Hyperbolic Fibrations ofPG(3,q)

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