Symmetric Tensor Rank and Scheme Rank: An Upper Bound in terms of Secant Varieties. (8th September 2013) - Ballico, E.
Abstract : Let X ⊂ ℙ r be an integral and nondegenerate variety. Let c be the minimal integer such that ℙ r is the c -secant variety of X, that is, the minimal integer c such that for a general O ∈ ℙ r there is S ⊂ X with # ( S ) = c and O ∈ 〈 S 〉, where 〈  〉 is the linear span. Here we prove that for every P ∈ ℙ r there is a zero-dimensional scheme Z ⊂ X such that P ∈ 〈 Z 〉 and deg ( Z ) ≤ 2 c ; we may take Z as union of points and tangent vectors of X reg .