Spectrum of $\mathbb{C}[x,y]^{\mathbb{C}^*}$

Question

Let $\mathbb{C}[x,y]$ the ring of polynomials with $\mathbb{C}$-coefficients. We can define an action $\phi: \mathbb{C}^* \times \mathbb{C}[x,y] \rightarrow \mathbb{C}[x,y]$ such that $\phi(\lambda,p(x,y))=p(\lambda x,\lambda^{-1}y)$. I have to calculate $\operatorname{Spec}(\mathbb{C}[x,y]^{\mathbb{C}^*})$, i.e. the prime ideals of $\mathbb{C}[x,y]$ invariant for the action of $\mathbb{C}^*$.

Answer 1

As Piotr and Martin point out in the comments, I think you're actually looking for the prime ideals of the invariant ring, not the prime ideals of the original ring that are fixed by the action.

If this is right, the first step is to identify the invariant ring.

A polynomial is invariant under $p(x,y)\mapsto p(\lambda x,\lambda^{-1}y)$ if and only if $x$ and $y$ occur to the same power in every term. Thus the invariants are actually the set of polynomials in $xy$. This ring is isomorphic to $\mathbb{C}[x]$ so the $\operatorname{Spec}$ should be the same.