Some characterizations of first neighborhood complete ideals in dimension two

Abstract

In this paper we study first neighborhood complete ideals in a two-dimensional normal Noetherian local domain (R,M) with algebraically closed residue field and the associated graded ring an integrally closed domain. It is shown that a complete quasi-one-fibered M-primary ideal I of R is a first neighborhood complete ideal if and only if e(I) = e(M)+ 1. This implies that (R,M) is a rational singularity if and only if a (every) first neighborhood complete ideal has minimal multiplicity. Moreover, if (R,M) has minimal multiplicity, then a complete quasi-one-fibered M-primary ideal I of order one is a first neighborhood complete ideal if and only if certain numerical data associated with I are minimal. This yields a simple proof of the fact that first neighborhood complete ideals in such a local ring R are projectively full. © 2012 Rocky Mountain Mathematics Consortium.