Resolutions of 2 and 3 dimensional rings of invariants for cyclic groups

Let $G$ be the cyclic group of order $n$ and suppose $\mathbf{F}$ is a field containing a primitive $n^{\text{th}}$ root of unity. We consider the ring of invariants $\mathbf{F}[W{]}^G$ of a three dimensional representation $W$ of $G$ where $G\subset \text{SL}(W)$. We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gr"obner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of $F[W{]}^G$. The case where $W$ is any two dimensional representation of $G$ is also handled.