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

Abstract

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