The G-Hilbert scheme for 1/r(1,a,r-a)

TL;DR

This paper provides an explicit combinatorial description of the G-Hilbert scheme for a cyclic group acting on three-dimensional space, linking its fan structure to the Euclidean algorithm.

Contribution

It offers a detailed description of the G-Hilbert scheme for cyclic groups with specific weights, extending prior work on abelian group Hilbert schemes.

Findings

Explicit fan description for the G-Hilbert scheme

Connection between fan properties and Euclidean algorithm

Generalization of previous abelian group results

Abstract

Following Craw, Maclagan, Thomas and Nakamura's work on Hilbert schemes for abelian groups, we give an explicit description of the G-Hilbert scheme for G equal to a cyclic group of order r, acting on C^3 with weights 1,a,r-a. We describe how the combinatorial properties of the fan of G-Hilbert scheme relates to the Euclidean algorithm.