Let $X$ be a complex manifold, not necessarily compact. Suppose $H$ is an analytic variety of codimension one that cannot be written as the zero set of a global holomorphic function; in other words, when $H$ is viewed as a nonzero divisor on $X$ it is nonprincipal.
There is a well-known holomorphic line bundle associated to $H$ denoted by $[H]$; this is given by taking local data $f_{\alpha}\in\mathcal{O}(U_{\alpha})$ of $H$ and associating to it the holomorphic line bundle which has transition function $g_{\alpha\beta}=f_{\alpha}/f_{\beta}$. The line bundle determined is independent of the local data of $H$ chosen.
Now it's not hard to see that the local data determines a global holomorphic section $s$ of $[H]$, since $f_{\alpha}=g_{\alpha\beta}f_{\beta}$. So $s^{-1}(0)=H$ for some global section. However, if $s'$ is any global holomorphic section of $[H]$, then it can be shown that the line bundle associated the divisor ${s'}^{-1}(0)$ is $[H]$. But two divisors determine the same line bundle if and only if they differ by a principal divisor, so ${s'}^{-1}(0)=H+(g)$, where $(g)$ is the divisor determined by a meromorphic function $g$ on $X$. (Here "+" denotes addition of divisors.)
My question is the following: Does this mean that every nontrivial global holomorphic section of [H] must contain $H$ as part of its zero locus? This seems awfully rigid --- am I making a mistake somewhere in my logic? I am rather new to divisors and line bundles, so I wouldn't be surprised if so. Any verification/debunking of this would be greatly appreciated.