Conical flocks, partial flocks, derivation, and generalized quadrangles

Abstract

L. Bader, G. Lunardon and J. A. Thas have shown that a flock ℱ0 of a quadratic cone in PG(3, q), q odd, determines a set ℱ={ℱ0,ℱ1,...,ℱq} of q+1 flocks. Each ℱj, 1≦j≦q, is said to be derived from ℱ0. We show that, by derivation, the flocks with q=3e arising from the Ganley planes yield an inequivalent flock for q≧27. Further, we prove that the Fisher flocks (q odd, q≧5) are the unique nonlinear flocks for which (q−1)/2 planes of the flock contain a common line. This result is used to show that each of the flocks derived from a Fisher flock is again a Fisher flock. Finally, we prove that any set of q−1 pairwise disjoint nonsingular conics of a cone can be extended to a flock. All these results have implications for the theory of translation planes.