文摘
A hyperplane of the symplectic dual polar space a144bfc69630b38688876bc4c52" title="Click to view the MathML source">DW(2n−1,F), ab241b0dfb64c5126c19aefad6c7526" title="Click to view the MathML source">n≥2, is said to be of subspace-type if it consists of all maximal singular subspaces of e90c2b1b1ef" title="Click to view the MathML source">W(2n−1,F) meeting a given a1092228dcac3addc9ef4246976" title="Click to view the MathML source">(n−1)-dimensional subspace of ab16" title="Click to view the MathML source">PG(2n−1,F). We show that a hyperplane of a144bfc69630b38688876bc4c52" title="Click to view the MathML source">DW(2n−1,F) is of subspace-type if and only if every hex F of a144bfc69630b38688876bc4c52" title="Click to view the MathML source">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 988cf78abcecff77cb8e6851c429f" title="Click to view the MathML source">F is a perfect field of characteristic 2, a stronger result can be proved, namely a hyperplane e6287fba811e9222a8df087df01a1af" title="Click to view the MathML source">H of a144bfc69630b38688876bc4c52" title="Click to view the MathML source">DW(2n−1,F) is of subspace-type or arises from the spin-embedding of e6a587e2c13447aaf45af289c61072" title="Click to view the MathML source">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.