# k-basis of $\Gamma(X,\mathscr{F})$ generates the sheaf $\mathscr{F}$?

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.

• $\mathcal{O}(1)$ 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. – Mohan May 29 at 12:57
• 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. – Inoue May 29 at 20:01
• 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 $\Gamma(X,L)\otimes \mathcal O_X \to L \to 0$. – Tabes Bridges May 29 at 20:09
• If $\phi^*u_i$s generate $L$, since they are linear combinations of $t_i$, $t_i$s must also generate $L$. – Mohan May 29 at 21:12