ABSTRACT

In the previous chapter, we brie‡y discussed focal-spreads and the Beutelspacher’s construction. Although there are a number of examples of such partitions, there has been no theory developed about general partitions. Part of our treatment in this text is modeled from the article by Jha and Johnson [77], and the reader is directed to this article for additional details. Although it is certainly possible to consider focal-spreads over arbi-

trary …elds, all of the material that is presented in this text is for …nite focal-spreads. The reader is directed to the open problem chapter 40 for more information. In this chapter, we begin to build a theory of partitions based on

focal-spreads and their obvious connections to translation planes. For general focal-spreads, however, nothing is known. For example, it is not known whether collineations that …x components pointwise or homologies are elations or whether an involution that does not …x a component pointwise becomes a Baer collineation or a kernel involution. In the ‘Handbook of Finite Translation Planes’[138], it is mentioned that there are not very many known partitions of vector spaces, in the sense that the partitions are not re…nements of Sperner t-spreads. Furthermore, the existence of such partitions has been established by Beutelspacher [14], who proves if the dimension of the vector space is n and it is required to …nd a partition, where the dimensions of the subspaces are ft1; t2; ::; tkg, where ti < ti+1; then if gcdft1; t2; ::; tkg = d and n > 2t1([tk=(dk)]+t2+:::+tk), a partition may be constructed with various subspaces of dimension ti. However, these partitions are basically constructed using focal-spreads and general spreads. We give some of the constructions later in the section on ‘towers of focal-spreads.’ The main point of this chapter is to use of ideas and theory of

…nite translation planes in order to develop a certain theory of focalspreads and, since these seem to be important building blocks of general spreads, such work might prove useful to the general theory of partitions.