Full embeddings of (α,β)-geometries in projective spaces—Part II Original Research Article

The incidence structures known as (α,β)-geometries are a generalization of partial geometries and semipartial geometries. In a previous paper, a classification of (α,β)-geometries fully embedded in PG(n,q), q odd and α>1, assuming that every plane of PG(n,q) containing an antiflag of S is either an α-plane or a β-plane, is given. The case that there is a so-called mixed plane and that β=q+1, is also treated there. In this paper we will treat the case β=q. This completes the classification of all proper (α,β)-geometries fully embedded in PG(n,q), q odd and α>1, such that PG(n,q) contains at least one α- or one β-plane. For q even, some partial results are obtained.