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Regulus-free Spreads of PG(3, q)

In: Designs and Finite Geometries

Author

Listed:
  • R. D. Baker

    (West Virginia State College, Department of Mathematics)

  • G. L. Ebert

    (University of Delaware, Department of Mathematical Sciences)

Abstract

An old conjecture of Brack and Bose is that every spread of Σ = PG(3, q) could be obtained by starting with a regular spread and reversing reguli. Although it was quickly realized that this conjecture is false, at least for q even, there still remains a gap in the spaces for which it is known that there are spreads which are regulus-free. In several papers Denniston, Bruen, and Bruen and Hirschfeld constructed spreads which were regulus-free, but none of these dealt with the case when p is a prime congruent to one modulo three. This paper closes that gap by showing that for any odd prime power p, spreads of PG(3, p) yielding nondesarguesian flag-transitive planes are regulus-free. The arguments are interesting in that they are based on elementary linear algebra and the arithmetic of finite fields.

Suggested Citation

  • R. D. Baker & G. L. Ebert, 1996. "Regulus-free Spreads of PG(3, q)," Springer Books, in: Dieter Jungnickel (ed.), Designs and Finite Geometries, pages 79-89, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-1395-3_5
    DOI: 10.1007/978-1-4613-1395-3_5
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