Lemma 53.3.2. Let $k$ be a field. Let $X$ be a nonsingular proper curve over $k$. Let $(\mathcal{L}, V)$ be a $\mathfrak g^ r_ d$ on $X$. Then there exists a morphism

$\varphi : X \longrightarrow \mathbf{P}^ r_ k = \text{Proj}(k[T_0, \ldots , T_ r])$

of varieties over $k$ and a map $\alpha : \varphi ^*\mathcal{O}_{\mathbf{P}^ r_ k}(1) \to \mathcal{L}$ such that $\varphi ^*T_0, \ldots , \varphi ^*T_ r$ are sent to a basis of $V$ by $\alpha$.

Proof. Let $s_0, \ldots , s_ r \in V$ be a $k$-basis. Since $X$ is nonsingular the image $\mathcal{L}' \subset \mathcal{L}$ of the map $s_0, \ldots , s_ r : \mathcal{O}_ X^{\oplus r + 1} \to \mathcal{L}$ is an invertible $\mathcal{O}_ X$-module for example by Divisors, Lemma 31.11.11. Then we use Constructions, Lemma 27.13.1 to get a morphism

$\varphi = \varphi _{(\mathcal{L}', (s_0, \ldots , s_ r))} : X \longrightarrow \mathbf{P}^ r_ k$

as in the statement of the lemma. $\square$

Comment #7489 by Hao Peng on

$\mathcal O_X^r$ should be replaced by $\mathcal O_X^{r+1}$

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