Pencilled regular parallelisms

Abstract

Over any field K, there is a bijection between regular spreads of the projective space PG(3, K) and 0-secant lines of the Klein quadric in PG(5, K). Under this bijection, regular parallelisms of PG(3, K) correspond to hyperflock determining line sets (hfd line sets) with respect to the Klein quadric. An hfd line set is defined to be pencilled if it is composed of pencils of lines. We present a construction of pencilled hfd line sets, which is then shown to determine all such sets. Based on these results, we describe the corresponding regular parallelisms. These are also termed as being pencilled. Any Clifford parallelism is regular and pencilled. From this, we derive necessary and sufficient algebraic conditions for the existence of pencilled hfd line sets.Mathematics Subject Classification: 51A15 51M30

Clifford-like parallelisms

“…When dealing with several parallelisms at the same time we add some subscript or superscript to the symbols and S. The seminal book [18] covers the literature about parallelisms up to the year 2010. For the state of the art, various applications, connections with other areas of geometry and historical remarks, we refer also to [1], [3], [7], [13], [20], [26], [32] and the references therein.…”

“…Thereby, it has to be assumed that ℓ,1 and ℓ,2 share a single parallel class. The piecewise Clifford parallelisms with two pieces are blends of ℓ,1 and ℓ,2 , but none of these is Clifford-like with respect to any double space structure on P. The proof of the last statement is beyond the scope of this article, since the methods utilised in [10] are totally different from ours.…”

“…In Section 4, it will be shown, by giving illustrative examples, that each of (13) and (14) is satisfied by some Clifford-like parallelisms. The situation in (15) does not occur due to Corollary 3.7 below.…”

“…We first turn to equation (13), that is, Γ = Γ ℓ . In any projective double space P(H F ), ℓ , r , this equation has two trivial solutions, namely = ℓ and, by ( 4), = r .…”