A probabilistic algorithm for the secant defect of Grassmann varieties

Details A probabilistic algorithm for the secant defect of Grassmann varieties A probabilistic algorithm for the secant defect of Grassmann varieties

2005-11-28

arxiv.org/abs/math/0511683v1

Mathematics

Algebraic Geometry

13 pages

Scientific paper

In this paper we study the higher secant varieties of Grassmann varieties in relation to Waring'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'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}.

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