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?
Best Answer
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 $g(X)=0$