Full embeddings of (\(\alpha\),\(\beta\))-geometries in projective spaces. II. (Q1810640)

The author continues the classification of proper \((\alpha,\beta)\)-geometries fully embedded into PG\((n,q)\) [Part I [Eur. J. Comb. 23, No. 6, 635--646 (2002; Zbl 1026.51003)] . The present paper deals with the case \(\beta=q\), odd, and \(\alpha>1\). It turns out that only two cases occur, namely \(\alpha=q-1\) and \(\alpha=q-\sqrt q\). In the case \(\alpha=q-1\) there are two types of geometries, which show up whether or not a certain type of planes containing an antiflag exists. In the case \(\alpha=q-\sqrt q\) only non-degenerate geometries are considered. One type of examples has \(n=3,\) or \(4\). For the other two possibilities (they must have \(n=3\)) it is not known whether examples exist. The section ``Conclusion'' wraps up the results of this and the previous paper [Part I, loc. cit.] to give a nice overview of what is known so far. (One of the rare instances where such a section makes sense.) Some partial results for the case \(q\) even are given, too.