Integrally closed ideals in regular local rings of dimension two

Let I be an integrally closed -primary ideal of a two-dimensional regular local ring . By Zariski’s unique factorization theorem, the ideal I factors as , where the si are positive integers and the Ii are distinct simple integrally closed ideals. For each integer i with 1≤i≤h, let Vi denote the unique Rees valuation ring associated with Ii, and let vi denote the Rees valuation associated with Vi. Assume that the field k is relatively algebraically closed in the residue field k(vi) of Vi. Let J=(a,b)R be a reduction of I and let denote the image of in the residue field k(vi) of Vi. We prove that . Assume that (e,f)R is a reduction of Ii. We prove that there exists a minimal generating set {gesi,gesi−1f,…,gfsi,ξsi+1,…,ξr} for I, where with and vi(ξp)>vi(I) for p with si+1≤p≤r. There exists such that Vi=R[It]Qi∩Q(R). We prove that the quotient ring R[It]/Qi is normal Cohen–Macaulay with minimal multiplicity at its maximal homogeneous ideal with this multiplicity being si. In particular, R[It]/Qi is regular if and only if si=1.