Weakly--exceptional quotient singularities

TL;DR

This paper classifies weakly--exceptional quotient singularities in dimensions 3 and 4, extending known results from dimension 2 and providing a comprehensive understanding of their structure.

Contribution

It provides the first classification of weakly--exceptional quotient singularities in dimensions 3 and 4, expanding the understanding of their geometric properties.

Findings

Dimension 2 weakly--exceptional quotient singularities are Dn, E6, E7, E8 types.

Complete classification of such singularities in dimensions 3 and 4.

Extension of Shokurov's results to higher dimensions.

Abstract

A singularity is said to be weakly--exceptional if it has a unique purely log terminal blow up. In dimension $2$, V. Shokurov proved that weakly--exceptional quotient singularities are exactly those of types $D_{n}$, $E_{6}$, $E_{7}$, $E_{8}$. This paper classifies the weakly--exceptional quotient singularities in dimensions $3$ and $4$.