A FINITENESS THEOREM FOR DUAL GRAPHS OF SURFACE SINGULARITIES

Consider a fixed connected, finite graph G and equip its vertices with weights pi 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 G 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 >= 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.