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Translation planes constructed by multiple hyper-regulus replacement

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Abstract

New classes of mutually disjoint hyper-reguli of order q n, for n > 2, are determined, which are not André hyper-reguli if n > 3. If n is odd, each hyper-regulus permits at least two replacements and if q is odd and (n, q−1) = 1, there are (q−1)/2 mutually disjoint hyper-reguli each of which may be replaced at least two ways. Each of the possible 2(q-1)/2 translation planes constructed is not André or generalized André. There are also new constructions of mixed subgeometry partitions.

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Authors and Affiliations

  1. Mathematics Department, Glasgow Caledonian University, Cowcaddens Road, Glasgow, G4 OBA, Scotland, UK

    Vikram Jha

  2. Mathematics Department, University of Iowa, Iowa City, IA, 52242, USA

    Norman L. Johnson

Authors
  1. Vikram Jha
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  2. Norman L. Johnson
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Correspondence to Vikram Jha.

Additional information

Many of the ideas of this article stem from work done while the second author was visiting Caledonian University in May of 2004, supported under a grant from the London Mathematical Society. The authors thank the University and the LMS for support on this work. Furthermore, the authors gratefully acknowledge the help of the referee in the writing of this article.

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Jha, V., Johnson, N.L. Translation planes constructed by multiple hyper-regulus replacement. J. Geom. 94, 59–87 (2009). https://doi.org/10.1007/s00022-009-0008-4

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  • DOI: https://doi.org/10.1007/s00022-009-0008-4

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