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