Definition of the quotient curve Given a smooth projective curve $C$ such that a group $G$ acts on it, how exactly is the quotient curve ${X/G}$ defined? I cannot seem to find any resources covering explicitly it's definition. I am aware of the fact that there are various categorical equivalences between geometrical objects (varieties) and algebraic objects (function fields) and maybe these are at work here, but I'm uncertain
 A: One easy way to define a quotient of a curve $C$ under the action of the finite group $G$ is as the curve with function field $K(C)^G$, the $G$-invariants of the function field $K(C)$ under the action of $G$. This works because the categories of regular curves over a base field $k$ with nonconstant morphisms and finitely generated field extensions of $k$ with transcendence degree one are canonically equivalent. To see that $K(X)^G$ is again a function field, note that $[K(X)^G:K(X)]$ is a finite Galois extension of degree $|G|$ - so $K(X)^G$ is again of transcendence degree one. For the claim that $K(X)^G$ is again finitely generated, see here on MO.
There are other ways: if $G$ is finite, then we can cover $X$ by $G$-invariant affine opens (take an affine open neighborhood $U$ of $x\in X$ and consider $\bigcap_{g\in G} gU$ which is open since $G$ is finite and affine as $X$ is separated) and compute the quotient on each of these affine pieces, then glue back together, using that for an affine scheme $\operatorname{Spec} R$, the quotient by $G$ is $\operatorname{Spec} R^G$.
