If $C$ is a complex Riemann Surface with positive genus, $D$ a divisor on $C$, $L$ a line bundle of $C$, with the term $L(D)$ what do we mean?
I have this idea:
$L(D)$ set of all sections of $L$ with poles on $D$. So it makes sense to define the exact sequence
$$0 \rightarrow L(D) \rightarrow L \rightarrow L/L(D) \rightarrow 0$$ where $L/L(D)$ is the set of all sections of $L$ modulo those with poles on $D$.