Spreads of T2(o), α-flocks and Ovals

We establish a representation of a spread of the generalized quadrangle T2(0), o 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 T2(o) for an oval o determined by the α-flock. This gives constructions of new spreads of T2(o), for certain ovals o, 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.

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