Definition (Divisor). A divisor is a formal sum of points on $C$, say

$$D=\sum _{P\in C}{n}_{p}P$$

where $n_p\in \mathbb{Z}$ and $n_p=0$ for all but finitely many $P\in C$.

We write $\mathrm{deg}D=\sum n_P$.

We say $D$ is effective (written $D\ge 0$) if $n_P\ge 0$ for all $P$.