A characterization of a class of hyperplanes of DW(2n−1,F)DW(2n−1,F)

A hyperplane of the symplectic dual polar space DW(2n−1,F)DW(2n−1,F), n≥2n≥2, is said to be of subspace-type if it consists of all maximal singular subspaces of W(2n−1,F)W(2n−1,F) meeting a given (n−1)(n−1)-dimensional subspace of PG(2n−1,F)PG(2n−1,F). We show that a hyperplane of DW(2n−1,F)DW(2n−1,F) is of subspace-type if and only if every hex FF of DW(2n−1,F)DW(2n−1,F) intersects it in either FF, a singular hyperplane of FF or the extension of a full subgrid of a quad. In the case FF is a perfect field of characteristic 2, a stronger result can be proved, namely a hyperplane HH of DW(2n−1,F)DW(2n−1,F) is of subspace-type or arises from the spin-embedding of DW(2n−1,F)≅DQ(2n,F)DW(2n−1,F)≅DQ(2n,F) if and only if every hex FF intersects it in either FF, a singular hyperplane of FF, a hexagonal hyperplane of FF or the extension of a full subgrid of a quad.