Asian Journal of Mathematics

On the Affine Homogeneity of Algebraic Hypersurfaces Arising from Goernstein Algebras

A. V. Isaev

Full-text: Open access

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


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