# Maps between riemann surfaces of degree 2

Suppose that $π : X → Y$ is a map of degree d between Riemann surfaces.
(a) For any point $p$ of $X$, explain why the ramification index $k_{p}$ cannot be bigger than $d$, i.e., that $k_{p} \leq d$.
(b) If $d = 2$ (a double cover) explain why a point $p$ of $X$ is either not a ramification point, or has ramification index exactly $2$.
(c) Again in the case that $d = 2$ explain why the number of branch points (on $Y$ ) is the same as the number of ramification points (on $X$).

For (a) $d=\sum k_{p_j}$, where $k_p$ is the ramification index of $p \in X$ so neither of the ramification indices can be greater than the degree. As a consequence of this, with $d=2$ the ramification index of a point can either be $1$ (in which case it is not a ramification point) or the ramification index could be $2$. I'm confused about part (c), is it because any ramification point in $X$ will have ramification index $2$ and image of a ramification point is a branch point?

• I think you're just supposed to show why two ramification points cannot land on the same branch point. Apr 3, 2017 at 21:33
• That makes sense! Thanks @ElizabethS.Q.Goodman Apr 3, 2017 at 21:49

Let $$\pi\colon X \rightarrow Y$$ is a proper non-constant holomorphic map between Riemann surfaces of degree $$d\leq 3$$. Denote the set of ramification points and branch points of $$\pi$$ by $$R$$ and $$B$$, respectively. Then the function $$R \rightarrow B;x\mapsto \pi(x)$$ is bijective.
It is clearly surjective. Now, assume for a contradiction that there exist distinct $$x_1,x_2\in R$$ such that $$\pi(x_1)=\pi(x_2)$$. Then $$3\geq\sum_{p\in\pi^{-1}(\pi(x_1))}\operatorname{mult}_p(\pi)\geq 4$$, which is absurd.