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.