Degree functions and projectively full ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

摘要

Let (R, m) be a 2-dimensional rational singularity with algebraically closed residue field and for which the associated graded ring is an integrally closed domain. According to Gohner, (R, m) satisfies condition (N): given a prime divisor nu, there exists a unique complete m-primary ideal A(nu) in R with T(A(nu)) = {nu} and such that any complete m-primary ideal with unique Rees valuation nu, is a power of A(nu). We use the theory of degree functions developed by Rees and Sharp as well as some results about regular local rings, to investigate the degree coefficients d(A(nu), nu). As an immediate corollary, we find that for a simple complete m(1)-primary ideal I-1 in an immediate quadratic transform (R-1, m(1)) of (R, m); the inverse transform of I-1 in R is projectively full.