A non-constant holomorphic map $F$ between riemann-surfaces is an isomorphism

Question

I want to show the following:

Let $F:X\rightarrow Y$ be a non-constant and holomorphic map between compact riemann surfaces with $genus(X)=genus(Y)\geq 2$.

In the above it holds that $F$ is an isomorphism.

I am pretty sure that I should use the formula of riemann-hurwitz which can be found here. But I wasnt able to proof the statement.

Hopefully someone can help me.

Answer 1

To make this disappear from the list of unanswered questions. The solution to this question is essentially given in the comments already.

Let $F\colon X \rightarrow Y$ be a non-constant holomorphic map between compact Riemann surfaces both of genus $g\geq 2$. The Riemann-Hurwitz formula yields $$(1-\operatorname{deg}(F))(2g-2)=\sum_{p \in X}(\operatorname{mult}_p(F)-1).$$ The right-hand side is non-negative, while, by assumption, the term $2g-2$ is strictly positive. Thus, we have $\operatorname{deg}(F)=1$ and $F$ is an isomorphism.