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.
This work was partially supported by NSA grant MDA 904–95-H-1013
This work was partially supported by NSA grant MDA 904–94-H-2033
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Dedicated to Hanfried Lenz on the occasion of his 80th birthday
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© 1996 Kluwer Academic Publishers, Boston
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Baker, R.D., Ebert, G.L. (1996). Regulus-free Spreads of PG(3, q). In: Jungnickel, D. (eds) Designs and Finite Geometries. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1395-3_5
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DOI: https://doi.org/10.1007/978-1-4613-1395-3_5
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