Invariants of the graded algebras associated to divisorial valuations dominating a rational surface singularity

Abstract

Let $(R,M)$ be a rational surface singularity and $\nu_E$ be a prime divisor of the second kind for $R$. Then $gr_{\nu_E} R$ is finitely generated over $R/M$. We recover information about the dual graph of the minimal resolution $\widetilde X$ of $\text{Spec } R$ from the set of all $gr_{\nu_E} R$. In particular we characterize those graded algebras corresponding to the exceptional curves in $\widetilde X$.