The blow-up of a simple ideal

Let R   be a two-dimensional regular local ring with maximal ideal mm and infinite residue field and let I   be a complete simple mm-primary residually rational ideal of R  ; let Σ=BlI(R)Σ=BlI(R) be the blow-up of I  , and R=:R0⊂R1⊂⋯⊂RnR=:R0⊂R1⊂⋯⊂Rn be the sequence determined by I  ; RiRi is a quadratic transform of Ri−1Ri−1. Σ is a normal surface; we show that Σ has one resp. two singular points if RnRn is free resp. not free. The singular points are rational singularities; we determine their multiplicities.