Blow-ups of three-dimensional toric singularities

TL;DR

This paper classifies the purely log terminal blow-ups of three-dimensional terminal toric singularities, detailing divisorial contractions under specific conditions related to toric structures and singularities.

Contribution

It provides a comprehensive description of divisorial contractions for three-dimensional toric terminal singularities with irreducible exceptional divisors and canonical singularities.

Findings

Classification of purely log terminal blow-ups

Description of divisorial contractions with toric structures

Conditions for canonical singularities in blow-ups

Abstract

The purely log terminal blow-ups of three-dimensional terminal toric singularities are described. The three-dimensional divisorial contractions $f\colon (Y,E)\to (X\ni P)$ are described provided that $\Exc f=E$ is an irreducible divisor, $(X\ni P)$ is a toric terminal singularity, $f(E)$ is a toric subvariety and $Y$ has canonical singularities.