The Poincare series of divisorial valuations in the plane defines the topology of the set of divisors

TL;DR

This paper demonstrates that the Poincare series of divisorial valuations in the plane uniquely determines the topological configuration of the divisors, linking algebraic invariants to geometric topology.

Contribution

It establishes that the Poincare series encodes the topology of divisors, providing a new way to recover the minimal resolution from algebraic data.

Findings

Poincare series coincides with the Alexander polynomial for plane curve singularities.

The Poincare series determines the minimal resolution topology of divisors.

A simplified proof of Yamamoto's theorem relating Alexander polynomial and topology.

Abstract

To a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincare series of this filtration turnes out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincare series of the corresponding multi-index filtration similar to the one associated to plane germs. Here we show that the Poincare series of a set of divisorial valuations on the ring of germs of functions of two variables defines "the topology of the set of the divisors" in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which proves that the Alexander polynomial is equivalent to the embedded topology.