The combinatorial properties of the hyperplanes of \(D\)W\((5,q)\) arising from embedding (Q1008995)

It was proved by \textit{B. De Bruyn} [Discrete Math. 309, No. 2, 304--321 (2009; Zbl 1160.51006)] that there are six isomorphism classes of hyperplanes in the dual polar space \(D\)W\((5, q), q\) even, which arise from its Grassmann-embedding. In the present paper the combinatorial properties of these hyperplanes are determined. Specifically, for each such hyperplane \(H\) the number of quads \(Q\) for which \({Q \cap H}\) is a certain configuration of points in \(Q\) and the number of points \({x \in H}\) for which \({x^\perp \cap H}\) is a certain configuration of points in \({x^\perp}\) is calculated. Using purely combinatorial techniques, it is also shown that the set of hyperplanes of \(D\)W\((5, q), q\) odd, which arise from its Grassmann-embedding can be divided into six subclasses if one takes only into account the above-mentioned combinatorial properties.