Some new classes of extended generalized quadrangles of order \((q+1,q-1)\)
The author constructs three new classes of extended generalized quadrangles of order \((q+1,q-1)\). An extended generalized quadrangle of order \((s,t)\) is a connected geometry consisting of points, lines and planes such that (1) point residues are generalized quadrangles of order \((s,t)\), (2) plane residues are isomorphic to the complete graph on \(s+2\) vertices, and (3) line residues are generalized digons. The paper starts with an overview of known examples of extended generalized quadrangles of order \((q+1,q-1)\). \textit{A. Blokhuis} and \textit{A. E. Brouwer} [J. Graph. Theory 13, No. 2, 229-244 (1989; Zbl 0722.05054)] constructed three examples for \(q=2\), while for \(q=3\), EGQ's have been found by \textit{F. Buekenhout} and \textit{X. Hubaut} [J. Algebra 45, 391-437 (1977; Zbl 0351.05021)] and \textit{D. V. Pasechnik} [J. Comb. Theory, Ser. A 72, No. 2, 232-242 (1995; Zbl 0845.51004)]. Recently, Del Fra and Pasini succeeded in constructing an infinite family of EGQ's of order \((q+1,q-1)\) for \(q\) odd. Using the projective space PG\((6,q)\) for \(q\) even and the Veronesean \(\xi : \text{PG}(2,q) \to \text{PG}(5,q)\), Yoshiara contributed another general construction of EGQ's, which is based on a set \(S\) of \(q+3\) planes \(\pi_i\) in PG\((5,q)\) satisfying (1) \(\pi_i \cap \pi_j\) is a point for any \(i \neq j\), (2) the set \(\{\pi_i \cap \pi_j\mid j\neq i\}\) is a hyperoval of \(\pi_i\), and (3) \(S\) generates PQ\((5,q)\). The author first defines new such sets \(S\) thus deriving new extended generalized quadrangles. Then a new construction in the spirit of Yoshiara is presented. He starts with a set of \(q+3\) planes in PG\((4,q)\) that satisfies conditions (1) and (2) above. (Clearly, condition (3) is satisfied trivially in PG\((4,q)\).) It turns out that Yoshiara's examples are \(q\)-fold covers of these new examples.
scientific article; zbMATH DE number 1266828