2011/01/02 by Alexander Isaev, Isaev, Alexander · 2 citations
Mathematics · #13H10 #14R20 #32V40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1101.0452
openalex publication_date 2011/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To every Gorenstein algebra A of finite dimension greater than 1 over a\nfield Bbb F of characteristic zero, and a projection \π on its maximal\nideal mathfrak m with range equal to the annihilator\n hboxAnn( mathfrak m) of mathfrak m, one can associate a certain\nalgebraic hypersurface S\π\⊂ mathfrak m. Such hypersurfaces\npossess remarkable properties. They can be used, for instance, to help decide\nwhether two given Gorenstein algebras are isomorphic, which for Bbb F= Bbb\nC leads to interesting consequences in singularity theory. Also, for Bbb\nF= Bbb R such hypersurfaces naturally arise in CR-geometry. Applications of\nthese hypersurfaces to problems in algebra and geometry are particularly\nstriking when the hypersurfaces are affine homogeneous. In the present paper we\nestablish a criterion for the affine homogeneity of S\π. This condition\nrequires the automorphism group hboxAut( mathfrak m) of mathfrak m\nto act transitively on the set of hyperplanes in mathfrak m complementary\nto hboxAnn( mathfrak m). As a consequence of this result we obtain the\naffine homogeneity of S\π under the assumption that the algebra A is\ngraded.\n