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

Projective aspects of the Geometry of Lagrangian Grassmannians and Spinor varieties