Lifting weighted blow-ups

Abstract

Let f\colon X \to Z be a local, projective, divisorial contraction between normal varieties of dimension n with \mathbb Q -factorial singularities. Let Y \subset X be a f -ample Cartier divisor and assume that f_{|Y}\colon Y \to W has a structure of a weighted blow-up. We prove that f\colon X \to Z , as well, has a structure of weighted blow-up. As an application we consider a local projective contraction f\colon X \to Z from a variety X with terminal \mathbb Q -factorial singularities, which contracts a prime divisor E to an isolated \mathbb Q -factorial singularity P\in Z , such that -(K_X + (n-3)L) is f -ample, for a f -ample Cartier divisor L on X . We prove that (Z,P) is a hyperquotient singularity and f is a weighted blow-up.