# Noncommutative cyclic isolated singularities

## TL;DR

This paper investigates when noncommutative graded quotient singularities are isolated, focusing on cyclic actions on skew polynomial rings, and extends previous results by establishing a partial dichotomy theorem.

## Contribution

It introduces a partial dichotomy theorem for isolatedness of noncommutative quotient singularities under cyclic actions, generalizing prior work in the field.

## Key findings

- Established a partial dichotomy theorem for isolatedness.
- Applied the theorem to cyclic actions on the (-1)-skew polynomial ring.
- Extended previous results by Bao, He, and others.

## Abstract

The question of whether a noncommutative graded quotient singularity $A^G$ is isolated depends on a subtle invariant of the $G$-action on $A$, called the pertinency. We prove a partial dichotomy theorem for isolatedness, which applies to a family of noncommutative quotient singularities arising from a graded cyclic action on the $(-1)$-skew polynomial ring. Our results generalize and extend some results of Bao, He and the third-named author and results of Gaddis, Kirkman, Moore and Won.