On the Affine Homogeneity of Algebraic Hypersurfaces Arising from Gorenstein Algebras

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To every Gorenstein algebra A of finite dimension greater than 1 over a field F of characteristic zero, and a 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 CR-geometry. 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 Sπ. This condition 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 Sπ under the assumption that the algebra A is graded.