2,004 research outputs found
On the Number of Affine Equivalence Classes of Spherical Tube Hypersurfaces
We consider Levi non-degenerate tube hypersurfaces in \CC^{n+1} that are
-spherical, i.e. locally CR-equivalent to the hyperquadric with Levi
form of signature , with . We show that the number of affine
equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in
the following cases: (i) , ;\linebreak (ii) , ;
(iii) . For all other values of and , except for , ,
the number of affine classes is known to be finite. The exceptional case ,
has been recently resolved by Fels and Kaup who gave an example of a
family of -spherical tube hypersurfaces that contains uncountably many
pairwise affinely non-equivalent elements. In this paper we deal with the
Fels-Kaup example by different methods. We give a direct proof of the
sphericity of the hypersurfaces in the Fels-Kaup family, and use the
-invariant to show that this family indeed contains an uncountable subfamily
of pairwise affinely non-equivalent hypersurfaces
A Criterion for Isomorphism of Artinian Gorenstein Algebras
Let be an Artinian Gorenstein algebra over an infinite field with
either or , where is the socle
degree of . To every such algebra and a linear projection on its
maximal ideal with range equal to the socle of
, one can associate a certain algebraic hypersurface
, which is the graph of a polynomial map
. Recently, the author and
his collaborators have obtained the following surprising criterion: two
Artinian Gorenstein algebras , are isomorphic if and only if any
two hypersurfaces and arising from and , respectively, are affinely equivalent. The proof is indirect and relies on
a geometric argument. In the present paper we give a short algebraic proof of
this statement. We also discuss a connection, established elsewhere, between
the polynomials and Macaulay inverse systems.Comment: To appear in the Journal of Commutative Algebra. arXiv admin note:
substantial text overlap with arXiv:1201.610
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