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.
Highlights
Let U be a unital of order n, i.e. a design with parameters 2−(n3 + 1, n + 1, 1)
If U is a subdesign of a projective plane π of order n2, i.e. a design with parameters 2−(n4 + n2 + 1, n2 + 1, 1), we call U an embedded unital
The design defined by a polar unital for which the ambient plane is the classical plane P G(2, n2) is called a classical unital
Summary
Let U be a unital of order n, i.e. a (unitary block) design with parameters 2−(n3 + 1, n + 1, 1) (see [3]). If U is a subdesign of a projective plane π of order n2, i.e. a design with parameters 2−(n4 + n2 + 1, n2 + 1, 1), we call U an embedded unital. If the points and lines of an embedded unital are respectively the absolute points and (restrictions of) non-absolute lines of a unitary polarity of the ambient plane, we call U a polar unital. A classical unital of order n can be embedded (as a polar unital) in the classical projective plane P G(2, n2). A natural question arises as to whether a classical unital can be embedded (as a polar unital) in a non-classical plane. At present this question seems to remain unanswered.
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