ABSTRACT
First, we o¤er a few words about why in a book on general spreads and parallelisms there would be included general theory of Baer groups. Of course, the previous chapter shows that partial Desarguesian parallelisms of de…ciency one in PG(3; q) are directly connected to translation planes admitting SL(2; q) generated by Baer collineations. Also, it is well known that given a ‡ock of a quadratic cone in PG(2; K), forK a …eld, there is an associated translation plane with spread in PG(3; K), such that the spread is a union of reguli that mutually share exactly one component `. The translation plane admits an elation group E with axis `, which …xes each regulus partial spread and acts transitively on the components not equal to `. Since such a translation plane is then derivable by any of the associated regulus nets, we see that any derived translation plane admits a ‘Baer group’that …xes the in…nite points of the derived net, …xes a Baer subplane, and …xes each original regulus partial spread, whose component orbits give the components of each regulus partial spread other than `. Conversely, if there is a translation plane with spread in PG(3; Z), for Z a skew…eld, such that there is a Baer group whose orbits union the …xed point subplane (the ‘Baer axis’) de…ne reguli, then there is a partial ‡ock of a quadratic cone with one conic missing; a partial ‡ock of a quadratic cone of de-…ciency one. This theory also works for ‡ocks of hyperbolic quadrics, where now there is a translation plane with spread in PG(3; Z), for Z a skew…eld, admitting a Baer group such that the net containing the Baer subplane in question (the ‘Baer axis’) admits a second Baer subplane left invariant by the Baer group (the ‘Baer coaxis’), and whose component orbits union the Baer axis and Baer coaxis are reguli, and again, in this case, we obtain a partial ‡ock of a hyperbolic quadric of de…ciency one. It is an extremely important problem to determine whether the net
de…ned by the Baer group in either the conical ‡ock type situation or the hyperbolic ‡ock type situation is a regulus net, for then upon derivation of the regulus net, we obtain a ‡ock. Hence, the question is whether partial ‡ocks of quadratic cones or of hyperbolic quadrics
of de…ciency one may be extended to ‡ocks. In the in…nite case, the existence of Baer groups of this type, implies K to be a …eld. In the …nite case, there are de…ciency one partial ‡ocks of hyperbolic quadrics, but the extension theory of Payne and Thas [173] shows that every partial ‡ock of a quadratic cone of de…ciency one in PG(3; q) may be extended uniquely to a ‡ock. Moreover, the recently emerging general theory of ‡ocks over arbitrary cones provides new translation planes admitting Baer groups, where now the spread is not in PG(3; q). So, the problem is to determine the structure of the components that do not intersect the corresponding Baer subplane. Much of the following material is taken more or less from our previ-
ous texts, and we include this here for the bene…t of the reader. Also, since whenever a Baer group exists, there is always the question of whether the corresponding Baer net de…ned by the components of the Baer subplane is derivable. We have previously discussed regular direct product nets and noted that they are, in fact, derivable nets. Therefore, a natural starting place for the study of Baer groups would be to study Baer subplanes in nets. However, now the subplane, which is Baer in the net might not be Baer in the a¢ ne plane containing the Baer net and even if the Baer net is derivable, in the general case there is no assurance that the a¢ ne plane is derivable. Indeed, in the in…nite case, if the kernel is a skew…eld that is not a …eld, there are problems with left and right vector spaces. We wish to include enough material on Baer groups and ‡ocks of
quadric sets so as to understand the nature of the problem both in the in…nite and …nite cases and also to give the theorem of Johnson, Payne-Thas mentioned above. This theorem can be generalized to more general settings such as ‡ocks of -cones, the theory of which we sketch in this text. Furthermore, we maintain that the interchange between …nite and in…nite point-line geometries provides considerable insights into both areas and whenever possible, we develop theory ostensible about …nite geometry somehow related to …nite …elds in a completely general manner in the in…nite case and somehow related to skew…elds. So we begin with the de…nition of a Baer subplane in the general
case. Of course, in a …nite projective plane of order n, a Baer subplane is simply a subplane of order
p n, thus requiring n to be a square.
Baer subplanes can also be characterized in the …nite case either as subplanes such that each point of the superplane is incident with a line of the subplane or that each line of the subplane is incident with a point of the subplane. The wonderful thing is, in the in…nite case, one of these properties does not imply the other. So, one property gives
what we call a ‘point-Baer’subplane and the other gives a ‘line-Baer’ subplane. If we have both properties, we obtain a ‘Baer subplane.’
