On the geometry of regular hyperbolic fibrations

Abstract

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. © 2006 Elsevier Ltd. All rights reserved.