Lemma 53.10.1. Let $X$ be a proper curve over a field $k$ with $H^0(X, \mathcal{O}_ X) = k$. If $X$ has genus $0$, then every invertible $\mathcal{O}_ X$-module $\mathcal{L}$ of degree $0$ is trivial.

**Proof.**
Namely, we have $\dim _ k H^0(X, \mathcal{L}) \geq 0 + 1 - 0 = 1$ by Riemann-Roch (Lemma 53.5.2), hence $\mathcal{L}$ has a nonzero section, hence $\mathcal{L} \cong \mathcal{O}_ X$ by Varieties, Lemma 33.44.12.
$\square$

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