Full embeddings of (α, β)-geometries in projective spaces

The incidence structures known as (alpha, beta)-geometries are a generalization of partial geometries and semipartial geometries. For an (alpha, beta)-geometry fully embedded in PG(n, q), the restriction to a plane turns out to be important. Planes containing an antiflag of the (alpha, beta)-geometry can be divided into a-planes, P-planes and mixed planes. In this paper (1, 0)-geometries fully embedded in PG(3, q) are classified under the assumption that PG(3, q) contains at least one 1-plane and at least one beta-plane. Next we classify (alpha, beta)-geometries fully embedded in PG(n, q), for alpha > 1 and q odd, under the assumption that every plane of PG(n, q) that contains an antiflag of S is either an alpha-plane or a beta-plane. We also treat the case that there is a mixed plane and that beta = q + 1. In a forthcoming paper we will treat the case beta = q. The cases beta = q and beta = q + 1 are the only cases that can occur under the assumptions that q is odd, alpha > 1 and that there is at least one beta-plane.