Regular hyperbolic fibrations

Abstract

A hyperbolic fibration is a set of qˇ 1 hyperbolic quadrics and two lines which together partition the points of PGO3; qU. The classical example of a hyperbolic fibration comes from a pencil of quadrics; however, several other families are now known. In this paper we begin the development of a general framework to study hyperbolic fibrations for odd prime powers q. One byproduct of hyperbolic fibrations is the 2 qˇ1 (not necessarily inequivalent) spreads of PGO3; qU they spawn via the selection of one ruling family of lines for each of the hyperbolic quadrics. We show how the hyperbolic fibration context can be used to unify the study of these spreads, especially those associated with j-planes. The question of whether a spread spawned from such a fibration could contain any reguli other than the ones it inherits from the fibration plays a significant role in the determination of its automorphism group, as well as being an interesting geometric question in its own right. This information is then used to address the problem of sorting out projective equivalences among the spreads spawned from a given hyperbolic fibration. Plucker coordinates are an important tool in most of these investigations.