Higher secants of spinor varieties

Let S_h be the even pure spinors variety of a complex vector space V of even dimension 2h endowed with a non degenerate quadratic form Q and let σ_k(S_h) be the k-secant variety of S_h. We decribe a probabilistic algorithm which computes the complex dimension of σ_k(S_h). Then, by using an inductive argument, we get our main result: σ_3(S_h) has the expected dimension except when h ∈ {7, 8}. Also we provide theoretical arguments which prove that S_7 has a defective 3-secant variety and S_8 has defective 3-secant and 4-secant varieties