**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 33.44.12 we see that $0 \leq \deg (\mathcal{L}') \leq d$. If $\deg (\mathcal{L}') = 0$, then the same lemma tells us $\mathcal{L}' \cong \mathcal{O}_ X$ and since we have two linearly independent sections we find we are in case (2). If $\deg (\mathcal{L}') > 0$ then $\varphi $ is nonconstant (since the pullback of an invertible module by a constant morphism is trivial). Hence

\[ \deg (\mathcal{L}') = \deg (X/\mathbf{P}^1_ k) \deg (\mathcal{O}_{\mathbf{P}^1_ k}(1)) \]

by Varieties, Lemma 33.44.11. This finishes the proof as the degree of $\mathcal{O}_{\mathbf{P}^1_ k}(1)$ is $1$.
$\square$

## Comments (2)

Comment #5100 by Tongmu He on

Comment #5309 by Johan on