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