Lifting Weighted Blow-ups

TL;DR

This paper proves that under certain conditions, a divisorial contraction with a weighted blow-up structure on a divisor extends this structure to the entire contraction, with applications to classifying singularities.

Contribution

It establishes that a local divisorial contraction with a weighted blow-up structure on a divisor also has a weighted blow-up structure globally, extending previous local results.

Findings

f: X -> Z has a weighted blow-up structure if its restriction to a divisor Y does

Z, P are hyperquotient singularities under certain conditions

f is a weighted blow-up when specific ampleness and singularity conditions are met

Abstract

Let f: X -> Z be a local, projective, divisorial contraction between normal varieties of dimension n with Q-factorial singularities. Let $Y \subset X$ be a f-ample Cartier divisor and assume that f|Y: Y -> W has a structure of a weighted blow-up. We prove that f: X -> Z, as well, has a structure of weighted blow-up. As an application we consider a local projective contraction f: X -> Z from a variety X with terminal Q-factorial singularities, which contracts a prime divisor E to an isolated 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.