On the nucleus of the Grassmann embedding of the symplectic dual polar space ,

Let n ≥ 3 and let F be a field of characteristic 2. Let DSp ( 2 n , F ) denote the dual polar space associated with the building of type C n over F and let G n − 2 denote the ( n − 2 ) -Grassmannian of type C n . Using the bijective correspondence between the points of G n − 2 and the quads of DSp ( 2 n , F ) , we construct a full projective embedding of G n − 2 into the nucleus of the Grassmann embedding of DSp ( 2 n , 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 = 3 and F is finite.