Hyperplanes of dual polar spaces of rank 3 with no subquadrangular quad (Q2783470)

Let \(\Delta\) be the point-line geometry arising from a thick dual polar space. Then \(\Delta\) is a near hexagon with quads. Let \(H\) be a geometric hyperplane of \(\Delta\). The intersection of \(H\) with any quad not contained in \(H\) is either the perp of a point in \(Q\) (the quad is singular), or a full subquadrangle (the quad is subquadrangular), or an ovoid of \(Q\) (the quad is ovoidal). If only one of these possibilities occurs for all quads \(Q\), then \(H\) is called uniform. Finite uniform hyperplanes have almost completely been classified by \textit{A. Pasini} and \textit{S. Shpectorov} for rank 3 dual polar spaces, see [J. Comb. Theory, Ser. A 94, No. 2, 276-288 (2001; Zbl 0989.51008)], using also some work of \textit{E. Shult} [Lond. Math. Soc. Lect. Note Ser. 165, 229-239 (1992; Zbl 0791.51003)].NEWLINENEWLINENEWLINEIn the paper under review the author considers hyperplanes of (not necessarily finite) thick dual polar spaces (arising from polar spaces of rank 3) such that no quad is subquadrangular. He obtains a classification into three subcases, although the classification itself is not explicit. We summarize the result. The first case that can occur is that the hyperplane is the union of perps of points of an ovoid of a quad (and since ovoids of classical generalized quadrangles are not classified -- in fact, there are free constructions -- this class cannot be further determined explicitly). The second case that can occur is that there exists a point \(p\) in the hyperplane \(H\) such that all quads through \(p\) not contained in \(H\) are singular and of type \(p^\perp\), and all other quads are ovoidal. Only an infinite example is known to the author. The third case is the most interesting one and is related to the split Cayley hexagon.NEWLINENEWLINENEWLINEThe proofs are purely geometric and beautiful. They provide an independent motivation to look at these kinds of questions.