Geometrical Constructions of Flock Generalized Quadrangles

AbstractWith 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 Knarr's construction from Thas' 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 Thas' 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)

Geometrical Constructions of Flock Generalized Quadrangles

AbstractWith 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 Knarr's construction from Thas' 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 Thas' 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)

Generalized quadrangles of orrder (s, s2), I

AbstractIn this paper generalized quadrangles of order (s, s2), s > 1, satisfying property (G) at a line, at a pair of points, or at a flag, are studied. Property (G) was introduced by S. E. Payne (Geom. Dedicata32 (1989), 93–118) and is weaker than 3-regularity (see S. E. Payne and J. A. Thas, “Finite Generalized Quadrangles,” Pitman, London, 1984). It was shown by Payne that each generalized quadrangle of order (s2, s), s > 1, arising from a flock of a quadratic cone, has property (G) at its point (∞). In particular translation generalized quadrangles satisfying property (G) are considered here. As an application it is proved that the Roman generalized quadrangles of Payne contain at least s3 + s2 classical subquadrangles Q(4, s). Also, as a by-product, several classes of ovoids of Q(4, s), s odd, are obtained; one of these classes is new. The goal of Part II is the classification of all translation generalized quadrangles satisfying property (G) at some flag ((∞), L)

k-arcs and partial flocks

AbstractUsing the relationship between partial flocks of the quadratic cone K in PG(3, q), q even, and arcs in the plane PG(2, q), new results on partial flocks and short proofs for known theorems on translation generalized quadrangles of order (q2, q) and on ovoids in PG(3, q) are obtained. It is shown that large partial flocks of K containing approximately q conics, q even, are always extendable to a flock, which improves a result by Payne and Thas. Then new and short proofs are given for a theorem of Johnson on translation generalized quadrangles and a theorem of Glynn on ovoids

A Family of Ovals with Few Collineations

A recently discovered [1] family of ovals in PG(2, q), q = 2e, e odd, is shown to have a cyclic collineation group of order 2e

On Ferri's characterization of the finite quadric Veronesean V24

AbstractWe generalize and complete Ferri's characterization of the finite quadric Veronesean V24 by showing that Ferri's assumptions also characterize the quadric Veroneseans in spaces of even characteristic