Isolated hypersurface singularities and polynomial realizations of affine quadrics

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.