A Result on Spreads of the Generalized QuadrangleT2(O), withOan Oval Arising from a Flock, and Applications

If F is a flock of the quadratic cone K of PG (3, q), q even, then the corresponding generalized quadrangle S(F) of order (q2,q ) has subquadrangles T2(O), with O an oval, of order q. We prove in a geometrical way that any such T2(O) has spreads S consisting of an element y ∈ O and the q2lines not in the plane PG (2,q ) of O of q quadratic cones Kx,x ∈ O − { y }, of the space PG (3, q) containing T2(O), where Kxhas vertex x, is tangent to PG (2, q) at xy and has nucleus line xn, with n the nucleus of O. We also show how the oval O can be directly constructed from the flock F.