Let $D$ be a Cartier divisor of a variety $X/K$ with associated line bundle $\mathcal{O}(D)$ and meromorphic section $s_D$. How do you define $s_D(P) \in K$ for $P \in X(K) \setminus \mathrm{supp}(D^{-1})$?

Perhaps one has to choose an embedding $\mathcal{O}(D) \hookrightarrow \mathcal{M}_X$ into the meromorphic functions. But this is not unique? Edit: No, $\mathcal{O}(D) = \{f \in \mathcal{M}_X \mid div(f) \geq -D\} \cup \{0\}$.

  • $\begingroup$ In general you can't, because $s_D$ can have poles outside of the support of $D$ (which is also the support of $D^{-1}$). Take for instance $D=0$ and $s_D$ not regular. Moreover, even if $P$ is not a pole of $s_D$, $s_D(P)$ depends on the choice of a basis of $\mathcal O(D)_P$. $\endgroup$ – Cantlog Oct 1 '13 at 21:51

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