A geometric construction of generalized quadrangles from polar spaces of rank three
This paper is devoted to the construction of generalized quadrangles from polar spaces of rank three. A generalized quadrangle is a thick polar space of rank two, that is an incidence structure such that: if a point \(x\) and a line \(L\) are not incident, then either all points of \(L\) are collinear with \(x\), or only one point of \(L\) is collinear with \(x\), furthermore, each line is incident with at least three points and each point is incident with at least three lines, and there is no point collinear with all the other points. The main result is a direct geometric construction of the generalized quadrangle from a polar space of rank three, that is a polar space whose maximal singular subspaces are projective planes. The results are interesting and, as the author points out, some generalized quadrangles, not associated with flocks, are obtained.
scientific article; zbMATH DE number 62174