# Surjectivity of a map

Let $C$ be a curve and $D\in\mathrm{Div}(C)$. Suppose $W$ is the canonical divisor. Let $g\in L(D)$ be such that $g\not\in L(D-P)$ for all $P\in C$ and $P\leq W-D$. Then one can show (given in Algebraic Curves by Fulton) that the following map is injective: $$\phi\colon L(W-D)/L(0)\longrightarrow L(W)/L(D)$$ given by $\phi ([f]):=[fg]$, where $[\cdot]$ denotes the residue class. I want to know when the map $\phi$ is surjective i.e. an isomorphism.

Clearly $\phi$ is an isomorphism when $D=0$ or $D=W$. Are there any other possibilities?

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What do you mean by $P\leq W-D$? –  Brenin Nov 14 '13 at 21:39
@Brenin: $P$ can be thought of as a divisor: $1$ at the $P$th position and zero elsewhere. –  pritam Nov 15 '13 at 15:14