The canonical modifications by weighted blow-ups

Abstract

In this paper we give a criterion for an isolated, hypersurface singularity of dimension $n\ (\geq 2)$ to have the canonical modification by means of a suitable weighted blow-up. Then we give a counter example to the following conjecture by Reid-Watanabe: For a 3-dimensional, isolated, non-canonical, log-canonical singularity $(X,x)$ of embedded dimension 4, there exists an embedding $(X,x)\subset ({\bC}^4, 0)$ and a weight ${\bw}=(w_0,w_1,\ldots ,w_n)$, such that the ${\bw}$-blow-up gives the canonical modification of $(X,x)$.