Geometrical Constructions of Flock Generalized Quadrangles

With any flock F of the quadratic cone K of PG(3, q) there corresponds a generalized quadrangle S(F) of order (q2, q). For q odd Knarr gave a pure geometrical construction of S(F) starting from F. Recently, Thas found a geometrical construction of S(F) which works for any q. Here we show how, for q odd, one can derive Knarrʹs construction from Thasʹ one. To that end we describe an interesting representation of the point-plane flags of PG(3, q), which can be generalized to any dimension and which can be useful for other purposes. Applying this representation onto Thasʹ model of S(F), another interesting model of S(F) on a hyperbolic cone in PG(6, q) is obtained. In a final section we show how subquadrangles and ovoids of S(F) can be obtained via the description in PG(6, q).