On the geometry of regular hyperbolic fibrations

Hyperbolic fibrations of PG ( 3 , q ) were introduced by Baker, Dover, Ebert and Wantz in [R.D. Baker, J.M. Dover, G.L. Ebert, K.L. Wantz, Hyperbolic fibrations of PG ( 3 , q ) , European J. Combin. 20 (1999) 1–16]. Since then, many examples were found, all of which are regular and agree on a line. It is known, via algebraic methods, that a regular hyperbolic fibration of PG ( 3 , q ) that agrees on a line gives rise to a flock of a quadratic cone in PG ( 3 , q ) , and conversely. In this paper this correspondence will be explained geometrically in a unified way for all q . Moreover, it is shown that all hyperbolic fibrations are regular if q is even, and (for all q ) every hyperbolic fibration of PG ( 3 , q ) which agrees on a line is regular.