Locally singular hyperplanes in thick dual polar spaces of rank 4
Journal of Combinatorial Theory Series A
The hyperplanes of DQ(2n,K) and DQ-(2n+1,q) which arise from their spin-embeddings
Journal of Combinatorial Theory Series A
On the hyperplanes of the half-spin geometries and the dual polar spaces DQ(2n,K)
Journal of Combinatorial Theory Series A
The structure of the spin-embeddings of dual polar spaces and related geometries
European Journal of Combinatorics
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Let @P be one of the following polar spaces: (i) a nondegenerate polar space of rank n-1=2 which is embedded as a hyperplane in Q(2n,K); (ii) a nondegenerate polar space of rank n=2 which contains Q(2n,K) as a hyperplane. Let @D and DQ(2n,K) denote the dual polar spaces associated with @P and Q(2n,K), respectively. We show that every locally singular hyperplane of DQ(2n,K) gives rise to a hyperplane of @D without subquadrangular quads. Suppose @P is associated with a nonsingular quadric Q^-(2n+@e,K) of PG(2n+@e,K), @e@?{-1,1}, described by a quadratic form of Witt-index n+@e-12, which becomes a quadratic form of Witt-index n+@e+12 when regarded over a quadratic Galois extension of K. Then we show that the constructed hyperplanes of @D arise from embedding.