SPREADS AND OVOIDS IN FINITE GENERALIZED QUADRANGLES

This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s, s2), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadrangles Q(4, 3e), e greater-than-or-equal-to 3, is constructed. Then, by the duality between Q(4, 3e) and the classical generalized quadrangle W(3e), we get line spreads of PG(3, 3e) and hence translation planes of order 3(2)e. These planes appear to be new. Note also that only a few classes of ovoids of Q(4, q) are known. Next we prove that each generalized quadrangle of order (q2, q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadrangles Q(5, q): If S is a generalized quadrangle of order (q, q2), q even, having a subquadrangle S' isomorphic to Q(4, q) and if in S' each ovoid consisting of all points collinear with a given point x of S/S' is an elliptic quadric, then S is isomorphic to Q(5, q).