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 qq. Moreover, it is shown that all hyperbolic fibrations are regular if qq is even, and (for all qq) every hyperbolic fibration of PG(3,q) which agrees on a line is regular.