Abstract
The finite Figueroa planes are non-Desarguesian projective planes of order q 3 for all prime powers q > 2. These planes were constructed algebraically in 1982 by Figueroa, and Hering and Schaeffer, and synthetically in 1986 by Grundhöfer. All Figueroa planes of finite square order are shown to possess a unitary polarity by de Resmini and Hamilton in 1998, and hence admit unitals. Using the result of O’Nan in 1971 on the non-existence of his configuration in a classical unital, and the intrinsic characterization by Taylor in 1974 of the notion of perpendicularity induced by a unitary polarity in the classical plane (introduced by Dembowski and Hughes in 1965), we show that these Figueroa polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unitary block design to be classical, and hence they are not classical.
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This work was partially supported by a grant from the Research Grant Council of the HKSAR, China (Project number: HKU7060/11P).
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Hui, M.W., Wong, P.P.W. Non-classical polar unitals in finite Figueroa planes. J. Geom. 103, 263–273 (2012). https://doi.org/10.1007/s00022-012-0121-7
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DOI: https://doi.org/10.1007/s00022-012-0121-7