Full Embeddings of (α, β) -Geometries in Projective Spaces

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