Lifting weighted blow-ups

Abstract

Let $f:X\to Z$ 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 $f_{|Y}:Y\to W$ has a structure of a weighted blow-up. We prove that $f:X\to Z$, as well, has a structure of weighted blow-up.

As an application we consider a local projective contraction $f:X\to Z$ 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.