On a projective scheme, every line bundle has a regular meromorphic section.

Lemma 31.15.12. Let $X$ be a Noetherian scheme which has an ample invertible sheaf. Then every invertible $\mathcal{O}_ X$-module is isomorphic to

$\mathcal{O}_ X(D - D') = \mathcal{O}_ X(D) \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(D')^{\otimes -1}$

for some effective Cartier divisors $D, D'$ in $X$. Moreover, given a finite subset $E \subset X$ we may choose $D, D'$ such that $E \cap D = \emptyset$ and $E \cap D' = \emptyset$. If $X$ is quasi-affine, then we may choose $D' = \emptyset$.

Proof. Let $x_1, \ldots , x_ n$ be the associated points of $X$ (Lemma 31.2.5).

If $X$ is quasi-affine and $\mathcal{N}$ is any invertible $\mathcal{O}_ X$-module, then we can pick a section $t$ of $\mathcal{N}$ which does not vanish at any of the points of $E \cup \{ x_1, \ldots , x_ n\}$, see Properties, Lemma 28.29.7. Then $t$ is a regular section of $\mathcal{N}$ by Lemma 31.15.1. Hence $\mathcal{N} \cong \mathcal{O}_ X(D)$ where $D = Z(t)$ is the effective Cartier divisor corresponding to $t$, see Lemma 31.14.10. Since $E \cap D = \emptyset$ by construction we are done in this case.

Returning to the general case, let $\mathcal{L}$ be an ample invertible sheaf on $X$. There exists an $n > 0$ and a section $s \in \Gamma (X, \mathcal{L}^{\otimes n})$ such that $X_ s$ is affine and such that $E \cup \{ x_1, \ldots , x_ n\} \subset X_ s$ (Properties, Lemma 28.29.6).

Let $\mathcal{N}$ be an arbitrary invertible $\mathcal{O}_ X$-module. By the quasi-affine case, we can find a section $t \in \mathcal{N}(X_ s)$ which does not vanish at any point of $E \cup \{ x_1, \ldots , x_ n\}$. By Properties, Lemma 28.17.2 we see that for some $e \geq 0$ the section $s^ e|_{X_ s} t$ extends to a global section $\tau$ of $\mathcal{L}^{\otimes e} \otimes \mathcal{N}$. Thus both $\mathcal{L}^{\otimes e} \otimes \mathcal{N}$ and $\mathcal{L}^{\otimes e}$ are invertible sheaves which have global sections which do not vanish at any point of $E \cup \{ x_1, \ldots , x_ n\}$. Thus these are regular sections by Lemma 31.15.1. Hence $\mathcal{L}^{\otimes e} \otimes \mathcal{N} \cong \mathcal{O}_ X(D)$ and $\mathcal{L}^{\otimes e} \cong \mathcal{O}_ X(D')$ for some effective Cartier divisors $D$ and $D'$, see Lemma 31.14.10. By construction $E \cap D = \emptyset$ and $E \cap D' = \emptyset$ and the proof is complete. $\square$

Comment #2599 by Rogier Brussee on

Suggested slogan: On a projective scheme, every line bundle has a meromorphic section.

Comment #3984 by Kestutis Cesnavicius on

It would be useful to mention in the statement that $D$ and $D'$ may be chosen to not pass through any of finitely many prespecified points of $X$, and also that one may choose $D' = 0$ when $X$ is quasi-affine (in this case one may choose $\mathcal{L} = \mathcal{O}_X$).

Comment #3985 by Laurent Moret-Bailly on

$\mathcal{N}$ is not defined.

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