I am not sure so I'll ask just in case. Let $X$ be a scheme and $\mathcal F$ a sheaf of modules on $X$ which is globally generated by $\{s_i\}$. Do these sections also generate $\Gamma(X,\mathcal F)$ as a module ?
Is it true if we suppose that there are finitely many generators, or more ?
In fact I tried to show that a morphism $\phi\colon\mathbb P^n_K\to \mathbb P^m_K$ with $n>m$ is constant. I argued by contradiction using that $\phi^*(x_1),\dots,\phi^*(x_m)$ were generators of $\phi^*\mathcal O_{\mathbb P^m_K}(1)$ (in fact they are global generators by Hartshorne II.7.1) and said that $\Gamma(\mathbb P^n_K,\mathcal O_{\mathbb P^n_K}(d))$ had a too big dimension for $\mathcal O_{\mathbb P^n_K}(d)$ to be isomorphic to $\phi^*\mathcal O_{\mathbb P^m_K}(1)$. But this argument might not work since it relies on saying that the dimension of $\Gamma(\mathbb P^n_K,\phi^*\mathcal O_{\mathbb P^m_K}(1))$ is at most $m$.
Best Answer
No. Take, for example, the sheaf $\mathcal O(2)$ on projective line with coordinates $(x_0:x_1)$. It is globally generated by sections $x_0^ 2$ and $x_1^2$, but they do not generate the space of global sections.