# What is known about finite morphisms from X to the projective line

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

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