{"title":"由$-2曲线和恰好一条$-3曲线组成的加权对偶图的分类","authors":"Stephen S.-T. Yau, Qiwei Zhu, Huaiqing Zuo","doi":"10.4213/im9337e","DOIUrl":null,"url":null,"abstract":"Let $(V, p)$ be a normal surface singularity. Let $\\pi\\colon (M, A)\\to (V, p)$ be a minimal good resolution of $V$. The weighted dual graphs $\\Gamma$ associated with $A$ completely describes the topology and differentiable structure of the embedding of $A$ in $M$. In this paper, we classify all the weighted dual graphs of $A=\\bigcup_{i=1}^n A_i$ such that one of the curves $A_i$ is a $-3$-curve, and all the remaining ones are $-2$-curves. This is a natural generalization of Artin's classification of rational triple points. Moreover, we compute the fundamental cycles of maximal graphs (see § 5) which can be used to determine whether the singularities are rational, minimally elliptic or weakly elliptic. We also give formulas for computing arithmetic and geometric genera of star-shaped graphs.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve\",\"authors\":\"Stephen S.-T. Yau, Qiwei Zhu, Huaiqing Zuo\",\"doi\":\"10.4213/im9337e\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $(V, p)$ be a normal surface singularity. Let $\\\\pi\\\\colon (M, A)\\\\to (V, p)$ be a minimal good resolution of $V$. The weighted dual graphs $\\\\Gamma$ associated with $A$ completely describes the topology and differentiable structure of the embedding of $A$ in $M$. In this paper, we classify all the weighted dual graphs of $A=\\\\bigcup_{i=1}^n A_i$ such that one of the curves $A_i$ is a $-3$-curve, and all the remaining ones are $-2$-curves. This is a natural generalization of Artin's classification of rational triple points. Moreover, we compute the fundamental cycles of maximal graphs (see § 5) which can be used to determine whether the singularities are rational, minimally elliptic or weakly elliptic. We also give formulas for computing arithmetic and geometric genera of star-shaped graphs.\",\"PeriodicalId\":54910,\"journal\":{\"name\":\"Izvestiya Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Izvestiya Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4213/im9337e\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Izvestiya Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4213/im9337e","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve
Let $(V, p)$ be a normal surface singularity. Let $\pi\colon (M, A)\to (V, p)$ be a minimal good resolution of $V$. The weighted dual graphs $\Gamma$ associated with $A$ completely describes the topology and differentiable structure of the embedding of $A$ in $M$. In this paper, we classify all the weighted dual graphs of $A=\bigcup_{i=1}^n A_i$ such that one of the curves $A_i$ is a $-3$-curve, and all the remaining ones are $-2$-curves. This is a natural generalization of Artin's classification of rational triple points. Moreover, we compute the fundamental cycles of maximal graphs (see § 5) which can be used to determine whether the singularities are rational, minimally elliptic or weakly elliptic. We also give formulas for computing arithmetic and geometric genera of star-shaped graphs.
期刊介绍:
The Russian original is rigorously refereed in Russia and the translations are carefully scrutinised and edited by the London Mathematical Society. This publication covers all fields of mathematics, but special attention is given to:
Algebra;
Mathematical logic;
Number theory;
Mathematical analysis;
Geometry;
Topology;
Differential equations.