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

Abstract

Let (A, m) be an excellent two-dimensional normal local domain. The geometric genus pg(A) is an important geometric invariant of A. A rational singularity is characterized by pg(A) = 0 and the integrally closed m-primary ideals of A are normal and well described by Cutkosky and Lipman. Later, Okuma, Watanabe and Yoshida characterized rational singularities through the pg-ideals. In this paper we define 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. A strongly elliptic singularity can be described by pg(A) = 1. We characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed m-primary ideals of A. Unlike pg-ideals, elliptic and strongly elliptic ideals are not necessarily normal and we give necessary and sufficient conditions for being normal. In the last section we discuss the existence (and the effective construction) of strongly elliptic ideals in any 2-dimensional normal local ring.