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Let $f:X\longrightarrow \mathbf{P}^1$ be a finite morphism of schemes. If $f$ is etale and $X$ is connected, we can show that $f$ is an isomorphism. That is, the projective line is simply connected. I would like to know if something more general can be said about $f$ if it is not assumed to be etale.

In this generality, it seems quite hard. But what if we assume $X$ to be also the projective line for example? Is it true that $f$ is then of the form $[x:y]\mapsto [x^n:y^n]$ in some sense?

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If $f$ is not etale then $X$ can be any complete curve. In fact if the base field is algebraically closed then just the choice of a non-constant element in the function field of a curve $X$ will give you a finite map to the projective line. –  Maharana Apr 12 '10 at 4:01

2 Answers 2

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No because $f$ can be ramified pretty much anywhere. Just think of a random rational function $f=p(x)/q(x)$ with $p,q$ coprime polynomials, $p$ non-constant and $q$ non-zero. That gives a finite morphism from the projective line to itself that is in general much more complicated than the map you suggest.

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Of course. Well it was nice hoping for it. –  Ari Apr 11 '10 at 16:33
    
If f has only two branch points and if the base field has characteristic 0 or bigger than deg f, then the result you expectd is true. –  Qing Liu Apr 12 '10 at 3:16

The question seems a bit vague, so let me also add one in addition to answers by Maharana, Kevin Buzzard and Qing Liu.

One simple but the most useful result for this kind of situation is the Riemann-Hurwitz formula, that relates the genus of X, genus of P^1 and the ramification points. See Hartshorne's chapter 4, or probably R. Narasimhan's book on Compact Riemann surfaces.

I encourage Ariyan to look at the general Riemann-Hurwitz formula for any nonconstant holomorphic maps from X to Y, where X and Y are both compact Riemann surfaces, and look at various consequences of this theorem.

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I already looked at Hartshorne. That's actually where I got the "inspiration" to ask the question. Basically, there the Hurwitz formula is applied to show that f is an isomorphism when it is etale. I do admit that the question is kind of vague. (There is a reason why I ask this question though. It has to do with push-forward maps in K-theory and intersection theory.) Anyway, thnx for the reference on Narasimhan's book. I'll take a look at it. –  Ari Apr 12 '10 at 9:47

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