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

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