Hyperplanes of dual polar spaces and the spin module (Q811662)

Let \(\Gamma\) be a dual polar space of type \(B_ n(F)\) defined by a quadric in \(PG(2n,F)\) of rank \(n\), where \(F\) is a field for which the generalized quadrangle \(W(3,F)\) of type \(Sp(4,F)\) possesses no ovoid (in particular \(F=GF(q)\), \(q\) odd). It is shown (1) that any embedding \(e: \Gamma\to P(V)\) satisfies \(dim V\leq 2^ n\), and (2) that if \(e\) is an embedding with \(dim V=2^ n\), then every geometric hyperplane arises from a hyperplane section of this embedding. As a consequence, any such embedding is universal.