Subregular Spreads of Pg(2n + 1; Q)

In this paper, we develop some of the theory of spreads of projective spaces with an eye towards generalizing the results of Bruck 2]. In particular , we wish to generalize the notion of a subregular spread to the higher dimensional case. Most of the theory here was anticipated by Bruck in 3], 4], and 5]; however, he never provided a detailed formulation. We ll this gap here by developing the connections between a regular spread of (2n + 1)-dimensional projective space and an n-dimensional circle geometry , which is the appropriate generalization of the Miquelian inversive plane. After developing this theory, we provide a fairly general method for constructing subregular spreads of PG(5; q). Finally, we explore a special case of this construction, which yields several examples of three-dimensional subregular translation planes which are not Andr e planes.