Existence of finite morphism to projective line

Question

As we all know, Belyi's theorem says: A complex curve $X$ is defined over a number field, if and only if there exists a finite morphism $t:X\to \mathbb{P}^1_\mathbb{C}$ of varieties over $\mathbb{C}$ with at most $3$ critical points. (I borrow the statement in this paper [1]) Throughout the paper, I cannot find the proof of the existence of finite morphism $t:X\to \mathbb{P}^1_\mathbb{C}$. Is it clear? I think that I missed something. Anyone who can give me a brief reason, it would be very helpful to me.

Answer 1

1) Choose a non-empty affine open subset $U\subset X$.
Then any regular non-constant function $t_0\in \mathcal O(U)$ extends to a rational function $f\in Rat(X)$ which gives you a finite morphism $t:X\to \mathbb P^1_\mathbb C$.

2) If you know Riemann-Roch, here is another proof: choose a point $x\in X$ and consider the divisor $D=(g+1)\cdot x$ ($g=$ genus of $X$) and its associated line bundle $\mathcal O(D)$.
Then by Riemann-Roch $ h^0(X,\mathcal O(D))=h^1(X,\mathcal O(D))+1-g+(g+1)\geq 2$.
Hence there exists a non constant $t\in H^0(X,\mathcal O(D))\subset Rat(X)$ and that $t$ will give you the desired finite morphism $t:X\to \mathbb P^1_\mathbb C$.