Non-classical polar unitals in finite Figueroa planes |
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Authors: | Man Wa Hui Philip P W Wong |
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Institution: | 1. Department of Mathematics, University of Hong Kong, Hong Kong, China
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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 Grundh?fer. 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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