Examples of Application of Nil-polynomials to the Biholomorphic Equivalence Problem for Isolated Hypersurface Singularities
by Alexander Isaev  

Vol. 8 No. 2 (2013) P.193~P.217

ABSTRACT

Let $V_1$, $V_2$ be hypersurface germs in $\mathbb{C}^m$ with $m\ge 2$, each having a quasi-homogeneous isolated singularity at the origin. In our recent article {\rm \cite{FIKK}} we reduced the biholomorphic equivalence problem for $V_1$, $V_2$ to verifying whether certain polynomials, called nil-polynomials, that arise from the moduli algebras of $V_1$, $V_2$ are equivalent up to scale by means of a linear transformation. In this paper we illustrate the above result by the examples of simple elliptic singularities of types $\tilde E_6$, $\tilde E_7$, $\tilde E_8$. The examples of singularities of types $\tilde E_6$, $\tilde E_7$ motivate a conjecture that states that just the highest-order terms of the corresponding nil-polynomials completely solve the equivalence problem in the homogeneous case. This conjecture was first proposed in our paper [EI] where it was established for plane curve germs defined by binary quintics and binary sextics. In the present paper we provide further evidence supporting the conjecture for binary forms of an arbitrary degree.

KEYWORDS
Isolated hypersurface singularities, equivalence problem, invariant theory

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 32S25, 13H10

MILESTONES

Received: 2012-07-20
Revised :
Accepted: 2013-09-12

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