On the Affine Homogeneity of Algebraic Hypersurfaces Arising from Gorenstein Algebras

TL;DR

This paper establishes a criterion for when algebraic hypersurfaces derived from Gorenstein algebras are affine homogeneous, linking this property to the action of automorphisms on certain hyperplanes, with implications in algebraic geometry and singularity theory.

Contribution

The paper provides a new criterion for the affine homogeneity of hypersurfaces associated with Gorenstein algebras, especially for graded algebras, connecting automorphism actions to geometric properties.

Findings

Criterion for affine homogeneity based on automorphism group action

Hypersurfaces are affine homogeneous if the algebra is graded

Automorphism group acts transitively on hyperplanes complementary to the annihilator

Abstract

To every Gorenstein algebra $A$ of finite dimension greater than 1 over a field ${\Bbb F}$ of characteristic zero, and a projection $\pi$ on its maximal ideal ${\mathfrak m}$ with range equal to the annihilator $\hbox{Ann}({\mathfrak m})$ of ${\mathfrak m}$, one can associate a certain algebraic hypersurface $S_{\pi}\subset{\mathfrak m}$. Such hypersurfaces possess remarkable properties. They can be used, for instance, to help decide whether two given Gorenstein algebras are isomorphic, which for ${\Bbb F}={\Bbb C}$ leads to interesting consequences in singularity theory. Also, for ${\Bbb F}={\Bbb R}$ 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 $S_{\pi}$. This condition requires the automorphism group $\hbox{Aut}({\mathfrak m})$ of ${\mathfrak m}$ to act transitively on the set of hyperplanes in ${\mathfrak m}$ complementary to $\hbox{Ann}({\mathfrak m})$. As a consequence of this result we obtain the affine homogeneity of $S_{\pi}$ under the assumption that the algebra $A$ is graded.