Spreads of T₂(o), α-flocks and ovals

Abstract

We establish a representation of a spread of the generalized quadrangle T 2(script O sign), script O sign an oval of PG(2, q), q even, in terms of a certain family of q ovals of PG(2, q) and investigate the properties of this representation. Using this representation we show that to every flock of a translation oval cone in PG(3, q) (α-flock), q even, there corresponds a spread of T 2(script O sign) for an oval script O sign determined by the α-flock. This gives constructions of new spreads of T 2(script O sign), for certain ovals script O sign, and in some cases solves the existence problem for spreads. It also provides a geometrical characterization of the ovals associated with a flock of a quadratic cone.