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.