Direct Constructions of Hyperplanes of Dual Polar Spaces Arising from Embeddings

Let e be one of the following full projective embeddings of a finite dual polar space Δ of rank n ≥ 2: (i) The Grassmann-embedding of the symplectic dual polar space Δ ≅ DW(2n – 1, q); (ii) the Grassmann-embedding of the Hermitian dual polar space Δ ≅ DH(2n – 1, q ²); (iii) the spin-embedding of the orthogonal dual polar space Δ ≅ DQ(2n, q); (iv) the spin-embedding of the orthogonal dual polar space Δ ≅DQ ⁻(2n + 1, q). Let H_e{mathcal{H}_{e}} denote the set of all hyperplanes of Δ arising from the embedding e. We give a method for constructing the hyperplanes of H_e{mathcal{H}_{e}} without implementing the embedding e and discuss (possible) applications of the given construction.