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…