On ideal filtrations for Newton nondegenerate surface singularities

TL;DR

This paper compares three filtrations of ideals on Newton nondegenerate surface singularities, establishing conditions under which they coincide and linking them to topological invariants like zeta functions.

Contribution

It introduces a cone within the Lipman cone where the three filtrations agree and connects their Hilbert series to topological zeta functions for certain singularities.

Findings

The three filtrations coincide within a specific cone in the Lipman cone.

The Hilbert series of these filtrations can be related to topological zeta functions.

For Newton nondegenerate suspension singularities, all three filtrations' zeta functions coincide.

Abstract

We compare three naturally occurring multi-indexed filtrations of ideals on the local ring of a Newton nondegenerate hypersurface surface singularity with rational homology sphere, which in many cases are all distinct. These are the divisorial, the order, and the image filtrations. These filtrations are indexed by the lattice associated with a toric partial resolution of the singularity, or equivalently, the free abelian group generated by the compact facets of the Newton polyhedron. We prove that there exists a top dimensional cone contained in the Lipman cone having the property that the three ideals indexed by order vectors from this cone coincide. As a corollary, if a periodic constant can be associated with the Hilbert series associated with these filtrations on the Lipman cone, then they coincide. In some cases, the Poincar\'e series associated with one of these filtrations has been shown to coincide with a zeta function associated with the topological type of the singularity. In the end of the article, we show that this is the case for all three filtrations considered in the case of a Newton nondegenerate suspension singularity. As a corollary, in this case, the zeta function provides a direct method of determining the Newton diagram from the link.