Hyperbolic unitals in the Hall planes
type: Item
instance of: scholarly article
title: Hyperbolic unitals in the Hall planes
zbMATH Open document ID: 0843.51015
DOI: 10.1007/BF01222862
author: Gloria Rinaldi
published in: Journal of Geometry
publication date: 1996-08-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 827399
zbMATH Keywords: Buekenhout unitals
zbMATH Keywords: Grüning unitals
zbMATH Keywords: Hall planes
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01222862
OpenAlex ID: W2088097808
description: scientific article; zbMATH DE number 827399