I am reading about Hurwitz's theorem

``Let $\varphi : C' \to C$ be a surjective morphism of irreducible smooth projective curves over $k$ of genus $g(C')$ and $g(C)$. Then we have $$2g(C') - 2 = \deg (\varphi ) (2g(C) - 2) + \deg (R_{\varphi}) $$ where $R_{\varphi}$ is the ramification divisor of $\varphi$.''

However, I haven't yet understood really about the ramification divisor and I had some difficulties to determinate $R_{\varphi}$. In particular, if $C' \equiv C \equiv \mathbb{P^1}$ then $R_{\varphi} = \sum\limits_{x \in C'} (e_x -1)$. Is this true?

My question: How is the dependence of the ramification divisor of surjective morphism to its ramification indexes?

Thank you very much!

  • 2
    $\begingroup$ I think that the discussion on this point in Hartshorne is very nice. Have a look. $\endgroup$ – aginensky Apr 4 at 15:44
  • 1
    $\begingroup$ That expression for the degree of the ramification divisor is true whenever $\varphi$ is separable. $\endgroup$ – Alex Youcis Apr 8 at 8:40

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