Gorenstein Quotient Singularities of Monomial Type in Dimension Three

TL;DR

This paper constructs crepant resolutions for certain three-dimensional quotient singularities of monomial type and proves a conjecture relating the Euler number of the resolution to the number of conjugacy classes.

Contribution

It introduces a method to resolve monomial type quotient singularities in dimension three and proves a key conjecture linking Euler numbers to conjugacy classes.

Findings

Constructed crepant resolutions for monomial quotient singularities in dimension three.

Proved the Euler number equals the number of conjugacy classes for these resolutions.

Extended ideas from trihedral singularities to monomial type cases.

Abstract

The purpose of this paper is to construct a crepant resolution of quotient singularities by finite subgroups of SL(3,C) of monomial type, and prove that the Euler number of the resolution is equal to the number of conjugacy classes. This result is a part of conjecture II in previous paper "Crepant resolution of trihedral singularities" (alg-geom 9404008). These singularities are different from trihedral, but main idea of the proof is based on the method of trihedral case.