A geometric description of the spin-embedding of symplectic dual polar spaces of rank 3

We give a geometrical description of the spin-embedding e sp of the symplectic dual polar space Δ ≅ DW ( 5 , 2 r ) by showing how the natural embedding of W ( 5 , 2 r ) into PG ( 5 , 2 r ) is involved in the Grassmann-embedding e gr of Δ. We prove that the map sending every quad of Δ to its nucleus realizes the natural embedding of W ( 5 , 2 r ) . Taking the quotient of e gr over the space spanned by the nuclei of the quadrics corresponding to the quads of Δ gives an embedding isomorphic to e sp .