A probabilistic algorithm for the secant defect of Grassmann varieties

In this paper we study the higher secant varieties of Grassmann varieties in relation to the functional Waring’s problem for alternating tensors and to the Alexander–Hirschowitz theorem. We show how to identify defective higher secant varieties of Grassmannians using a probabilistic method involving Terracini’s lemma, and we describe an algorithm which can compute, by numerical methods, dim (SsG(k, n)) for n ⩽ 14. Our main result is that, except for Grassmannians of lines, if n ⩽ 14 and (if n = 14 we have studied the case k ⩽ 5) there are only the four known defective cases: S2G(2, 6), S2G(3, 7), S3G(3, 7) and S3G(2, 8).