A complex-analytic proof of a criterion for isomorphism of Artinian Gorenstein algebras

Abstract: Let A be an Artinian Gorenstein algebra over a field k of characteristic zero with dimkA>1. To every such algebra and a linear projection π on its maximal ideal [InlineEquation not available: see fulltext.] with range equal to the socle Soc(A) of A, one can associate a certain algebraic hypersurface [InlineEquation not available: see fulltext.], which is the graph of a polynomial map P_π: kerπ→Soc(A)≃k. Recently, the following surprising criterion has been obtained: two Artinian Gorenstein algebras A and [InlineEquation not available: see fulltext.] are isomorphic if and only if any two hypersurfaces S_π and [InlineEquation not available: see fulltext.] arising from A and [InlineEquation not available: see fulltext.], respectively, are affinely equivalent. In the present paper, we focus on the cases [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.] and explain how in these situations the above criterion can be derived by complex-analytic methods. The complex-analytic proof for [InlineEquation not available: see fulltext.] has not previously appeared in print but is foundational for the general result. The purpose of our paper is to present this proof and compare it with that for [InlineEquation not available: see fulltext.], thus highlighting a curious connection between complex analysis and commutative algebra.