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

TL;DR

This paper presents a method for constructing generating sequences for valuations on two-dimensional quotient singularities, computes the semigroup of invariant rings, and analyzes module finiteness properties.

Contribution

It introduces a new approach to generate sequences for valuations on 2D singularities and computes associated semigroups for invariant rings under group actions.

Findings

Constructed generating sequences for valuations on quotient singularities.

Computed semigroups of values for invariant rings under finite Abelian group actions.

Identified conditions for the semigroup of the original ring to be finitely generated over the invariant ring's semigroup.

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.