Does the global section of a (quasi-coherent) sheaf always exist?

Question

I just come up with this question while studying Vakil's $The\space Rising\space Sea$. There are quite a lot of places discussing (quasi-coherent) sheaves of $\mathcal{O}_X$-modules. And I would like to ask is there always a nontrivial global section of such sheaves?

Answer 1

No, take for example $X = \mathbb{P}^n$ and consider $\mathcal{O}_{\mathbb{P}^n}(m)$ for some negative integer $m$. These coherent $\mathcal{O}_{\mathbb{P}^n}$-modules have trivial global sections.