Affine rigidity of Levi degenerate tube hypersurfaces

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On the Number of Affine Equivalence Classes of Spherical Tube Hypersurfaces

We consider Levi non-degenerate tube hypersurfaces in \CC^{n+1} that are $(k,n-k)$-spherical, i.e. locally CR-equivalent to the hyperquadric with Levi form of signature $(k,n-k)$, with $n\le 2k$. We show that the number of affine equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in the following cases: (i) $k=n-2$, $n\ge 7$;\linebreak (ii) $k=n-3$, $n\ge 7$; (iii) $k\le n-4$. For all other values of $k$ and $n$, except for $k=3$, $n=6$, the number of affine classes is known to be finite. The exceptional case $k=3$, $n=6$ has been recently resolved by Fels and Kaup who gave an example of a family of $(3,3)$-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 $j$-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 $A$ be an Artinian Gorenstein algebra over an infinite field $k$ with either $\hbox{char}(k)=0$ or $\hbox{char}(k)>\nu$, where $\nu$ is the socle degree of $A$. To every such algebra and a linear projection $\pi$ on its maximal ideal ${\mathfrak m}$ with range equal to the socle $\hbox {Soc}(A)$ of $A$, one can associate a certain algebraic hypersurface $S_{\pi}\subset{\mathfrak m}$, which is the graph of a polynomial map $P_{\pi}:\hbox{ker}\,\pi\to \hbox{Soc}(A)\simeq k$. Recently, the author and his collaborators have obtained the following surprising criterion: two Artinian Gorenstein algebras $A$, $\tilde A$ are isomorphic if and only if any two hypersurfaces $S_{\pi}$ and $S_{\tilde\pi}$ arising from $A$ and $\tilde A$, 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 $P_{\pi}$ and Macaulay inverse systems.Comment: To appear in the Journal of Commutative Algebra. arXiv admin note: substantial text overlap with arXiv:1201.610