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 $\Delta\cong DW(5,2^r)$ by showing how the natural embedding of $W(5,2^r)$ into $\PG(5,2^r)$ is involved in the Grassman-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}.