A Characterization of \((\alpha, s + 1)\)-Geometries, \(1<\alpha<s+1\), Satisfying the Axiom of Pasch

Abstract

In this paper, a characterization of two classes of \((q, q+1)\)-geometries, that are fully embedded in a projective space \(PG(n, q)\), is obtained. The first class is the one of the \((q,q+1)\)-geometry \(H^{n,m}_q\), having points the points of \(PG(n, q)\) that are not contained in an \(m\)-dimensional subspace \(\Pi[m]\) of \(PG(n, q)\), for \(0 \leq m \leq n-3\), and lines the lines of \(PG(n, q)\) skew to \(\Pi[m]\). The second class is the one of the \((q,q+1)\)-geometry \(SH^{n,m}_q\), having the same point set as \(H^{n,m}_q\), but with \(-1 \leq m \leq n-3\), and lines the lines skew to \(\Pi^{n,m}_q\) that are not contained in a certain partition of the point set of \(SH^{n,m}_q\). Our characterization uses the axiom of Pasch, which is also known as axiom of Veblen-Young. It is a generalization of the characterization for partial geometries satisfying the axiom of Pasch by J. A. Thas and F. De Clerck. A characterization for \(H^{n,m}_q\) was already proved by H. Cuypers. His result however does not include \(SH^{n,m}_q\).