A Construction of Extended Generalized Quadrangles Using the Veronesean

Given a finite generalized quadrangle Q, an extended generalized quadrangle is called an extension of Q if each point-residue is isomorphic to Q. There are several known examples of extended generalized quadrangles [4, §1.1], among which those admitting flag-transitive automorphism groups have been extensively studied in their close relations with group theory. Those flag-transitive geometries are usually discovered first as coset geometries of groups constructed by amalgamating certain triples of subgroups. On the other hand, not so many geometric constructions are known for extensions of generalized quadrangles even when they are flag-transitive. Recently, Del Fra et al. [4] constructed an extension of the Tits quadrangle T ∗ 2 (O) for each hyperoval O in PG(2, q) with even q as well as its covering if O is regular (or classical). Among them only two geometries for q = 2, 4 are flag-transitive [4, Theorem 7, 9]. In this note, for every regular hyperoval O of PG(2, q) with q even, we construct an extended generalized quadrangle G with point-residues isomorphic to the dual of the Tits quadrangle T ∗ 2 (O). The main idea is to construct a family of q + 3 planes in PG(5, q) with the property that any two of them intersect at a point (see Section 2.1) as the images of the duals of points of O under the map ζ sending PG(2, q) to the Veronese variety V2 in PG(5, q) together with the nucleus of V2 (see Section 3.4)†. The extended generalized quadrangle G is then constructed by suspending these planes and their intersections from various points outside from PG(5, q) (see Section 2.3). In particular, the dual of G is a subgeometry of the affine 1-, 2and 4-subspaces in the affine space AG(6, q). The extended generalized quadrangle G turns out to be flag-transitive iff q = 2 or 4 (see Sections 4.5 and 4.6). If q = 2, G is the double cover of the Cameron–Fisher extension C F−(4) of the 4 × 4 grid (see Section 4.6). If q = 4, G is the half-quotient of the simply connected geometry G(0) constructed in [10, §4.2, 3] as a coset geometry. For q > 4, the simple connectedness cover of G has recently been established by the author.