The hyperplanes of DQ(2n,K) and DQ−(2n+1,q) which arise from their spin-embeddings

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