Quasi-one-fibered and projectively full ideals in 2-dimensional Muhly rational singularities

Abstract

Let (R, m) be a 2-dimensional rational singularity. Assume that R is a Muhly domain: (R, m) is an integrally closed Noetherian local domain with algebraically closed residue field and the associated graded ring $$\mathrm{gr}_mR$$ is an integrally closed domain. Let I be a complete m-primary ideal of R and let $$v_1, \ldots , v_n$$ be the Rees valuations of I. For every $$v_i \ne \mathrm{ord}_R$$ , there is a complete m-primary ideal $$I_i$$ of R such that $$I_i$$ is quasi-one-fibered (i.e. $$I_i$$ has exactly one Rees valuation different from $$\mathrm{ord}_R$$ ) and the degree coefficient of $$I_i$$ with respect to $$v_i$$ is $$d(I_i,v_i)=1$$ . We show the following formula: for some $$s, u_1, \ldots , u_n \geqslant 0$$ . We then prove that if $$\mathrm{ord}_R$$ is not a Rees valuation of I and if , then I is projectively full.