ABSTRACT

Given any spread of PG(3; K), take the duality of the projective space. What is obtained is called a ‘dual spread,’ and, as we have mentioned before, if the spread is represented by a set of matrices, the dual spread is represented by the set of transposes to these matrices. When we deal with K as a skew…eld, everything becomes more complicated and we have been using the notation of PG(3; K)R and PG(3; K)L to represent the right and left projective spaces arising from a 4-dimensional vector space over K. For all of the known in…nite classes of parallelisms of PG(3; K), all spreads are either known to be dual spreads or suspected to be dual spreads. Now assume that we have a parallelism of PG(3; K)R, where the dual spreads (which, by the way, are in PG(3; K)L), also form a parallelism. The question is, are all of these dual parallelisms isomorphic to the original? Again, for all of the in…nite classes of parallelisms, the answer is uniformly no!