Poincare series of filtrations corresponding to ideals on surfaces

TL;DR

This paper introduces a new type of filtration on surface singularities associated with ideals, computes their Poincare series, and unifies previous filtration concepts in the case of the complex plane.

Contribution

It defines a novel filtration related to ideals on surface singularities and computes its Poincare series, bridging existing filtration types in the complex plane case.

Findings

Defined a filtration corresponding to ideals on surface singularities.

Computed the Poincare series for this filtration in specific cases.

Unified previous filtration concepts in the complex plane context.

Abstract

Earlier the authors considered and, in some cases, computed Poincare series of two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular on the complex plane). A filtration from the first class was defined by a curve (with several branches) on the surface singularity. The other one (so called divisorial filtration) was defined by a set of components of the exceptional divisor of a modification of the surface singularity. Here we define a filtration corresponding to an ideal or to a set of ideals in the ring of germs of functions on a surface singularity and compute the corresponding Poincare series in some cases. For the complex plane this notion unites the two classes of filtrations described above.