# the image of $\text{Div}(X)\to \text{Pic}(X)$ is generated by those line bundles with $H^{0}(X,L) \ne 0$

Let $$X$$ be complex manifold, the image of $$\text{Div}(X)\to \text{Pic}(X)$$ is generated by those line bundles with $$H^{0}(X,L) \ne 0$$ ( i.e. it admits a global non trivial holomorphic section).

The proof for the image of a divisor $$D$$, gives a non trivial global section goes as follows:

Proof: Any divisor $$D=\sum a_i\left[Y_i^i\right] \in \operatorname{Div}(X)$$ can be written as $$D=\sum a_i^{+}\left[Y_i^{+}\right]-\sum a_j^{-}\left[Y_j^{-}\right]$$with $$a_k^{\pm} \geq 0$$. Hence, $$\mathcal{O}(D) \cong \mathcal{O}\left(\sum a_i^{+}\left[Y_i\right]\right) \otimes$$ $$\mathcal{O}\left(\sum a_j^{-}\left[Y_j\right]\right)^*$$ and the line bundles $$\mathcal{O}\left(\sum a_i^{+}\left[Y_i\right]\right)$$ and $$\mathcal{O}\left(\sum a_j^{-}\left[Y_j\right]\right)$$ are both associated to effective divisors and, therefore, admit non-trivial global sections.

My question is we have proved that $$\mathcal{O}\left(\sum a_j^{-}\left[Y_j\right]\right)$$ admits non trivial global section however $$\mathcal{O}(D)$$ tensors a dual bundle of it, it's not clear that the dual bundle admits non trivial global section?

• And we know if $X$ is compact , if both line bundle and its dual admits non trivial global section then the bundle should be trivial? Nov 20, 2022 at 12:33

It is enough to consider effective divisors. Write a divisor $$D$$ as $$D_1-D_2$$, with $$D_1,D_2$$ both effective, and let $$L_1:=\mathcal O(D_1)$$, $$L_2:=\mathcal O(D_2)$$ be the associated bundles. Then $$\mathcal O(D)=\mathcal O(D_1)\otimes \mathcal O(-D_2)= L_1\otimes L_2^* \in \langle L_1,L_2\rangle$$, with $$H^0(X,L_1)\neq 0$$ and $$H^0(X,L_2)\neq 0$$, because $$D_1,D_2$$ are effective. Therefore, the image of the map $$\mathrm{Div}(X)\to \mathrm{Pic}(X)$$ is generated by line bundle having global sections, i.e. arising from effective diviors.
For the question in the comment, it is also correct: if both $$L=\mathcal O(D)$$ and $$L^*= \mathcal O(-D)$$ have non trivial global section, there exist divisor $$A\in |D|$$ and $$B\in |-D|$$. But $$A+B$$ is a principal divisor, and $$A$$ and $$B$$ are bot effective, whence $$A=B=0$$.
• thank you @Desperado , how to prove that $\mathrm{Pic}(X)$ is generated by line bundle having global section? Nov 20, 2022 at 13:23
• Sorry, I meant the image of the map $Div(X) \to Pic (X)$. I edit my answer accordingly Nov 20, 2022 at 18:18
• A non effective divisor has no nontrivial global sections at all. But $L_1:=\mathcal O (\sum a_i^+[Y_i])$ and $L_2:=\mathcal O (\sum a_j^-[Y_j])$ are both effective and they do have nontrivial global sections. So $L=L_1\otimes L_2^*$ belongs to the subgroup of $\mathrm{Pic} (X)$ generated by $L_1$ and $L_2$, so in a subgroup generated by line bundles with nontrivial global sections. Nov 21, 2022 at 19:40
• thank you, Desperado, excellent solution! I didn't realize that as a subgroup $L_2$ will generate $L_2^*$. Nov 22, 2022 at 0:23