The hyperplanes of \(DQ(2n, \mathbb K)\) and \(DQ^{ - }(2n+1,q)\) which arise from their spin-embeddings (Q880009)

In this paper the author proves the following three facts: First, the locally singular hyperplanes of the dual polar space \(DQ(2n,\mathbb K)\), for \(n\) greater than 2, are precisely the hyperplanes of \(DQ(2n,\mathbb K)\) which arise from the spin-embedding. Second, the hyperplanes of the dual polar space \(DQ^-(2n+1,q)\), for \(n\) greater than 2, which arise from the spin-embedding are precisely the hyperplanes \(H\) of \(DQ^-(2n+1,q)\) which satisfy the following property: if \(Q\) is an ovoidal quad, then \(Q\cap H\) is a classical ovoid of \(Q\). Finally, the hyperplanes \(DQ^-(7,q)\) which arise from its spin-embedding are precisely the singular hyperplanes, the extensions of the classical ovoids of the quads and the hexagonal hyperplanes.