Let $X$ be a scheme and $D$ be a Cartier divisor on $X$. Then $D$ determines a line bundle $\mathcal{O}(D)$ on $X$. Under which condition, is the converse true? That is, when does a line bundle come from a Cartier divisor. This is equivalen to saying when does a line bundle have a meromorphic section?

I know that when $X$ is a non-projective manifold line bundles do not have sections in general.

  • $\begingroup$ Look at Hartshorne chapter II section 6. It is all about this. If $X$ is an integral scheme, then the Cartier class divisor group is isomorphic to the group of line bundles (up to isomorphism), i.e. $CaCl(X)\to Pic(X)$ is an isomorphism which maps $D\mapsto \mathcal{O}(D)$. I don't think it's equivalent to having a section, because Cartier divisors are locally principal and hence don't have to be given by the zero locus of a global section. $\endgroup$ – Matt Oct 11 '12 at 21:33
  • $\begingroup$ See arxiv.org/pdf/math/0001104 $\endgroup$ – Andrew Oct 11 '12 at 21:35
  • $\begingroup$ @Matt: he/she meant rational sections (see the title). $\endgroup$ – user18119 Oct 11 '12 at 21:37
  • $\begingroup$ Matt If your line bundle is given by a Cartier divisor, then take a trivial section on one local trivialization and extends to a rational section of the line bundle. On the other hand, if there is a rational, its zero and poles gives a divisor which gives back the original line bundle to you. $\endgroup$ – M. K. Oct 12 '12 at 4:56

The map you describe $Cacl(X)\to Pic(X)$ sending the linear equivalence class $[D]$ of a Cartier divisor $D$ to the line bundle $\mathcal O(D)$ is always injective.
It is very often surjective: it is the case if $X$ is integral or if $X$ is projective over a field.
However Kleiman has given a complicated example of a complete non-projective 3-dimensional irreducible scheme on which there is a line bundle not having any non-zero rational section and thus not coming from a Cartier divisor.
The scheme $X$ is obtained from Hironaka's complete, integral, non-singular, non projective variety of dimension 3 (which is already a strange beast!) by adding nilpotents to the local ring of just one point.
The details can be found in Hartshorne's Ample Subvarieties of Algebraic Varieties , Chapter I, Example 1.3, page 9.

Here is a picture (in blue) of Hironaka's strange beast . The description is on page 185 of Shafarevich's book.

  • $\begingroup$ This is an old answer. But as Andrews have mentioned above, this construction by Kleiman does not work, however it can be modified to achieve a counterexample. $\endgroup$ – user40276 Jul 8 '16 at 11:25

The most general assertion I know is:

Any invertible sheaf on $X$ is isomorphic to a $O_X(D)$ when $X$ is locally noetherian and if the associated points of $X$ are contained in an affine open subset of $X$ (EGA IV.21.3.4).

The condition on the associated points is satisfied if for instance $X$ is quasi-projective over a noetherian ring (then any finite subset of $X$ is contained in an affine open subset), or if $X$ is noetherian and reduced.


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