Normal Hilbert coefficients and elliptic ideals in normal two-dimensional singularities

TL;DR

This paper characterizes elliptic and strongly elliptic singularities in two-dimensional normal local rings using Hilbert coefficients and explores conditions for their normality and existence.

Contribution

It provides a new characterization of strongly elliptic singularities via Hilbert coefficients and discusses the normality and construction of elliptic ideals.

Findings

Characterization of strongly elliptic singularities using Hilbert coefficients.

Necessary and sufficient conditions for elliptic and strongly elliptic ideals to be normal.

Existence and explicit construction methods for strongly elliptic ideals in two-dimensional normal rings.

Abstract

Let $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper we study the elliptic and the strongly elliptic ideals of $A$ with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and by Yau. In analogy with the rational singularities, in the main result we characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed $\mathfrak m$-primary ideals of $A$. Unlike $p_g$-ideals, elliptic ideals and strongly elliptic ideals are not necessarily normal and necessary and sufficient conditions for being normal are given. In the last section we discuss the existence (and the effective construction) of strongly elliptic ideals in any two-dimensional normal local ring.