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Let $C$ be a reduced curve (over $k$). As the title says, which requirements do we need to be sure that there is some finite (non-contant) morphism onto $\mathbb{P}_k^1$ where $k$ is any field.

The question is therefore some kind of generalization of Hartshorn IV Exercise 1.6. (because of his definition of curve)

I do have in mind that the curve should be allowed to be singular.

There should be some reference, maybe in EGA, but I didn't find one. I do really want to find a reference.

Thank you in advance!

Jesko Hüttenhain
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windsheaf
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  • You can always do that. – Jesko Hüttenhain Jun 30 '17 at 15:46
  • Dear @JeskoHüttenhain Thank you, but is there a reference which I could refer to? Like the Kedlaya paper mentioned in the comments of the link you provided. Actually, in that paper Kedlaya says that the case for $n=1$ has become a folk theorem. Hence there should be a paper/book I could refer to. – windsheaf Jul 01 '17 at 08:03
  • I do not have the precise reference you want, that's why I only commented - you could of course reference Shafarevich's book and the Kedlaya paper both and argue that the proof in Shafarevich only relies on the field being infinite, not algebraically closed. Or, simply include the proof for infinite fields. – Jesko Hüttenhain Jul 01 '17 at 08:30

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