Flocks, ovoids and generalized quadrangles

Let (∞, π∞) be an incident point-plane pair of PG(3, q). A tetrad with respect to (∞, π∞) is a set {X,Y, Z,W} of points of PG(3, q) \ π∞ such that {∞, X, Y, Z,W} is a cap of PG(3, q), ∞ ∈ 〈X,Y, Z〉 and W 6∈ 〈X,Y, Z〉. A set Θ of ovoids of PG(3, q) is tetradic with respect to (∞, π∞) if each ovoid contains ∞, has tangent plane π∞ and is such that each tetrad with respect to (∞, π∞) is contained in a unique ovoid of Θ. From this definition we are able to prove that such a set Θ has a rich and constrained structure, related to flocks and Laguerre planes. In fact, we shall see that a tetradic set of ovoids always gives rise to a generalized quadrangle (GQ) of order (q, q2) satisfying Property (G) at a flag, and conversely, suggested by a construction of J. A. Thas from 1999. In the case where each element of the tetradic set Θ is an elliptic quadric, the existence of the set Θ is equivalent to the existence of a flock of a quadratic cone and the corresponding flock GQ. Using these connections Barwick, Brown and Penttila showed that a GQ satisfying Property (G) at a pair of points and whose associated ovoids are all elliptic quadric must be a dual flock GQ. In the more general ovoid case, when q is even, Brown showed that a tetradic set with the property that the set of dual ovoids arising from the tangent planes to ovoids of the set, is also a tetradic set, comprises elliptic quadrics. Hence a GQ satisfying Property (G) at a line must be the dual of flock GQ (as conjectured by J. A. Thas at Combinatorics ’98). We shall discuss all of the above results.