Simple Elliptic Hypersurface Singularities: A New Look at the Equivalence Problem

Abstract

Let $V_1, V_2$ be hypersurface germs in $\mathbb C^m$, with $m \geq 2$, each having a quasi-homogenous isolated singularity at the origin. In our recent joint article with G. Fels, W. Kaup and N. Kruzhilin we reduced the biholomorphic equivalence problem for $V_1, V_2$ to verifying whether certain polynomials arising from the moduli algebras of $V_1, V_2$ are equivalent up to scale by means of a linear transformation. In the present we illustrate this result by the examples of simple elliptic singularities of $types \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$ and compare our method with that due to M. G. Eastwood who has also introduced certain polynomials that distinguish non-equivalent singularities within each of these three types.