The Stacks project

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 either

  1. is nonconstant and has degree $\leq d$, or

  2. factors through a closed point of $\mathbf{P}^1_ k$ and in this case $H^0(X, \mathcal{O}_ X) \not= k$.

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

It seems that the morphism in 53.3.4 might be constant. Since might not be algebraically closed, even if the global sections of are linearly independent over , they might not be algebraically independent over . A possible consequence is that the morphism induced by these two sections might factor through a -rational point of where is a nontrivial field extension.

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