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

✍ Scribed by Sara Cauchie; Frank De Clerck; Nicholas Hamilton

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 118 KB
• Volume: 23
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.2002.0593

✦ Synopsis

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 a β-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.