Let $X$ be a smooth variety over $\mathbb{C}$, and let $D$ be a divisor on $X$. What is the condition on $D$ so that we can speak of a canonical section $s$ on $H^0(X,D)$ such that $D$ is the zero locus of $s$?
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If I understand this correctly, the condition is for $\mathcal O(D)$ to be a globally generated sheaf of principal ideals. This is the case if and only if $D=-E$ is the negative of an effective cartier divisor $E$ which is locally given by a system $\{(U,s_U)\}_{U}$ with $s_U\in\mathcal O_X(U)$. In other words, $D$ is locally given by $\{(U,s_U^{-1})\}_{U}$. Note that since the $s_U\in \mathcal O_X(U)$ agree on intersections, you can glue them to a global section $s\in H^0(X,\mathcal O_X(-E)) \subseteq H^0(X,O_X)$. |
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