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Pencilled regular parallelisms

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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.

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Authors and Affiliations

  1. Institut für Diskrete Mathematik und Geometrie, Technische Universität, Wiedner Hauptstraße 8–10/104, A-1040, Wien, Austria

    H. Havlicek

  2. Patrizigasse 7/14, A-1210, Wien, Austria

    R. Riesinger

Authors
  1. H. Havlicek
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  2. R. Riesinger
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Correspondence to H. Havlicek.

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In memoriam Walter Benz

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Havlicek, H., Riesinger, R. Pencilled regular parallelisms. Acta Math. Hungar. 153, 249–264 (2017). https://doi.org/10.1007/s10474-017-0742-2

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  • DOI: https://doi.org/10.1007/s10474-017-0742-2

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