NORMAL HILBERT COEFFICIENTS AND ELLIPTIC IDEALS IN NORMAL TWO-DIMENSIONAL SINGULARITIES

TOMOHIRO OKUMA,MARIA EVELINA ROSSI,KEN-ICHI YOSHIDA,KEI-ICHI WATANABE

doi:10.1017/nmj.2022.5

Abstract

AbstractLet $(A,\mathfrak m)$ 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 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.

Highlights

Introduction and NotationsLet (A, m) be an excellent two-dimensional normal local ring and let I be an m-primary ideal of A
The integral closure Iof I is the ideal consisting of all solutions z of some equation with coeffi√cients ci ∈ Ii: Zn + c1Zn−1 + c2Zn−2 + · · · + cn−1Z + cn = 0
Inspired by a paper by the first author [22], we investigate the integrally closed m-primary ideals of elliptic singularities

Summary

Introduction and Notations

Let (A, m) be an excellent two-dimensional normal local ring and let I be an m-primary ideal of A. It is known that A is a rational singularity (see [1]) if and only if every integrally closed m-primary ideal I of A is normal (see [16] and [4]), equivalently e2(I) = 0, that is I is a pg-ideal, as proved in [23, 24]. Okuma proved that if A is an elliptic singularity, thenr(I) = 2 for any integrally closed m-primary ideal of A, see [22, Theorem 3.3]. (See Proposition 3.3.) If A is not a rational singularity, for any m-primary integrally closed ideal I of A, In is either a pg-ideal or an elliptic ideal for every n ≥ pg(A). If A is strongly elliptic and I is not a pg-ideal, Proposition 3.16 and Theorem 3.23 give necessary and sufficient conditions for being I normal. We present an effective geometric construction, see Example 4.9

Preliminaries and normal reduction number

The existence of strongly elliptic ideals