Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve

S. S.-T. Yau ; Qiwei Zhu ; Huaiqing Zuo

Izvestiya. Mathematics , Tome 87 (2023) no. 5, pp. 1078-1116.

Résumé

Let $(V,p)$ be a normal surface singularity. Let $\pi :(M,A)\to (V,p)$ be a minimal good resolution of $V$. The weighted dual graphs $\mathrm{\Gamma }$ 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.

Détail

Mots-clés : normal singularities, topological classification, weighted dual graph.

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