What is a $G$-Galois Branched Cover

Question

What is, in the language of Schemes, a $G$-galois branched cover?

Answer 1

I don't think this phrase has a single precise translation into the general language of schemes, but it will mean something like the following:

a finite surjective morphism $X \to Y$ of integral schemes such that the corresponding extension of function fields $K(Y) \subset K(X)$ is Galois with Galois group $G$. It would also be reasonable to require that $X$ and $Y$ be normal.

To understand what it means geometrically, one should imagine that $X$ and $Y$ are projective varieties over some field. What this will mean then is that $X \to Y$ is surjection with finite fibres, that $G$ acts as a group of automorphisms of $X$ over $Y$, and that if we remove the branch locus (i.e. the closed subset of $Y$ along which the fibres contain points with multiplicity $> 1$) then each fibre is acted on faithfully and transitively by $G$ (so, away from the ramification locus, there are $|G|$ sheets of $X$ over $Y$, which are permuted by $G$).

A concrete example is given by (the projectivization of) the map $(x,y) \mapsto x$ from the elliptic curve $E$ defined by $y^2 = x^3 - x$ to $\mathbb P^1$.

The group $G$ is cyclic of order two, acting by $(x,y) \mapsto (x,-y)$, and the branch points are precisely the points where $y = 0$ (four of them; three three finite points $(0,0), (\pm 1,0)$, and also one at infinity).

In this context, an important result is the Zariski--Nagata purity theorem, which says (under mild hypotheses, e.g. that $Y$ and $X$ are smooth, or more generally, that $Y$ is smooth and $X$ is normal) that the set of branch points has pure codimension one, i.e. is a divisor.