The blow-up of a simple ideal

Abstract

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