Let $L(D)$ denote the space of meromorphic functions $f$ on $M$ (complex manifold), s.t. $D+(f)\ge0$. Then we can find a global meromorphic section of $\mathcal{O}(D)$ $s_0$ with $(s_0)=D$. Then we have the identification $L(D)\xrightarrow{\otimes s_0} H^0(M,\mathcal{O}(D))$.
On Griffiths, Harris p.138, it says that if $D=\sum a_iV_i$ is an effective divisor, more generally, if $E$ is any holomorphic vector bundle on $M$, $\mathcal{E}$ its sheaf of holomorphic sections, $\mathcal{E}(D)$ for the sheaf of meromorphic sections of $E$ with poles of order $\le a_i$ on $V_i$. Again, tensoring with $s_0$ gives identification $\mathcal{E}\xrightarrow{\otimes s_0}\mathcal{O}(E\otimes [D])$. (Here $[D]$ is the line bundle of $D$)
Now I wonder why is the condition effective necessary? In particular, in $L(D)\xrightarrow{\otimes s_0} H^0(M,\mathcal{O}(D))$, we don't require $D$ to be effective.
And in particular, take $E$ to be the trivial line bundle, I think I can conclude that to any divisor $D$, (may not be effective) $\mathcal{O}(D)$ can be identified as the sheaf of meromorphic functions s.t. $D+(f)\ge 0$ on $V_i$, is that right?
Moreover, we know when $D$ is effective we have the short exact sequence:
$0\to \mathcal{O}(-D)\to \mathcal{O}_X\to \mathcal{O}_D\to 0$
I wonder why we require $D$ to be effective? Since to any divisor $D$, $\mathcal{O}(-D)$ can be identified as the sheaf of meromorphic functions s.t. $-D+(f)\ge 0$ on $V_i$, and $\mathcal{O}_D$ is the cokernel sheaf.
Best Answer
The hypothesis that the divisor be effective is used in giving the interpretation that the sections have poles of order $\le a_i$ on $V_i$. If you modify the "description" to say that when $a_i<0$, we require a zero of order $\ge |a_i|$ along $V_i$, then it works fine.
In the case of your final short exact sequence, you need $D$ effective here because when $V$ is a hypersurface, there is no sheaf injection $\mathscr O_X(V) \to \mathscr O_X$. (Locally, a meromorphic function with a pole along $V$ is not a holomorphic function.) Also, we interpret $\mathscr O_V$ as the structure sheaf of the hypersurface $V$, and I've never seen this in the context of the structure sheaf of the "formal" $-V$.