Hypersurface exceptional singularities

TL;DR

This paper investigates hypersurface exceptional singularities in complex space, establishing a finiteness result for weighted homogeneous cases and classifying all Brieskorn type exceptional singularities in dimension three.

Contribution

It proves the finiteness of weights for weighted homogeneous exceptional singularities and classifies all Brieskorn type exceptional singularities in three dimensions.

Findings

Number of weights for weighted homogeneous exceptional singularities is finite

Complete classification of Brieskorn type exceptional singularities in dimension 3

Establishment of equivalence between canonical and weighted homogeneous exceptional singularities

Abstract

This paper studies hypersurface exceptional singularities in $\mathbb C^n$ defined by non-degenerate function. For each canonical hypersurface singularity, there exists a weighted homogeneous singularity such that the former is exceptional if and only if the latter is exceptional. So we study the weighted homogeneous case and prove that the number of weights of weighted homogeneous exceptional singularities are finite. Then we determine all exceptional singularities of the Brieskorn type of dimension 3.