Ranks and sets evincing them for the tangent developable

Abstract

Let X ⊆ PP^r be an integral and non-degenerate variety. We study the complement in the k-secant variety of X of the set of all points with X-rank k. In the case k=2 this is contained in the tangent developable τ(X) of X and we study its X-ranks and the sets evincing them when X is a smooth OADP, i.e. X=n, r=n+1 and a general point of PP²ⁿ⁺¹ is contained in a unique secant line of X.