Polarities and ovals in the Hughes plane

Abstract

SummaryIn 1946 Baer (Polarities infinite projective planes, Bull. Am. Math. Soc. 52, 77–93) showed that the absolute points of a polarity in a finite projective plane of odd non-square order always form an oval, that is, in a plane of order n there are exactly n+ 1 absolute points and no three are collinear. It is well known that the absolute points of polarities in planes of odd square order form ovals in some cases.If the oval is a subset of the set of absolute points, then the oval itself determines the polarity, and this makes it appear unlikely that the oval could be a proper subset. Among other results in the paper it is to be proved that in the regular Hughes plane there is a polarity which is determined by an oval which is a relatively small subset of the set of absolute points. Explicitly, if Ω is the Hughes plane of order q2 and A is the central subplane of order q, then every conic in Δ can be extended to an oval in Ω, and this oval determines a polarity in which there are ½(q3–q) additional absolute points.