Let $X$ be a smooth complex projective curve. Let $D$ be any divisor and let $p$ be a point. Rick Miranda claims this sequence is exact on page 285 of his book on Riemann surfaces:
$$ 0 \to \mathcal O(D-p) \to \mathcal O(D) \xrightarrow{\text{eval}_p} \mathbb C_p \to 0 $$ where the last map is supposed to be evaluation at $p$ and $\mathbb C_p$ is the skyscraper sheaf of $\mathbb C$ at $p$.
- Why does the last map make sense? A meromorphic function in $\mathcal O(D)$ can have a pole at $p$.
- Perhaps one way to fix this is to realize that (Weil) divisors on a smooth curve are in fact locally principal, so we can find some neighborhood $U$ of $p$ with $D \mid_U = div(\phi)$. Then we can evaluate $f\phi$ (which is regular because its divisor is effective). But this depends on choice of $\phi$ so the map is no longer natural.