On geometric SDPS-sets of elliptic dual polar spaces

Let n∈N∖{0,1} and let K and K′ be fields such that K′ is a quadratic Galois extension of K. Let Q−(2n+1,K) be a nonsingular quadric of Witt index n in PG(2n+1,K) whose associated quadratic form defines a nonsingular quadric Q+(2n+1,K′) of Witt index n+1 in PG(2n+1,K′). For even n, we define a class of SDPS-sets of the dual polar space DQ−(2n+1,K) associated to Q−(2n+1,K), and call its members geometric SDPS-sets. We show that geometric SDPS-sets of DQ−(2n+1,K) are unique up to isomorphism and that they all arise from the spin embedding of DQ−(2n+1,K). We will use geometric SDPS-sets to describe the structure of the natural embedding of DQ−(2n+1,K) into one of the half-spin geometries for Q+(2n+1,K′).