On the Affine Homogeneity of Algebraic Hypersurfaces Arising from Gorenstein Algebras

Abstract

To every Gorenstein algebra $A$ of finite dimension greater than 1 over a field $\mathbb{F}$ of characteristic zero, and a projection $\pi$ on its maximal ideal $\mathfrak{m}$ with range equal to the annihilator $Ann(\mathfrak{m})$ 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 $\mathbb{F}=\mathbb{C}$ leads to interesting consequences in singularity theory. Also, for $\mathbb{F}=\mathbb{R}$ 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 $Aut(\mathfrak{m})$ of $\mathfrak{m}$ to act transitively on the set of hyperplanes in $\mathfrak{m}$ complementary to $Ann(\mathfrak{m})$. As a consequence of this result we obtain the affine homogeneity of $S_{\pi }$ under the assumption that the algebra $A$ is graded.

13 pages

References in corpus (2)

• Nilpotent algebras and affinely homogeneous surfaces
• Isolated hypersurface singularities and polynomial realizations of affine quadrics

Cited by in corpus (1)

• Nilpotent algebras and affinely homogeneous surfaces