\(C_ 2.c\) geometries and generalized quadrangles of order \((s - 1, s + 1)\)
type: Item
instance of: scholarly article
title: \(C_ 2.c\) geometries and generalized quadrangles of order \((s - 1, s + 1)\)
zbMATH Open document ID: 0768.51005
DOI: 10.1007/BF03323102
author: Antonio Pasini
author: Alberto Del Fra
published in: Results in Mathematics
publication date: 1993-01-12T00:00:00Z
review text: In this well written and interesting paper the authors give generalizations of several well known geometries belonging to the diagram \(C_ 2\cdot c\) and study the relationship among various properties (axioms) that may or may not hold. One example of a main result is a characterization of the partial geometry \(S=T^*_ 2(K)\), where \(K\) is a maximal arc of degree \(d\) in \(PG(2,q)\). This partial geometry is realized in \(PG(3,q)\) as follows. Identify \(PG(2,q)\) with a plane \(H\) of \(PG(3,q)\). The points of \(S\) are the points of \(PG(3,q)\) not in \(H\) and lines of \(S\) are the lines of \(PG(3,q)\) not in \(H\) that meet the arc \(K\). Incidence is the natural one inherited from \(PG(3,q)\). A certain Buekenhout geometry \(pG\).\(L\) can be obtained from \(S\) by adding as planes those planes of \(PG(3,q)\) other than \(H\) that meet \(K\) in \(d\) points. This geometry \(pG\).\(L\) satisfies both the Sharp Cramming Property (given a pointline pair \((a,L)\), there exists a plane incident with both \(a\) and \(L)\) and the Weak Linearity Condition (if \(u,v,w\) are distinct planes with \(w\) containing two distinct points of \(u\cap v\), then \(u\cap v\subseteq w)\). These two properties are shown to characterize \(T^*_ 2(K)\) among those geometries having an appropriate Buekenhout diagram.
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51E24
zbMATH DE Number: 86366
zbMATH Keywords: generalized quadrangles
zbMATH Keywords: partial geometry
zbMATH Keywords: Buekenhout geometry
reviewed by: Stanley E. Payne
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 86366