Switching Sets Regolari E Cubiche Rigate DiP G(5,q) Piani Di Tralsazione Ad Essi AssociatiG(5,q) Piani Di Tralsazione Ad Essi Associati

Abstract

In a recent paper, the authors studied some algebraic hypersurfaces of the third order in the projective spacePG(5,q) and they called them ruled cubics, since they possess three systems of planes. Any two of these constitute a regular switching set and furthermore, if Σ is a given regular spread ofPG(5,q), one of the three systems is contained in Σ.

The subject of this note is to prove, conversely, that every regular switching set (Φ, Φ′) with Φ ⊂ Σ is a ruled cubic and to construct, for a generic choice of the projective reference system inP G(5,q), the quasifield which coordinatizes the translation plane Π associated with the spread (Σ − Φ) ∪ Φ′.

The planes Π, of orderq ³, are a generalization of the finite Hall planes.