The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings

We characterize the hyperplanes of the dual polar space D W ( 2 n − 1 , q ) which arise from projective embeddings as those hyperplanes H of D W ( 2 n − 1 , q ) which satisfy the following property: if Q is an ovoidal quad, then Q ∩ H is a classical ovoid of Q . A consequence of this is that all hyperplanes of the dual polar spaces D W ( 2 n − 1 , 4 ) , D W ( 2 n − 1 , 16 ) and D W ( 2 n − 1 , p ) ( p prime) arise from projective embeddings.