Acta Mathematica Hungarica

, Volume 153, Issue 1, pp 249–264 | Cite as

Pencilled regular parallelisms

  • H. HavlicekEmail author
  • R. Riesinger


Over any field \({\mathbb{K}}\), there is a bijection between regular spreads of the projective space \({{\rm PG}(3,\mathbb{K})}\) and 0-secant lines of the Klein quadric in \({{\rm PG}(5,\mathbb{K})}\). Under this bijection, regular parallelisms of \({{\rm PG}(3,\mathbb{K})}\) 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.

Key words and phrases

pencilled regular parallelism hyperflock determining line set Clifford parallelism linear flock 

Mathematics Subject Classification

51A15 51M30 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und GeometrieTechnische UniversitätWienAustria
  2. 2.WienAustria

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