Generating sequences and Poincar\'e series for a finite set of plane divisorial valuations

TL;DR

This paper investigates the structure of divisorial valuations in a 2D local ring using semigroups and graded algebras, providing minimal generators, decompositions, and relations to Poincaré series and Alexander polynomials.

Contribution

It establishes finite generation of the value semigroup, computes minimal generators, and links Poincaré series to Alexander polynomials, extending understanding of plane curve singularities.

Findings

Semigroup of values is finitely generated.

Minimal generating sequences for valuations and curves are obtained.

Poincaré series relates to Alexander polynomial and can be expressed via limits.

Abstract

Let $V$ be a finite set of divisorial valuations centered at a 2-dimensional regular local ring $R$. In this paper we study its structure by means of the semigroup of values, $S_V$, and the multi-index graded algebra defined by $V$, $\gr_V R$. We prove that $S_V$ is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in $V$, the approximation of a reduced plane curve singularity $C$ by families of sets $V^{(k)}$ of divisorial valuations, and the relationship between the value semigroup of $C$ and the semigroups of the sets $V^{(k)}$, allow us to obtain the (finite) minimal generating sequences for $C$ as well as for $V$. We also analyze the structure of the homogeneous components of $\gr_V R$. The study of their dimensions allows us to relate the Poincar\'e series for $V$ and for a general curve $C$ of $V$. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincar\'e series of $V$. Moreover, the Poincar\'e series of $C$ could be seen as the limit of the series of $V^{(k)}$, $k\ge 0$.