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Given a closed subvariety X in a projective space, the rank with respect to Xof a point p in this projective space is the least integer r such that p liesin the linear span of some r points of X. Let W_k be the closure of the set ofpoints of rank with respect to X equal to k. For small values of k such lociare called secant varieties. This article studies the loci W_k for values of klarger than the generic rank. We show they are nested, we bound theirdimensions, and we estimate the maximal possible rank with respect to X inspecial cases, including when X is a homogeneous space or a curve. The theoryis illustrated by numerous examples, including Veronese varieties, the Segreproduct of dimensions (1,3,3), and curves. An intermediate result provides alower bound on the dimension of any GL_n orbit of a homogeneous form.

On the locus of points of high rank