Syzygies of some invariant rings with cyclic group actions

TL;DR

This paper studies the algebraic structure of invariant rings under cyclic group actions on polynomial rings, providing classifications for specific cases related to their minimal free resolutions and generating invariants.

Contribution

It offers a complete classification of invariant rings for certain cyclic group actions, especially when the defining ideal has codimension two or multidegrees lie on two lines.

Findings

Classified cases with codimension two defining ideals

Analyzed multidegrees of invariants on two lines

Provided explicit descriptions of minimal free resolutions

Abstract

We investigate actions of cyclic groups on polynomial rings with two variables, and the minimal free resolution of the corresponding invariant ring. In particular, we fully classify several cases, including the case the defining ideal has codimension two, and when the multidegrees of generating invariants lie on the union of two lines.