\(C_ 2.c\) geometries and generalized quadrangles of order \((s - 1, s + 1)\)
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.
scientific article; zbMATH DE number 86366
scientific article