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 $n\ge 2$. We prove that if this singularity is of type c$A/n$, then any such contraction is a weighted blowup; and that if otherwise, then $f$ is either a weighted blowup of a singularity of type c$D/2$ 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.