Quasi-one-fibered ideals of order one in dimension two

Abstract

Simple complete $\M$-primary ideals are fundamental in the Zariski-Lipman theory of complete ideals in a two-dimensional regular local ring $(R,\M)$. Complete $\M$-primary ideals of order one constitute a particular class of such ideals containing, for example, all the first neighborhood ideals of $R$. A number of interesting properties of complete $\M$-primary ideals of order one have been proved by several authors. For example, these ideals have only one Rees valuation (and hence are one-fibered) and they have the very simple form $(x_1^n,x_2)R$, $n\in\n_{+}$, with $x_1,x_2$ a minimal ideal basis of $\M$. In the present paper we investigate how far some results concerning these complete $\M$-primary ideals of order one can be extended to complete quasi-one-fibered $\M$-primary ideals of order one in a natural generalization of $R$: a two-dimensional normal Noetherian local domain with algebraically closed residue field and the associated graded ring an integrally closed domain.