Two aspects of log terminal surface singularities
type: Item
instance of: scholarly article
title: Two aspects of log terminal surface singularities
zbMATH Open document ID: 0874.14020
DOI: 10.1007/BF02940775
author: V. Pereyra
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1997-11-10T00:00:00Z
review text: Log canonical singularities of a surface \(X\) have an index, i.e. a smallest positive integer \(I\) such that \(I\cdot K_X\) is Cartier. If \(\sigma:\widetilde{X}\to X\) is a good resolution of \(X\), \(E_i\) being the exceptional curves, \(I\) is obtained from the codiscrepancy \(\Delta\), where \(\sigma^*K_X= K_{\widetilde{X}}+ \Delta\) and \(\Delta=\sum \delta_iE_i\) and \(I=\min\{n\in \mathbb{N}_+\mid \forall i:n\cdot\delta_i\in \mathbb{Z}\}\). This paper studies log canonical and log terminal singularities, i.e. such singularities with \(\delta_i\leq 1\) for all \(i\) (\(\delta_i<1\), respectively), for a fixed number \(I\). There is a classification given in theorem A: The set of log terminal singularities consists of finitely many series (the number of series is explicitly given for each \(I\)) and in each series the singularities are indexed by natural numbers and ``behave very regular''. Essential for this investigation is a new numerical characterization of cyclic quotient singularities: The Hirzebruch-Jung continued fraction and the description of Riemenschneider, respectively, are compared with a triplet \((I,m,\alpha)\) involving the index \(I\), where \(n=I\cdot m\), \(q=m\cdot\alpha-1\) if the singularity is of type \((n,q)\). A series of cyclic quotient singularities is obtained simply in the form \((I,m,\alpha), (I,m+I,\alpha), (I,m+2I,\alpha),\dots\) -- The theorem provides an algorithm to obtain all log canonical singularities of given index \(I\). The author points out that his recent implementation on a computer proved useful to find log surfaces of prescribed index and Kodaira dimension 0. In theorem B, the weights \(b_i:= E_i^2\) of the singularity are considered. They are shown to measure ``how singular'' is the singularity for the case of log terminal singularities, whereas this is not the case in general. -- As an application, a new bound for the maximal number of weights \(\neq 2\) of a log terminal singularity of index \(I\) is given. Fu…
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 706663
zbMATH Keywords: index of singularity
zbMATH Keywords: codiscrepancy
zbMATH Keywords: Cartier divisor
zbMATH Keywords: log terminal singularities
zbMATH Keywords: cyclic quotient singularities
zbMATH Keywords: Hirzebruch-Jung continued fraction
zbMATH Keywords: algorithm
zbMATH Keywords: log canonical singularities
zbMATH Keywords: weights
reviewed by: Marko Roczen
MaRDI profile type: Publication
cites work: FRACTIONAL INDICES OF LOG DEL PEZZO SURFACES
cites work: Two two-dimensional terminations
cites work: CRITICAL POINTS OF SMOOTH FUNCTIONS AND THEIR NORMAL FORMS
cites work: Q3134882
cites work: Q4324887
cites work: Rationale Singularitäten komplexer Flächen
cites work: Q772397
cites work: Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces
cites work: 2-dimensionale Singularitäten und Differentialformen. (2-dimensional singularities and differential forms)
cites work: Q3136464
cites work: The topology of normal singularities of an algebraic surface and a criterion for simplicity
cites work: DEL PEZZO SURFACES WITH LOG-TERMINAL SINGULARITIES. I
cites work: Q3897139
cites work: Deformationen von Quotientensingularitäten (nach zyklischen Gruppen)
cites work: Q3775664
full work available at URL: https://doi.org/10.1007/bf02940775
OpenAlex ID: W2318066665
description: scientific article; zbMATH DE number 706663