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 Grassmann-embedding 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. © 2007 Elsevier Inc. All rights reserved.