A finiteness theorem for dual graphs of surface singularities

TL;DR

This paper proves a finiteness result for the coefficients of the canonical cycle in numerically Gorenstein surface singularities with a given dual graph, and characterizes when infinite possibilities occur for self-intersection numbers.

Contribution

It establishes a finiteness theorem for the coefficients of the canonical cycle in surface singularities with fixed dual graph and describes conditions for infinite possibilities of self-intersection numbers.

Findings

Finite possibilities for canonical cycle coefficients when dual graph is fixed.

Finite number of self-intersection configurations for non-cyclic dual graphs.

Characterization of cases with infinite self-intersection options.

Abstract

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 $\Gamma$.