Cohomology of ideals 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 $1$.