On an Arithmetical Property of the Normalization of Regular Systems of Weights

Abstract

The weights of a two-dimensional weighted homogeneous polynomial $f$ of degree $h$ corresponding to an isolated singularity are arithmetically characterized by Prof. K. Saito and are called a regular system of weights. Let $m_0$ be the dimension of the vector space of the elements of degree $h$ of the Jacobi ring of $f$. It is shown that $m_0$ is determined by weights and is estimated from below by using the genus of the central curve and the number of branches of a minimal good resolution of the corresponding singularity.