A probabilistic algorithm for the secant defect of Grassmann varieties

Abstract

In this paper we study the higher secant varieties of Grassmann varieties in relation to Waring&#39;s problem for alternating tensors and to Alexander-Hirschowitz theorem. We show how to identify defective higher secant varieties of Grassmannians using a probabilistic method involving Terracini&#39;s Lemma, and we describe an algorithm which can compute, by numerical methods, dim(G(k,n)^{s}) for n<=14. Our main result is that, except for Grassmannians of lines, if n<=14 and k<=(n-1)/2 (if n=14 we have studied the case k<=5) there are only the four known defective cases: G(2,6)^{3}, G(3,7)^{3}, G(3,7)^{4} and G(2,8)^{4}.