Cohomology of ideals in elliptic surface singularities

TL;DR

This paper introduces the normal reduction number for two-dimensional normal singularities, proving elliptic singularities have a reduction number of two and characterizing certain non-rational singularities as Gorenstein with specific ideal properties.

Contribution

It defines the normal reduction number for surface singularities and characterizes maximally elliptic singularities of degree one among non-rational Gorenstein singularities.

Findings

Elliptic singularities have normal reduction number two.

Non-rational Gorenstein singularities with p_g-ideals are maximally elliptic of degree 1.

Characterization of ideal properties in elliptic surface singularities.

Abstract

We introduce the the normal reduction number of two-dimensional normal singularities and prove that elliptic singularity has normal reduction number two. We also prove that for a two-dimensional normal singularity which is not rational, it is Gorenstein and its maximal ideal is a $p_g$-ideal if and only if it is a maximally elliptic singularity of degree $1$.