On the third secant variety
arXiv:1111.7005
Abstract
We determine normal forms and ranks of tensors of border rank at most three. We present a differential-geometric analysis of limits of secant planes in a more general context. In particular there are at most four types of points on limiting trisecant planes for cominuscule varieties such as Grassmannians. We also show the singular locus of the first two secant varieties of all triple Segre products has codimension at least two.
22 pages; v2 exposition improved in accordance with referee's suggestions, corrected a few minor mistakes in the proofs. To appear in Journal of Algebraic Combinatorics