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

摘要

We give a geometrical description of the spin-embedding e(sp) of the symplectic dual polar space Delta congruent to 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 Delta. We prove that the map sending every quad of Delta 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 Delta gives an embedding isomorphic to e(sp). (C) 2007 Elsevier Inc. All rights reserved.