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