Spreads of T-2(O), alpha-flock and ovals
We establish a representation of a spread of the generalized quadrangle T-2(O), 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) (alpha-flock), q even, there corresponds a spread of T-2(O) for an oval O determined by the alpha-flock. This gives constructions of new spreads of T-2(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.
spread flock oval generalized quadrangle