Three-fold divisorial contractions to singularities of higher indices

Abstract

We complete the explicit study of a three-fold divisorial contraction whose exceptional divisor contracts to a point by treating the case where the point downstairs is a singularity of index n≥2. We prove that if this singularity is of type cA/n, then any such contraction is a weighted blowup; and that if otherwise, then f is either a weighted blowup of a singularity of type cD/2 embedded into a cyclic quotient of a smooth five-fold, or a contraction with discrepancy 1/n, 1, or 2. Every such exceptional case of discrepancy 1 or 2 has an example. The erratum to our previous article [13] appears in the appendix.