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This is the question.

Sorry for asking a repeating question. My problem is just the same as this one. But I don`t understand the answer.

The answer states"f is uniquely determined by the branch locus." whether it's for a fixed curve X? If so I can understand. However I think it means "X and f are determined uniquely",right? Then I'm confused.

If there are two curves. The morphism determined by canonical divisor K gives the same points in P^1. How can I say they're isomorphic?

I'm really confused.Any help is appreciated.By the way,it`s my first time to raise a question here.Please tell me if I do something wrong…

HELPQAQ
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community May 14 '22 at 15:43
  • I don't understand where it is shown that $X,X_2$ are isomorphic if they come with two maps $f,f_2$ to $\Bbb{P}^1$ with the same ramification. If $k=\Bbb{C}$ then I can prove it using that the field extensions are Galois and that they are the field of algebraic functions with prescribed monodromy. – reuns May 14 '22 at 16:00
  • @reuns Yes it's what I'm confused.(Thank you for your reply. – HELPQAQ May 14 '22 at 16:08
  • If someone has the answer perhaps they should post it in the linked question instead of here – reuns May 14 '22 at 16:08
  • @reuns The linked question has been solved. I thought nobody will continue to answer it so I asked it here.I don't understand "field of algebraic functions with prescribed monodromy". Can you explane it?Do you mean the measure at the last of the linked question? I think It's right although it's not my question. – HELPQAQ May 14 '22 at 16:24

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