On the nucleus of the Grassmann embedding of the symplectic dual polar space DSp(2n,F), char(F)=2

Let n≥3n≥3 and let FF be a field of characteristic 2. Let DSp(2n,F) denote the dual polar space associated with the building of type Cn over FF and let Gn−2Gn−2 denote the (n−2)(n−2)-Grassmannian of type Cn. Using the bijective correspondence between the points of Gn−2Gn−2 and the quads of DSp(2n,F), we construct a full projective embedding of Gn−2Gn−2 into the nucleus of the Grassmann embedding of DSp(2n,F). This generalizes a result of an earlier paper [I. Cardinali, G. Lunardon, A geometric description of the spin-embedding of symplectic dual polar spaces of rank 3, J. Combin. Theory Ser. A (in press)] which contains an alternative proof of this fact in the case when n=3n=3 and FF is finite.