Two classes of hyperplanes of dual polar spaces without subquadrangular quads

Let Π 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 Δ and DQ(2n,K) denote the dual polar spaces associated with Π and Q(2n,K), respectively. We show that every locally singular hyperplane of DQ(2n,K) gives rise to a hyperplane of Δ without subquadrangular quads. Suppose Π is associated with a nonsingular quadric Q−(2n+ϵ,K) of PG(2n+ϵ,K), ϵ∈{−1,1}, described by a quadratic form of Witt-index , which becomes a quadratic form of Witt-index when regarded over a quadratic Galois extension of K. Then we show that the constructed hyperplanes of Δ arise from embedding.