Conical, ruled and deficiency one translation planes
Vikram Jha, Norman Lloyd Johnson · Bulletin of the Belgian Mathematical Society - Simon Stevin · 1999
Flocks of deficiency one of quadratic cones and hyperbolic quadrics in P G(3, K), for K a finite field, correspond to translation planes admitting certain collineation groups that fix Baer subplanes pointwise.In this article, this theory is extended to the general situation where K is an arbitrary field.Flocks of quadratic cones and of hyperbolic quadrics in P G(3, q) correspond to spreads in P G(3, q) which are unions of q or q + 1 reguli respectively.A more general theory of the analogous conical and ruled spreads is developed for spreads in P G(3, K) and K an arbitary skewfield. Introduction.In this article, some generalizations of the connections with flocks of quadric sets in P G(3, q) are developed.For a complete history of the problems and theory of geometries related to flocks as well as certain fundamental definitions, the reader is referred to the survey article by Johnson and Payne [26].Let F be a flock of a quadratic cone in P G(3, q).It is now well known that there is a corresponding spread in P G(3, q) which is a union of q reguli sharing a common line.Furthermore, there is a corresponding generalized quadrangle of order (q 2 , q) obtained via an associated q -clan.When F is a flock of a hyperbolic quadric in P G(3, q), there is an associated spread in P G(3, q) which is a union of q + 1 reguli sharing two common lines.