I've watched a lecture on K3 surfaces (although K3 surfaces are not the point of this question) where the following example is given:
Let $\pi:S\stackrel{2:1}{\to}\Bbb{P}^2$ be the branched double cover ramified over a smooth sextic curve $C\subset\Bbb{P}^2$. Then $S$ is a K3 surface.
The lecturer assumes everyone is familiar with those terms and goes on saying:
"[…] by the properties of double covers, $K_S=\pi^*K_{\Bbb{P}^2}+R$, where $R$ is the ramification. Furthermore, $\pi^*C=2R$"
I'm having a hard time trying to find out what "branched cover" and "ramification" mean, how the map $\pi$ is constructed, let alone figuring out why those pullback properties are true.
I've looked for precise definitions in books (Hartshorne, Shafarevich, Harris, Görtz-Wedhorn, Beauville) and internet-based material (Vakil, Gathmann, wikipedia), couldn't find it anyhere.
Many of these references eventually mention "branch" or "ramification" in passing or loosely, as if assuming the reader knows about it.
So my questions are:
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What are the definitions of "branched covering" and "ramification"?
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What is the map $\pi$ explicitly?
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Is there a code of ethics among algebraic geometers to make simple things harder for newcomers? I hope I won't get anyone in trouble with the ethics commitee.
Best Answer
Let's tackle ramified/unramified first, since that's something that's pretty uniform across the literature.
Definition (ref): A morphism of schemes $f:X\to S$ is unramified at $x\in X$ if there exists an affine open neighborhood $\operatorname{Spec} A=U\subset X$ of $x$ and an affine open $\operatorname{Spec} R=V\subset S$ so that $f(U)\subset V$ and the induced ring map $R\to A$ is of finite type and the module of Kahler differentials $\Omega_{A/R}$ is zero. A morphism of schemes is unramified if it is unramified at every point.
Equivalently, $f:X\to S$ is unramified if it is locally of finite type and $\Omega_{X/S}=0$. One may find an overview of relevant results and alternate characterizations at the Stacks Project section on unramified morphisms.
Perhaps somewhat expectedly, if something isn't unramified, then it's ramified. The best intuition one can have for this sort of morphism is provided at the Wikipedia page on the ramification with the following diagram:
Ramification is where there are "fewer points than you expect" because some branches "came together", like the marked points on $X$ on top in this image (this characterization assumes that your map behaves with respect to the included finiteness condition, because otherwise you're truly out of luck). To be precise, the ramification locus is the locus of points $x\in X$ where $(\Omega_{X/Y})_x\neq 0$, and the branch locus is it's image in $Y$ under the map $f$.
A branched covering is maybe a little more "geometrically" defined in the literature, so let us talk about that. The goal is to get maps which are covering maps "most places" - that is, we allow some amount of defect outside of a dense open subset. The moral of the story (and what I would take as my definition if I were in charge) is that a branched covering is a finite surjective morphism which is generically unramified and thus generically etale. (Generically unramified is automatic in characteristic zero, so frequently one may omit this condition if that's the only scenario one is concerned with - this is somewhat common, though decidedly not universal.)
To construct $\pi$ explicitly, the idea is that one wants to emulate the construction of the square-root function as a double cover of the complex plane ramified at the origin. The intuitive way to do this is to construct a square-root of the equation of the curve $C$, and this actually works: if our sextic is cut out by a homogeneous degree-six equation $f(x,y,z)$, then the equation $w^2=f(x,y,z)$ inside the weighted projective space $\Bbb P(1,1,1,3)$ with coordinates $x,y,z,w$ will cut out our ramified double cover.
As for part 3, no, there's no conspiracy that I'm aware of. It does happen sometimes in mathematics that there are things that "everybody knows" which can be pretty frustrating when you're not among the "everybody". This problem is not unique to algebraic geometry - in fact, in some ways, algebraic geometry has done a lot of work to get rid of this sort of thing via resources like Vakil's online notes and Stacks Project, though neither are full and complete references. I've found that the best way to resolve things like this is to start googling, going to the library (though, uh, with the way the world is right now, this might need some adjustment), and asking people who know more than I do what's up.