# Revista Matemática Iberoamericana

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**Volume 34, Issue 4, 2018, pp. 1809–1820**

**DOI: 10.4171/rmi/1044**

Published online: 2018-12-06

Lifting weighted blow-ups

Marco Andreatta^{[1]}(1) Università di Trento, Povo (Trento), Italy

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.

*Keywords: *Contractions, weighted blow-up, Q-factorial terminal singularities

Andreatta Marco: Lifting weighted blow-ups. *Rev. Mat. Iberoam.* 34 (2018), 1809-1820. doi: 10.4171/rmi/1044