ABSTRACT
In Chapter 2 on the basic theory of focal-spreads, we showed that additive focal-spreads are equivalent to additive partial t-spreads in vector spaces of dimension 2t, wherein lives, of course, translation planes of order qt. It is still an open question whether every focal-spread of type (t; k) arises as a k-cut of a t-spread of a translation plane. In the additive case, if the ‘companion’partial t-spread is not maximal over the prime …eld, then there is a semi…eld plane of order qt that extends the partial spread and the dual semi…eld plane then extends the additive focal-spread. There are maximal additive partial spreads in PG(3; q) (see, e.g., Jha and Johnson [90]), and although there is no semi…eld with spread in PG(3; q) that extends the additive partial spreads, it is not known whether there are spreads in larger dimensional projective spaces within which an additive partial spread in PG(3; q) may be extended to a semi…eld spread (the reader is directed to the open problem section for this partial spread). In this chapter, we construct a number of very interesting additive
partial t-spreads. What makes these partial spreads interesting involves the ‘subplane dimension question’: Given any translation plane of order ps and any a¢ ne subplane of order pz must z divide s? It is easy to show that any a¢ ne subplane of a (a¢ ne) translation plane is also a translation plane of the same characteristic. The reader probably recalls the existence of Fano subplanes (subplanes of order 2) in translation planes of odd characteristic, but these subplanes are subplanes of the ‘projective translation plane.’ There are a few situations where the subplane dimension question
can be answered in the a¢ rmative. For example, if is a Desarguesian plane coordinatized by a …nite …eld isomorphic to GF (pt); then any a¢ ne subplane 0 is also Desarguesian and may be coordinatized by a sub…eld isomorphic to GF (pk), so k does divide t in this case. More generally, if a …nite translation plane of order pt admits a collineation of order p that …xes an a¢ ne subplane 0 of order pk pointwise, then Foulser [52] has shown that k must, in fact, divide t. However, this is the extent of the knowledge regarding the subplane dimension question.
So, how does this idea impact what we are trying to say with regards to additive partial t-spreads and their companion additive focal-spreads of type (t; k)? We begin by showing how to construct a tremendous variety of additive partial t-spreads that admit extremely unusual a¢ ne subplanes, ones for which the subplane dimension question is answered in a resounding ‘no’! If any of these additive partial t-spreads can be extended they may also be extended to semi…eld t-spreads admitting the same a¢ ne subplanes. What this means is that either there are undiscovered semi…eld planes about which we basically know nothing or there are additive maximal partial spreads about which nothing is known. Of course, the problem is, so far we can’t be certain which alternative might be valid. It is apparently not known that additivity of partial spreads can
be arranged to be inherited. That is, given an additive partial spread that is not maximal, there is, of course, a partial spread containing the original partial spread, but we can also arrange it so that super partial spread is also additive.
