Normal reduction number of normal surface singularities

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 $\tilde{X}\to X$ of $(X,o)$ and cohomology groups of different line bundles $\mathcal{L}\in \mathrm{Pic}(\tilde{X})$. 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. The present note was partially motivated by the open problems formulated by the third author (2021). Here we answer several of them.