International Journal of Mathematics c © World Scientific Publishing Company A FINITENESS THEOREM FOR DUAL GRAPHS OF SURFACE SINGULARITIES

Consider a fixed connected, finite graph Γ and equip its vertices with weights pi which are non-negative integers. We show that there are a finite number of possibilities for the coefficients of the canonical cycle of a numerically Gorenstein surface singularity having Γ as the dual graph of the minimal resolution, the weights pi of the vertices being the arithmetic genera of the corresponding irreducible components. As a consequence we get that if Γ is not a cycle, then there are a finite number of possibilities of self-intersection numbers which one can attach to the vertices which are of valency ≥ 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 Γ