Why there is no meromorphic function of degree $d=1$ on any compact Riemann surface of positive genus?

Question

We have the Riemann-Hurwitz formula:

$$ 2g_X-2=d(2g_Y-2)+\sum_{x\in X}(e_x-1) $$

It is said that from this we can deduce that there is no meromorphic function of degree $d=1$ on any compact Riemann surface of positive genus.

I wonder how?

If I let $d=1$, I can get $$ 2(g_X-g_Y)=\sum_{x\in X}(e_x-1) $$

but what's next? Maybe I lack some knowledge about meromorphic function on Riemann surface, anyone can help?

Answer 1

Let $f:X\to \mathbb C$ a meromorphic function or equivalently a holomorphic function $f:X\to \mathbb P^1$ . Suppose d=1. Then f is bijective holomophic map and therefore a biholomorphic map. Follow that $g(X)=g(\mathbb P^1)$ because $g$ is a topological invariant.

another way:

Like you said : $$2( g(X)-g(\mathbb P^1))=\sum(e_x-1)$$

but $e_x=1$ for all x because $d=1$ and as we know $g(\mathbb P^1)=0$

Therefore $g(X)=0$