On the affine homogeneity of algebraic hypersurfaces arising from gorenstein algebras

To every Gorenstein algebra A of finite vector space dimension greater than 1 over a field F of characteristic zero, and a linear projection π on its maximal ideal m with range equal to the annihilator Ann(m) of m, one can associate a certain algebraic hypersurface Sπ ⊂ m. Such hypersurfaces possess remarkable properties. They can be used, for instance, to help decide whether two given Gorenstein algebras are isomorphic, which for F = C leads to interesting consequences in singularity theory. Also, for F = R such hypersurfaces naturally arise in CRgeometry. Applications of these hypersurfaces to problems in algebra and geometry are particularly striking when the hypersurfaces are affine homogeneous. In the present paper we establish a criterion for the affine homogeneity of S_π. This criterion requires the automorphism group Aut(m) of m to act transitively on the set of hyperplanes in m complementary to Ann(m). As a consequence of this result we obtain the affine homogeneity of S_π; under the assumption that the algebra A is graded.