The hyperplanes of and which arise from their spin-embeddings

We characterize the hyperplanes of the dual polar spaces DQ ( 2 n , K ) and DQ − ( 2 n + 1 , q ) which arise from their respective spin-embeddings. The hyperplanes of DQ ( 2 n , K ) which arise from its spin-embedding are precisely the locally singular hyperplanes of DQ ( 2 n , K ) . The hyperplanes of DQ − ( 2 n + 1 , q ) which arise from its spin-embedding are precisely the hyperplanes H of DQ − ( 2 n + 1 , q ) which satisfy the following property: if Q is an ovoidal quad, then Q ∩ H is a classical ovoid of Q.