On the geometry of regular hyperbolic fibrations

Hyperbolic fibrations of PG(3, q) were introduced by Baker, Dover, Ebert and Wantz in [1]. 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.