On an arithmetical property of the normalization of regular systems of weights
type: Item
instance of: scholarly article
title: On an arithmetical property of the normalization of regular systems of weights
zbMATH Open document ID: 0774.14003
author: Yoshika Ueda
published in: Tokyo Journal of Mathematics
publication date: 1993-01-17T00:00:00Z
review text: The weights of a weighted homogeneous polynomial \(f\in\mathbb{C}[x,y,z]\) with isolated singularity form a regular system of weights: four positive integer \((a,b,c;h)\), whose characteristic function \[ \chi(T)=T^{- h}{(T^ h-T^ a)(T^ h-T^ b)(T^ h-T^ b)\over(T^ a-1)(T^ b- 1)(T^ c-1)} \] is regular on \(\mathbb{C}\backslash 0\). Write \(\chi(T)=\sum a_ nT^ n\). The coefficient \(a_ 0\) is equal to the genus of the central curve in the resolution of \(f\) (there are also \(r\) branches), and \(m_ 0:=a_{h-a-b-c}\) is the dimension of the degree \(h\) part of the Jacobi ring of \(f\). This paper gives an arithmetical proof (i.e. without singularity theory) of the inequality \(m_ 0\geq r-3+a_ 0\), with equality if \(a_ 0=0\).
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32S25
zbMATH DE Number: 97907
zbMATH Keywords: isolated singularity
zbMATH Keywords: regular system of weights
reviewed by: Jan Stevens
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270130248
OpenAlex ID: W2091774083
DOI: 10.3836/TJM/1270130248
description: scientific article; zbMATH DE number 97907