Lemma 53.3.4. Let $k$ be a field. Let $X$ be a nonsingular proper curve over $k$. Let $(\mathcal{L}, V)$ be a $\mathfrak g^1_ d$ on $X$. Then the morphism $\varphi : X \to \mathbf{P}^1_ k$ of Lemma 53.3.2 has degree $\leq d$.

**Proof.**
By Lemma 53.3.2 we see that $\mathcal{L}' = \varphi ^*\mathcal{O}_{\mathbf{P}^1_ k}(1)$ has a nonzero map $\mathcal{L}' \to \mathcal{L}$. Hence by Varieties, Lemma 33.43.12 we see that $\deg (\mathcal{L}') \leq d$. On the other hand, we have

by Varieties, Lemma 33.43.11. This finishes the proof as the degree of $\mathcal{O}_{\mathbf{P}^1_ k}(1)$ is $1$. $\square$

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## Comments (1)

Comment #5100 by Tongmu He on