Multi-graded algebras associated to surface singularities

In this paper we consider a multi-graded algebra T-X associated to the exceptional divisors of a resolution X of a germ of a surface singularity S. We show that the finite generation of T-X for all resolutions X is equivalent to the rationality of the singularity S. In the final Section 5, we survey results from our joint work with Juergen Herzog and Ana Reguera on Hilbert functions and Poincare series associated to T-X.