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

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    I recently tried to learn what a cyclic cover was and got confused. I'd like to see a good answer to this. It would probably help me understand.2010-10-16

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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.

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    Thanks.That's why I asked here this question, I couldn't find in the books a precise definition of it.2010-10-16
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    Just a short addendum: in the arithmetic theory of coverings, sometimes **G-Galois cover** means slightly more: what Matt E said plus a particular isomorphism between the abstract group $G$ and the automorphism group of the covering. One might wonder whether this matters at all, and the answer (unfortunately!) is yes it does, sometimes, in the definitions of fields of moduli of coverings. Tread carefully: this is a subtle subject.2010-10-17