Higher secant varieties of the Segre varieties image

Let image (t-copies) embedded in image via the Segre embedding. Let (Vt)s be the subvariety of image which is the closure of the union of all the secant imageʹs to Vt. The expected dimension of (Vt)s is min st+(s-1),N . This is not the case for (V4)3, which we conjecture is the only defective example in this infinite family. We prove (Theorem 2.3): if image and st=et-δt then (Vt)s has the expected dimension, except possibly when s=st+1. Moreover, whenever t=2k-1, (Vt)s has the expected dimension for every s.