Higher secant varieties of the Segre varieties P1×⋯×P1

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 et=[2tt+1]≡δt(mod2) 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.