ask the same questions as in the case of arbitrary tensors. These are translated into algebraic geometry problems on secant varieties of Veronese varieties and Grassmannians
Non-regular cases, i.e., where the Hilbert function of the scheme of two-fat points is not as expected, have to be analyzed case by case; regular cases can be proven by induction: (a) The list of non-regular cases corresponds to defective Veronese varieties and is very classical; see Section 2.1.3, page 11 and [36] for the list of all papers where all these cases were investigated
Notice that the above definition of skew-symmetric apolarity works well for computing the dimension of secant varieties to Grassmannians since it defines the apolar of a subspace that is exactly what is needed for Terracini’s lemma, but if one would like to have an analogous definition of apolarity for skew symmetric tensors, there are a few things that have to be done
Summary
When considering finite dimensional vector spaces over a field k (which for us, will always be algebraically closed and of characteristic zero, unless stated otherwise), there are three main functors that come to attention when doing multilinear algebra:. The first interest in the secant variety σ2(X) of a variety X ⊂ PN lies in the fact that if σ2(X) = PN, the projection of X from a generic point of PN into PN−1 is an isomorphism This goes back to the XIX Century with the discovery of a surface X ⊂ P5, for which σ2(X) is a hypersurface, even though its expected dimension is five. Not directly addressed to the study of secant varieties, they confirmed the conjecture that, apart from the quadratic Veronese varieties and a few well-known exceptions, all the Veronese varieties have higher secant varieties of the expected dimension In a sense, this result completed a project that was underway for over 100 years (see [2,3,9])
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