Non-principle divisor of degree $0$ on Riemann surface

Question

For a Riemann surface, I want to know when there exists a non-principle divisor of degree $0$. In another word, when do we have $\text{Pic}^0=0$?

Thanks!

Answer 1

If $\mathrm{Pic}^0=0$ for a Riemann surface $X$ then any divisor of $X$ degree zero is principal. In particular, let $a,b \in X$ be distinct, $a-b$ has divisor zero so there is a meromorphic function $f$ on $X$ of which $a$ is the unique simple root and $b$ is the unique simple pole.

But it follows that $f: X \rightarrow \mathbb{P}^1$ has degree one, so is injective, continuous and open so a biholomrphism. Thus $X \cong \mathbb{P}^1$.