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Let $k$ be an algebraically closed field, and let $X$ be a projective scheme, and let $\phi \colon X \to \mathbb{P}^n_k$ be a morphism and let $\mathscr{F}$ be the Serre's twisting sheaf $\mathcal{O}(1)$ on $\mathbb{P}^n_k$.

Then $L = \phi^* \mathscr{F}$ is an invertible sheaf on $X$ and $\Gamma (X,L)$ is a finite dimensional k-vector space. Let $\{t_0, \cdots , t_m \}$ be a basis of $\Gamma (X,L)$.

My Question:

Do global sections $\{t_0, \cdots , t_m \}$ generate the $\mathcal{O}_X$-module $L$ ?
Can we define the morphism $\varphi \colon X \to \mathbb{P}^m_k$ associated to $\{t_0, \cdots , t_m \}$?

Thank you.

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    $\begingroup$ $\mathcal{O}(1)$ on the projective space is generated by global sections, say $u_0,\ldots, u_n$, then $\phi^*u_i\in\Gamma(X,L)$ generate $L$ and so $t_i$s generate it too. $\endgroup$ – Mohan May 29 at 12:57
  • $\begingroup$ I don'r know why " if global sections $\{ u_0, \cdots , u_n \}$ generate $\mathcal{O}(1)$, then $\phi^* u_i \in \Gamma(X, L)$ generate $L$ " implies this. $\endgroup$ – Inoue May 29 at 20:01
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    $\begingroup$ Global generation is given by the surjection $\Gamma(\mathbb P^m, \mathcal O(1)) \otimes \mathcal O \to \mathcal O(1) \to 0$. Since pullback preserves right exactness, one obtains the corresponding surjection $\Gamma(X,L)\otimes \mathcal O_X \to L \to 0$. $\endgroup$ – Tabes Bridges May 29 at 20:09
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    $\begingroup$ If $\phi^*u_i$s generate $L$, since they are linear combinations of $t_i$, $t_i$s must also generate $L$. $\endgroup$ – Mohan May 29 at 21:12

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