## Asian Journal of Mathematics

- Asian J. Math.
- Volume 15, Number 4 (2011), 631-640.

### On the Affine Homogeneity of Algebraic Hypersurfaces Arising from Goernstein Algebras

#### Abstract

To every Gorenstein algebra $A$ of finite vector space dimension greater than 1 over a field $\mathbb{F}$ of characteristic zero, and a linear projection $\pi$ on its maximal ideal $\mathfrak{m}$ with range equal to the annihilator $\operatorname{Ann}(\mathfrak{m})$ 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 criterion requires the automorphism group $\operatorname{Aut}(\mathfrak{m})$ of $\mathfrak{m}$ to act transitively on the set of hyperplanes in m complementary to $\operatorname{Ann}(\mathfrak{m})$. As a consequence of this result we obtain the affine homogeneity of $S_{\pi}$ under the assumption that the algebra $A$ is graded.

#### Article information

**Source**

Asian J. Math., Volume 15, Number 4 (2011), 631-640.

**Dates**

First available in Project Euclid: 12 March 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ajm/1331583351

**Mathematical Reviews number (MathSciNet)**

MR2853652

**Zentralblatt MATH identifier**

1273.14128

**Subjects**

Primary: 14R20: Group actions on affine varieties [See also 13A50, 14L30] 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 32V40: Real submanifolds in complex manifolds

**Keywords**

Gorenstein algebras affine homogeneity

#### Citation

Isaev, A. V. On the Affine Homogeneity of Algebraic Hypersurfaces Arising from Goernstein Algebras. Asian J. Math. 15 (2011), no. 4, 631--640. https://projecteuclid.org/euclid.ajm/1331583351