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 $\sigma_k(S_h)$ be the $k$-secant variety of $S_h$. We decribe a probabilistic algorithm which computes the complex dimension of $\sigma_k(S_h)$. Then, by using an inductive argument, we get our main result: $\sigma_3(S_h)$ has the expected dimension except when $h\in \{7,8\}$. 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.