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?
• That expression for the degree of the ramification divisor is true whenever $\varphi$ is separable. – Alex Youcis Apr 8 at 8:40