The Stacks project

Lemma 33.44.12. Let $k$ be a field. Let $X$ be a proper curve over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module.

  1. If $\mathcal{L}$ has a nonzero section, then $\deg (\mathcal{L}) \geq 0$.

  2. If $\mathcal{L}$ has a nonzero section $s$ which vanishes at a point, then $\deg (\mathcal{L}) > 0$.

  3. If $\mathcal{L}$ and $\mathcal{L}^{-1}$ have nonzero sections, then $\mathcal{L} \cong \mathcal{O}_ X$.

  4. If $\deg (\mathcal{L}) \leq 0$ and $\mathcal{L}$ has a nonzero section, then $\mathcal{L} \cong \mathcal{O}_ X$.

  5. If $\mathcal{N} \to \mathcal{L}$ is a nonzero map of invertible $\mathcal{O}_ X$-modules, then $\deg (\mathcal{L}) \geq \deg (\mathcal{N})$ and if equality holds then it is an isomorphism.

Proof. Let $s$ be a nonzero section of $\mathcal{L}$. Since $X$ is a curve, we see that $s$ is a regular section. Hence there is an effective Cartier divisor $D \subset X$ and an isomorphism $\mathcal{L} \to \mathcal{O}_ X(D)$ mapping $s$ the canonical section $1$ of $\mathcal{O}_ X(D)$, see Divisors, Lemma 31.14.10. Then $\deg (\mathcal{L}) = \deg (D)$ by Lemma 33.44.9. As $\deg (D) \geq 0$ and $= 0$ if and only if $D = \emptyset $, this proves (1) and (2). In case (3) we see that $\deg (\mathcal{L}) = 0$ and $D = \emptyset $. Similarly for (4). To see (5) apply (1) and (4) to the invertible sheaf

\[ \mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{N}^{\otimes -1} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{N}, \mathcal{L}) \]

which has degree $\deg (\mathcal{L}) - \deg (\mathcal{N})$ by Lemma 33.44.7. $\square$

Comments (2)

Comment #8769 by Paola on

In order to prove (5), if you apply (4) as it is stated, I think you only get that and are isomorphic, not that the given morphism is an isomorphism. However, from the proof of (4) one sees that the following stronger claim holds: if and is a nonzero section of , then the morphism mapping to is an isomorphism. Am I right?

There are also:

  • 4 comment(s) on Section 33.44: Degrees on curves

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B40. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 33.44.12 as pdf