Hilbert desingularizations for three dimensional canonical cyclic quotient singularities

TL;DR

This paper investigates the Hilbert property and resolutions of three-dimensional canonical cyclic quotient singularities, establishing the existence of Hilbert desingularizations for these singularities.

Contribution

It demonstrates the existence of Hilbert desingularizations for all three-dimensional canonical cyclic quotient singularities using classifications and resolution techniques.

Findings

Existence of Hilbert desingularizations for all such singularities.

Application of Fujiki-Oka resolutions and their iterations.

Utilization of classification by Ishida and Iwashita.

Abstract

In this paper, we shall discuss Hilbert property of ${\rm Hilb}^{G}(\mathbb{C}^{3})$, Fujiki-Oka resolutions and iterated Fujiki-Oka resolutions for three dimensional canonical cyclic quotient singularities by using the classification shown by Ishida and Iwashita\cite{II}. Finally, we shall prove that there exists a Hilbert desingularization for any three dimensional canonical cyclic quotient singularity.