A construction of extended generalized quadrangles using the Veronesean
In this paper the author constructs new examples of \(c.C_2\)-geometries as follows. Consider a set \({\mathcal S}\) of \(q+3\) planes in \(\text{PG} (5,q)\) mutually intersecting in a point and spanning \(\text{PG} (5,q)\), with the property that, for all \(A\in{\mathcal S}\), the set of intersecting points of \(A\) with the other members of \({\mathcal S}\) forms a hyperoval in \(A\). Such a set of planes can be given by considering the image under the Veronesean mapping \(\zeta: \text{PG} (2,q) \leftarrow \text{PG} (5,q)\), \(q\) even of a dual hyperoval, together with the nucleus of the corresponding quadratic Veronesean variety \({\mathcal V}_2^4\). All hyperovals in the members of \({\mathcal S}\) induced by the other members are isomorphic to the regular hyperoval \({\mathcal O}\) (a conic union its nucleus). Now one embeds \(\text{PG} (5,q)\) as a hyperplane in \(\text{PG} (6,q)\). The points of the extended generalized quadrangle \({\mathcal Q}\) are the three-dimensional subspaces of \(\text{PG} (6,q)\) meeting \(\text{PG} (5,q)\) precisely in one of the elements of \({\mathcal S}\); the lines of \({\mathcal Q}\) are the lines of \(\text{PG} (6,q)\) meeting \(\text{PG} (5,q)\) precisely in a point of intersection of two members of \({\mathcal S}\); the blocks of \({\mathcal Q}\) are all points of \(\text{PG} (6,q)\) not in \(\text{PG} (5,q)\). The residue of a point is clearly isomorphic to the Tits quadrangle \(T^*({\mathcal O})\). The author discusses automorphisms and collinearity graphs of these examples. In particular, he shows that there is a flag-transitive automorphism group if and only if \(q=2\) or \(q=4\) (and in these cases the geometry \({\mathcal Q}\) is the double cover, respectively the half quotient, of some other known geometry).
scientific article; zbMATH DE number 1088578