Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 50, No. 2, pp. 483-493 (2009)

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Embeddings of projective Klingenberg planes in the projective space PG(5,$\K$)

Dirk Keppens and Hendrik Van Maldeghem

Department of Industrial Engineering, KAHO Sint--Lieven, Gebr. Desmetstraat 1, B--9000 Gent, Belgium, e-mail:; Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, B--9000 Gent, Belgium, e-mail:

Abstract: In this paper embeddings of projective Klingenberg planes in a 5-dimensional projective space are classified. It is proved that if a PK-plane is fully embedded in $\PG(5,\K$), for some skewfield $\K$, then it is either isomorphic to the Desarguesian projective Klingenberg plane (projective Hjelmslev plane for bijective $\sigma$) $\PH(2,\D(\K,\sigma))$ over a ring of ordinary or twisted dual numbers or it is a subgeometry of an ordinary projective plane. As a consequence we have in the finite case that, if a projective Klingenberg plane of order $(qt,t)$ is embedded in PG(5,$q$), then it is a projective Hjelmslev plane $\PH(2,\D(q,\sigma))$ over a ring of ordinary or twisted dual numbers over the Galois field $\GF(q)$. The embeddings related to the twisted case are new.

Keywords: projective Klingenberg plane, projective Hjelmslev plane, embedding, dual numbers

Classification (MSC2000): 51C05, 51A45

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Electronic version published on: 28 Aug 2009. This page was last modified: 28 Jan 2013.

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