Quasigroups and Groups Arising from Cubic Surfaces

Abstract

We shall discuss Manin's structure theorems about cubic hypersurfaces. Under some conditions, a suitable factor of the set of non-singular points of a projective cubic hypersurface may be endowed with a structure of symmetric terentropic quasigroup in which any square is idempotent; it is therefore isotopic to an exponent 6 Commutative Moufang Loop. This generalizes the classical construction of an abelian group from a plane curve. But in the case of a surface there are still several open questions. Some properties of the terentropic quasigroups and of the cubic quasigroups are stated. Several concrete examples are presented. The multiplication groups of the cubic quasigroups are described.