Identifiability and singular locus of secant varieties to spinor varieties

In this work we analyze the $Spin(V)$-structure of the secant variety of lines $\sigma_{2}(\mathbb{S})$ to a Spinor variety $\mathbb{S}$ minimally embedded in its spin representation. In particular, we determine the poset of the $Spin(V)$-orbits and their dimensions. We use it for solving the problems of identifiability and tangential-identifiability in $\sigma_2(\mathbb S)$, and for determining the second Terracini locus of $\mathbb{S}$. Finally, we show that the singular locus $Sing(\sigma_{2}(\mathbb{S}))$ contains the two $Spin(V)$-orbits of lowest dimensions and it lies in the tangential variety $\tau(\mathbb{S})$: we also conjecture what it set-theoretically is.
Comment: 32 pages