Hyperbolic Fibrations of PG(3, q)

A hyperbolic bration is set of q ?1 hyperbolic quadrics and two lines which together partition the points of PG(3; q). The classical example of a hyperbolic bration comes from a pencil of quadrics; however, several other families are known. In this paper we construct a new family of hyperbolic brations for odd prime powers q. As an application of hyperbolic brations, we note that they can be used to construct 2 q?1 (not necessarily inequivalent) spreads of PG(3; q) by choosing one ruling family from each of the hyperbolic quadrics in the bration. For our new bration we discuss some properties of the spreads obtained in the above manner.