Higher secant varieties of the Segre varieties P1x•••xP1

Let V-t = P-1 x ... x P-1 (t-copies) embedded in p(N)(N = 2(t) - 1) via the Segre embedding. Let (V-t)(s) be the subvariety of pN which is the closure of the union of all the secant Ps-1's to V-t. The expected dimension of (V-t)(s) is min{st + (s - 1), N}. This is not the case for (V-4)(3), which we conjecture is the only defective example in this infinite family. We prove (Theorem 2.3): if e(t) = [2'/t+1] = delta(t)(mod2) and s(t) = e(t) - delta(t) then (V-t)(s) has the expected dimension, except possibly when s = s(t) + 1. Moreover, whenever t = 2(k) - 1, (V-t)(s) has the expected dimension for every s. (c) 2005 Elsevier B.V. All rights reserved.