Classification of \((1,s)\)-geometries fully embedded in PG(\(n,s\)), for \(s{\neq}\)2
A partial linear space of order \((s,t)\) is a connected incidence structure \({\mathcal S} = (P,{\mathcal L}, I)\) such that any two distinct point are incident with at most one line, each line is incident with exactly \(s+1\), \(s>1\), points and each point is incident with exactly \(t+1\), \(t>1\), lines. The incidence number \(i(x,L)\) of some antiflag \((x,L)\) is the number of points collinear with \(x\) and incident with \(L\). An \((\alpha,\beta)\)-geometry is a partial linear space \({\mathcal S}\) of some order \((s,t)\) such that \(i({\mathcal A}) =\{\alpha, \beta\}\), where \({\mathcal A}\) is the set of antiflags of \({\mathcal S}\). In this setting, a generalized quadrangle is a \((1,1)\)-geometry. An \((\alpha,\beta)\)-geometry \({\mathcal S}\) is said to be fully embedded into projective space PG\((n,q)\), if the sets of points and of lines of \({\mathcal S}\) are subsets of the respective sets of PG\((n,q)\), the incidence relation \(I\) is inherited from the one of PG\((n,q)\), and \(s=q\). The author classifies full embeddings of \((1,s)\)-geometries in PG\((n,s)\) for \(s\neq 2\). \textit{F. De Clerck} and \textit{J. A. Thas} have obtained such a classification for \((\alpha,\alpha)\)-geometries already in [Arch. Math. 30, 537-540 (1978; Zbl 0367.05019)], while for \((0,\alpha)\)-geometries with \(\alpha>1\) this was done by De Clerck, Thas and Debroey in two papers published in 1983 and 1984 [\textit{F. De Clerck} and \textit{J. A. Thas}, Ann. Discrete Math. 18, 229-240 (1983; Zbl 0503.51004) and \textit{J. A. Thas, I. Debroey} and \textit{F. De Clerck}, Discrete Math. 51, 283-292 (1984; Zbl 0548.51006)]. Embeddings of generalized quadrangles have been fully classified by Hirschfeld in 1991 [\textit{J. W. P. Hirschfeld} and \textit{J. A. Thas}, `General Galois geometries' (Oxford Mathematical Monographs. Clarendon Press, Oxford) (1991; Zbl 0789.51001)]. In two previous papers the author proved some results about full embeddings of \((\alpha,\beta)\)-geometries if \(\alpha>1\), and for \(\alpha=1\) and \(n=3\). The classification result which is proved in the paper under review says that every proper \((1,s)\)-geometry \(\mathcal S\) (i.e. \(s>2\)) fully embedded into some PG\((n,s)\) has its points on some cone \(\Pi[n-m-1]\)GQ for \(m\in\{3,4,5\}\) that are not contained in the vertex \(\Pi[n-m-1]\). The lines of \(\mathcal S\) are the lines of this cone and contain \(s+1\) points of \(\mathcal S\). The proof is for the most part elementary and self-contained though at the same time quite sophisticated.
scientific article; zbMATH DE number 1945004