Projective aspects of the Geometry of Lagrangian Grassmannians and Spinor varieties

We study the projective behavior, mainly with respect to osculating spaces and secant varieties, of Lagrangian Grassmannians and Spinor varieties. We prove that these varieties have osculating dimension smaller than expected. Furthermore, we give numerical conditions ensuring the non secant defectivity of Lagrangian Grassmannians in their Pl\"ucker embedding and of Spinor varieties in both their Pl\"ucker and Spinor embeddings.