Recently I am reading Griffiths' Principle of Algebraic Geometry. On page 136 he mentioned this:
Now suppose $M$ is compact. For every $D' \in |D|$ there exists $f \in \mathfrak{L}(D)$ such that $$ D' = D + (f) $$ and conversely any two such functions $f,f'$ differ by a nonzero constant.
The result seens reasonable on a compact mfd but I fail to show it. What I know is that on a cpt mfd any divisor is globally finite and $D'-D$ is a principal divisor given by a meromorphic function $f$ on $M$ which means the line bundle of $D'-D$ is a trivial bundle. Moreover $ord_{Y_i}f = b_i - a_i$ if we assume $D = \sum_{i=1}^N a_i Y_i, D = \sum_{i=1}^N b_i Y_i$. But if $f \in \mathfrak{L}(D)$, we will conclude that $b_i \geq 0$ by definition and hence $D'$ is principal. Where is the problem?