Generating Sequences and Semigroups of Valuations on 2-Dimensional Normal Local Rings

Abstract

In this paper we develop a method for constructing generating sequences for valuations dominating the ring of a two dimensional quotient singularity. Suppose that $K$ is an algebraically closed field of characteristic zero, $K[X,Y]$ is a polynomial ring over $K$ and $\nu$ is a rational rank 1 valuation of the field $K(X,Y)$ which dominates $K[X,Y]_{(X,Y)}$. Given a finite Abelian group $H$ acting diagonally on $K[X,Y]$, and a generating sequence of $\nu$ in $K[X,Y]$ whose members are eigenfunctions for the action of $H$, we compute a generating sequence for the invariant ring $K[X,Y]^H$. We use this to compute the semigroup $S^{K[X,Y]^H}$ of values of elements of $K[X,Y]^H$. We further determine when $S^{K[X,Y]} (\nu)$ is a finitely generated $S^{K[X,Y]^H} (\nu)$ -module.