Higher Secant Varieties of the Segre varieties P1x...xP1

Let V_t be image of the Segre embedding of P1x...xP1 (t-copies) and let (V_t)^s be its s-secant variety, which is the closure of the union of all the (s-1)-planes which are s-secant to V_t. The expected dimension of (V_t)^s is min {st+(s-1),2^t-1} . 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)that under certain conditions on t,s (which leave only one value of s uncovered for all t) that the expected dimension is actually achieved