Asian Journal of Mathematics

Volume 27 (2023)

Number 2

A criteria for classification of weighted dual graphs of singularities and its application

Pages: 261 – 300

DOI: https://dx.doi.org/10.4310/AJM.2023.v27.n2.a4

Authors

Stephen S.-T. Yau (Department of Mathematical Sciences, Tsinghua University, Beijing, China; and Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing, China)

Qiwei Zhu (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Huaiqing Zuo (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Abstract

Let $(V, p)$ be a normal surface singularity. Let $\pi : (M,E) \to (V, p)$ be a minimal good resolution of $V$, such that the irreducible components $E_i$ of $E = \pi^{-1} (p)$ are nonsingular and have only normal crossings. There is a natural weighted dual graph $\Gamma$ associated to $E$. Along with the genera of the $E_i$, $\Gamma$ fully describes the topology and differentiable structure of the embedding of $E$ in $M$. Intuitively, normal surface singularity has simplest topology if all the irreducible curves in the exceptional set are smooth rational curves with self-intersection number $-2$. It can be shown that these are necessary ADE-singularities. In our previous work we classify all the weighted dual graphs of $E = \cup^n_{i =1} E_i$ such that one of the curves $E_i$ is $-3$ curve, and the rest all are $-2$ curves. This is a natural generalization of Artin’s classification of rational triple points. However there is no general method to classify or examine all possible weighted dual graphs of $E = \cup^n_{i =1} E_i$. In this article, we introduce a new concept, component factor, which is useful and computable for classifying weighted dual graphs. Based on it, we present a criteria for verifying whether a graph is the weighted dual graph associated to $E$. As a result, we give a complete classification of weighted dual graphs consist of $-2$ curves and exactly one $-4$ curve.

Keywords

normal singularities, topological classification, weighted dual graph

2010 Mathematics Subject Classification

14B05, 32S25

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Yau is supported by Tsinghua University Education Foundation fund (042202008) and NSFC Grant 11961141005.

Zuo is supported by NSFC Grants 12271280, 11961141005, and Tsinghua University Initiative Scientific Research Program.

Received 14 July 2022

Accepted 21 February 2023

Published 12 October 2023