Complete ideals and multiplicities in two-dimensional regular local rings

Let I be a complete m-primary ideal of a two-dimensional regular local ring (R,m). The beautiful theory developed by Zariski about complete ideals of R implies that the Rees valuation rings V of I are in a natural one-to-one correspondence with the minimal primes P of the ideal mR[It] in the Rees algebra R[It]. In the previous work of Huneke, Sally and the authors, the structure of R[It]/P is considered in the case where the residue field R/m=k is relatively algebraically closed in the residue field kv of V. In this paper we consider the structure of R[It]/P without the assumption that k is relatively algebraically closed in kv and obtain the following results: we give necessary and sufficient conditions for R[It]/P to be normal; we determine the multiplicity of R[It]/P; we examine the Cohen–Macaulay property of R[It]/P; and we describe implications for affine components of the blowup of I.