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 $g{r}_{{\nu }_E}R$ is finitely generated over $R/M$. We recover information about the dual graph of the minimal resolution $\tilde{X}$ of $\text{Spec }R$ from the set of all $g{r}_{{\nu }_E}R$. In particular we characterize those graded algebras corresponding to the exceptional curves in $\tilde{X}$.