文摘
A hyperplane of the symplectic dual polar space 876bc4c52" title="Click to view the MathML source">DW(2n−1,F), b0dfb64c5126c19aefad6c7526" 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 959961e90c2b1b1ef" title="Click to view the MathML source">W(2n−1,F) meeting a given 8dcac3addc9ef4246976" title="Click to view the MathML source">(n−1)-dimensional subspace of PG(2n−1,F). We show that a hyperplane of 876bc4c52" title="Click to view the MathML source">DW(2n−1,F) is of subspace-type if and only if every hex ae0f1d2f782679f261892fc1" title="Click to view the MathML source">F of 876bc4c52" title="Click to view the MathML source">DW(2n−1,F) intersects it in either ae0f1d2f782679f261892fc1" title="Click to view the MathML source">F, a singular hyperplane of ae0f1d2f782679f261892fc1" title="Click to view the MathML source">F or the extension of a full subgrid of a quad. In the case e6851c429f" 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 876bc4c52" 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 ae0f1d2f782679f261892fc1" title="Click to view the MathML source">F intersects it in either ae0f1d2f782679f261892fc1" title="Click to view the MathML source">F, a singular hyperplane of ae0f1d2f782679f261892fc1" title="Click to view the MathML source">F, a hexagonal hyperplane of ae0f1d2f782679f261892fc1" title="Click to view the MathML source">F or the extension of a full subgrid of a quad.