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Let $G \subset SL_2(\mathbb{C})$ be a finite subgroup acting linearly on $\mathbb{C}[X, Y]$. Then it is claimed that the ring of invariants $\mathbb{C}[X, Y]^G$ is always a hypersurface. I am not able to see its proof by myself. Please help.

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I don't think you can tell this a priori without actually computing a presentation of the invariants.

If you do that, then it is a matter of checking that you can generate each of those rings with three elements, and that these satisfy exactly one polynomial relation. Finding the invariants is easy in most cases, but hard for the binary icosahedral group. Everything can be done by computer, using Groebner bases; you can spot the relations by looking hard at generators (and harder for the binary icosahedral group...), and then checking that you have them all is a matter of using Molien's formula. ALl this was done before von Neumann was born, so it is quite possible to do by hand, of course!

The description of generators and relations is done in Klein's Lectures on the icosahedron, and in several more modern places. I think, for example, that this is done explicitly in Dolgashev's notes on McKay correspondence, which Google will find for you. See also and the references listed there.

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