The embedding of \((0,\alpha)\)-geometries in \(\mathrm{PG}(n,q)\). II
For a positive integer \(\alpha >1,\) a finite (0,\(\alpha)\)-geometry is a connected incidence structure \(S=(P,B,I)\) with a symmetric incidence relation satisfying the fllowing axioms: (i) two distinct points are incident with at most one line, (ii) if a point x and a line L are not incident, then there are 0 or \(\alpha\) points which are collinear with x and incident with L, and (iii) each line is incident with at least two points, and each point is incident with at least two lines. By the connectedness of S one can prove that there exists a positive integer \(s>1\) such that every point is incident with \(s+1\) lines and, duality, that each line is incident with \(s+1\) points. In the paper under review the authors continue the study of (0,\(\alpha)\)- geometries previously initiated in [\textit{J. A. Thas} and \textit{F. De Clerck}, Ann. Discrete Math. 18, 229-240 (1983; Zbl 0503.51004)]. First, they determine all (0, \(\alpha)\)-geometries with \(q+1\) points on a line embeddable in PG(n,q), where \(n>3\) and \(q>2\) is assumed. Second, they prove some theorems about the embedding of (0,2)-geometries in PG(n,2). In particular, they determine all (0,2)-geometries in PG(4,2) and describe an unusual embedding of \(U_{2,3}(9)\) in PG(5,2).
scientific article; zbMATH DE number 3874085
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