A finiteness theorem for dual graphs of surface singularities

Details A finiteness theorem for dual graphs of surface singularities A finiteness theorem for dual graphs of surface singularities

2008-05-13

arxiv.org/abs/0805.1842v2

International Journal of Mathematics 20, no. 8 (2009), 1057-1068

Mathematics

Complex Variables

10 pages

Scientific paper

Consider a fixed connected, finite graph $\Gamma$ and equip its vertices with weights $p_i$ which are non-negative integers. We show that there is a finite number of possibilities for the coefficients of the canonical cycle of a numerically Gorenstein surface singularity having $\Gamma$ as the dual graph of the minimal resolution, the weights $p_i$ of the vertices being the arithmetic genera of the corresponding irreducible components. As a consequence we get that if $\Gamma$ is not a cycle, then there is a finite number of possibilities of self-intersection numbers which one can attach to the vertices which are of valency $\geq 2$, such that one gets the dual graph of the minimal resolution of a numerically Gorenstein surface singularity. Moreover, we describe precisely the situations when there exists an infinite number of possibilities for the self-intersections of the component corresponding to some fixed vertex of $\Gamma$.

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