Isolated hypersurface singularities and polynomial realizations of affine quadrics

TL;DR

This paper reduces the biholomorphic equivalence problem for hypersurface germs with isolated singularities to a linear equivalence problem for associated quadratic and cubic forms, simplifying the classification process.

Contribution

It establishes a reduction from the biholomorphic equivalence problem of hypersurface germs to a linear equivalence problem for specific polynomial pairs derived from moduli algebras.

Findings

Biholomorphic equivalence is characterized by linear equivalence of polynomial pairs.

The associated polynomials are determined by their quadratic and cubic terms.

The approach simplifies classification of hypersurface singularities.

Abstract

Let $V$, $\tilde V$ be hypersurface germs in $\CC^m$, each having a quasi-homogeneous isolated singularity at the origin. We show that the biholomorphic equivalence problem for $V$, $\tilde V$ reduces to the linear equivalence problem for certain polynomials $P$, $\tilde P$ arising from the moduli algebras of $V$, $\tilde V$. The polynomials $P$, $\tilde P$ are completely determined by their quadratic and cubic terms, hence the biholomorphic equivalence problem for $V$, $\tilde V$ in fact reduces to the linear equivalence problem for pairs of quadratic and cubic forms.