The Cyclic Parallelisms of PG(3,5)

AbstractWe determine, by a computer search, all the cyclic parallelisms of PG(3,5). There are 45 of them, up to projective equivalence. In particular, there are two cyclic regular parallelisms of PG(3,5). Previously, no example was known of a regular parallelism in PG(3,q), forqodd

Regular parallelisms on PG(3,R) from generalized line stars: The oriented case

Using the Klein correspondence, regular parallelisms of PG(3,R) have been described by Betten and Riesinger in terms of a dual object, called a hyperflock determining (hfd) line set. In the special case where this set has a span of dimension 3, a second dualization leads to a more convenient object, called a generalized star of lines. Both constructions have later been simplified by the author. Here we refine our simplified approach in order to obtain similar results for regular parallelisms of oriented lines. As a consequence, we can demonstrate that for oriented parallelisms, as we call them, there are distinctly more possibilities than in the non-oriented case. The proofs require a thorough analysis of orientation in projective spaces (as manifolds and as lattices) and in projective planes and, in particular, in translation planes. This is used in order to handle continuous families of oriented regular spreads in terms of the Klein model of PG(3,R). This turns out to be quite subtle. Even the definition of suitable classes of dual objects modeling oriented parallelisms is not so obvious

Characterising Clifford parallelisms among Clifford-like parallelisms

We recall the notions of Clifford and Clifford-like parallelisms in a $3$-dimensional projective double space. In a previous paper the authors proved that the linear part of the full automorphism group of a Clifford parallelism is the same for all Clifford-like parallelisms which can be associated to it. In this paper, instead, we study the action of such group on parallel classes thus achieving our main results on characterisation of the Clifford parallelisms among Clifford-like ones.Comment: 2 Figure

Parallelism

EnProblems involving the idea of parallelism occur in finite geometry and in graph theory. This article addresses the question of constructing parallelisms with some degree of "symmetry". In particular, can we say anything on parallelisms admitting an automorphism group acting doubly transitively on "parallel classes"