Hyperbolic unitals in the Hall planes
This interesting paper is based on the following construction of projective planes by \textit{P. Quattrocchi} and \textit{L. A. Rosati} [Geom. Dedicata 44, No. 2, 233-240 (1992; Zbl 0773.51007)]: let \(P\) be a projective plane, \(F\) a family of lines of \(P\) and \(\alpha\) a permutation on the points of \(P\) such that the line \(pq\) is in \(F\) if and only if \(p^\alpha q^\alpha\) is in \(F\). A new projective plane \(P'\) is defined as follows: both point and line set are the respective sets of \(P\), a point \(p\) is incident with a line \(\ell\) if and only if \(p\) is incident with \(\ell\) in \(P\) and \(\ell \not \in F\), or \(p^\alpha\) is incident with \(\ell\) in \(P\) and \(\ell \in F\). The author now transforms unitals of \(P\) into unitals of \(P'\), rediscovering the unitals of Grüning in (the dual) Hall plane, and also settling some isomorphism questions concerning Grüning's unitals and Buekenhout's unitals.
scientific article; zbMATH DE number 827399