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Journal of Geometry

, Volume 107, Issue 2, pp 287–303 | Cite as

Clifford parallelisms and external planes to the Klein quadric

  • Hans HavlicekEmail author
Article

Abstract

For any three-dimensional projective space \({\mathbb{P}(V)}\), where V is a vector space over a field F of arbitrary characteristic, we establish a one-one correspondence between the Clifford parallelisms of \({\mathbb{P}(V)}\) and those planes of \({\mathbb{P} (V \wedge V)}\) that are external to the Klein quadric representing the lines of \({\mathbb{P}(V)}\). We also give two characterisations of a Clifford parallelism of \({\mathbb{P}(V)}\), both of which avoid the ambient space of the Klein quadric.

Keywords

Projective double space Clifford parallelism Klein quadric Plücker embedding geometric hyperplane condition of crossed pencils 

Mathematics Subject Classification

Primary 51A15 Secondary 51J15 

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Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und GeometrieTechnische UniversitätWienAustria

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