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.

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

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

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

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

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.

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