Geodesic
Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve
Izvestiya. Mathematics , Tome 87 (2023) no. 5, pp. 1078-1116

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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.
Keywords: normal singularities, topological classification, weighted dual graph. 
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     title = {Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve},
     journal = {Izvestiya. Mathematics },
     pages = {1078--1116},
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     number = {5},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2023_87_5_a13/}
}
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S. S.-T. Yau; Qiwei Zhu; Huaiqing Zuo. Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve. Izvestiya. Mathematics , Tome 87 (2023) no. 5, pp. 1078-1116. http://geodesic.mathdoc.fr/item/IM2_2023_87_5_a13/