What is the relation between $\mathbb{C}[M]$ and $\mathbb{C}[M/U]$.

Question

Let $M$ be a variety and let $U$ be a group. By definition, $M/U$ is the space of all $U$-orbits of $M$. Now we take coordinate rings $\mathbb{C}[M]$ and $\mathbb{C}[M/U]$. What is the relation between these two rings? What are the elements in $\mathbb{C}[M/U]$ look like? Thank you very much.

Answer 1

If $U$ acts on $M$, then it acts on $\mathbf{C}[M]$ too. $\mathbf{C}[M/U]$ "ought to be" the subring of fixed points of $U$; think of them as functions on $M$ that are constant when restricted to orbits of $U$. I don't know if there are any technical subtleties that will cause problems in the algebraic geometry setting.

Answer 2

Given your action of $U$ on $M$, there exists in the category of topological spaces a quotient topological space $M/U $and a continuous map $\pi: M\to M/U$ with all sorts of nice properties.
The catch, however is that in general there is no reasonable way to endow $M/U$ with the structure of complex manifold:
A counterexample is $M=\mathbb C^n$ and $U=GL_n(\mathbb C)$ acting in the obvious way.
The quotient topological space has two points, one corresponding to the set $\mathbb C^n \setminus \{0\}$ of non-zero vectors and the other to the singleton set $\{0\}$.
It is connected but not Hausdorff and there is no reasonable structure of complex manifold on that quotient.
A reasonable sufficient for a quotient manifold to exist is that $U$ be a Lie group and that the action be free and proper.
Examples are $U=$discrete group, which yields a covering space, but also the non discrete $U=\mathbb C^*$ acting on $\mathbb C^n \setminus \{0\}$ and yielding projective space $\mathbb P^ {n-1}(\mathbb C)$.
The case of algebraic varieties (or rather schemes) is a whole branch of algebraic geometry: Mumford wrote a book on the subject which earned him a Fields medal in 1974.

Answer 3

If $M$ is an affine variety and $U$ is a finite group then $M/U$ is also an affine variety (this is proved by Hilbert - Noether in a more general settings). More generally, if $U$ is a reductive group then $M/U$ is an affine variety (this result is due to Nagata). In this case, $\mathbb{C}[M/U] = \mathbb{C}[M]^U,$ the invariant subring of $\mathbb{C}[M].$ But for general group $U,$ the invariant subring $\mathbb{C}[M]^U$ may not be finitely generated algebra over $\mathbb{C}$ and hence it will not have any variety structure. For the case of projective variety, there are notions of good quotient and geometric quotient. This branch of algebraic geometry is called Geometric Invariant Theory. You can look at "Introduction to Moduli Problems and Orbit Spaces" by P. E. Newstead.