Normal reduction number of normal surface singularities

TL;DR

This paper studies the normal reduction number of complex surface singularities, providing topological bounds and linking it to cohomology and Abel map stability, advancing understanding of ideal stabilization in singularity theory.

Contribution

It introduces topological upper bounds for the normal reduction number and connects it with cohomology and Abel map stability in surface singularities.

Findings

Topological bounds for the normal reduction number.

Connection between reduction number and cohomology groups.

Stability properties of iterated Abel maps related to singularity invariants.

Abstract

Let $(X,o)$ be a complex analytic normal surface singularity and let ${\mathcal O}_{X,o}$ be its local ring. We investigate the normal reduction number of ${\mathcal O}_{X,o}$ and related numerical analytical invariants via resolutions $\widetilde{X}\to X$ of $(X,o)$ and cohomology groups of different line bundles ${\mathcal L}\in {\rm Pic}(\widetilde{X})$. The normal reduction number is the universal optimal bound from which powers of certain ideals have stabilization properties. Here we combine this with stability properties of the iterated Abel maps. Some of the main results provide topological upper bounds for both stabilization properties.