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Acta Mathematica Hungarica

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

Pencilled regular parallelisms

  • H. HavlicekEmail author
  • R. Riesinger
Article

Abstract

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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References

  1. 1.
    W. Benz, Classical Geometries in Modern Contexts, Birkhäuser (Basel, 2007).Google Scholar
  2. 2.
    Betten A.: The packings of PG(3,3). Des. Codes Cryptogr., 79, 583–595 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. Betten and R. Löwen, Compactness of the automorphism group of a topological parallelism on real projective 3-space, Results Math., (2017), doi: 10.1007/s00025-017-0674-8.
  4. 4.
    Betten D., Riesinger R.: Topological parallelisms of the real projective 3-space. Results Math., 47, 226–241 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Betten D., Riesinger R.: Constructing topological parallelisms of \({\rm PG}(3,\mathbb{R})\) via rotation of generalized line pencils. Adv. Geom., 8, 11–32 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Betten D., Riesinger R.: Hyperflock determining line sets and totally regular parallelisms of \({\rm PG}(3,\mathbb{R})\). Monatsh. Math., 161, 43–58 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Betten D., Riesinger R.: Clifford parallelism: old and new definitions, and their use. J. Geom., 103, 31–73 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Betten D., Riesinger R.: Automorphisms of some topological regular parallelisms of \({{\rm PG}(3,\mathbb{R})}\). Results Math., 66, 291–326 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Blunck and A. Herzer, Kettengeometrien – Eine Einführung, Shaker Verlag (Aachen, 2005).Google Scholar
  10. 10.
    Blunck A., Pasotti S., Pianta S.: Generalized Clifford parallelisms. Quad. Sem. Mat. Brescia, 20, 1–13 (2007)zbMATHGoogle Scholar
  11. 11.
    Blunck A., Pasotti S., Pianta S.: Generalized Clifford parallelisms. Innov. Incidence Geom., 11, 197–212 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    T. E. Cecil, Lie Sphere Geometry, Springer (New York, 2008).Google Scholar
  13. 13.
    Cogliati A.: Variations on a theme: Clifford’s parallelism in elliptic space. Arch. Hist. Exact Sci., 69, 363–390 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Havlicek H.: Spheres of quadratic field extensions. Abh. Math. Sem. Univ. Hamburg, 64, 279–292 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Havlicek H.: A note on Clifford parallelisms in characteristic two. Publ. Math. Debrecen, 86, 119–134 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Havlicek H.: Clifford parallelisms and external planes to the Klein quadric. J. Geom., 107, 287–303 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    J. W. P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford University Press (Oxford, 1985).Google Scholar
  18. 18.
    J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Springer (London, 2016).Google Scholar
  19. 19.
    Johnson N. L.: Parallelisms of projective spaces. J. Geom., 76, 110–182 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    N. L. Johnson, Combinatorics of Spreads and Parallelisms, vol. 295 of Pure and Applied Mathematics (Boca Raton), CRC Press (Boca Raton, 2010).Google Scholar
  21. 21.
    H. Karzel and H.-J. Kroll, Geschichte der Geometrie seit Hilbert, Wissenschaftliche Buchgesellschaft (Darmstadt, 1988).Google Scholar
  22. 22.
    N. Knarr, Translation Planes, volume 1611 of Lecture Notes in Mathematics, Springer (Berlin, 1995).Google Scholar
  23. 23.
    Löwen R.: Regular parallelisms from generalized line stars in \({P_3\mathbb{R}}\): a direct proof. J. Geom., 107, 279–285 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    R. Löwen, A characterization of Clifford parallelism by automorphisms, Preprint, arXiv:1702.03328 (2017).
  25. 25.
    G. Pickert, Analytische Geometrie, Akademische Verlagsgesellschaft Geest & Portig (Leipzig, 1976).Google Scholar
  26. 26.
    J. G. Semple and G. T. Kneebone, Algebraic Projective Geometry, The Clarendon Press, Oxford University Press (New York, 1998).Google Scholar
  27. 27.
    Topalova S., Zhelezova S.: New regular parallelisms of PG(3,5). J. Combin. Des., 24, 473–482 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    H. Van Maldeghem, Generalized Polygons, Birkhäuser (Basel, 1998).Google Scholar

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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