I have a very dumb question. Let $X = \mathbb{P}^2_k = Proj(k[x,y,z])$ where $k$ is algebraically closed. We have an invertible sheaf $\mathcal{O}(2)$ on $X$. Its space of global sections contains the elements $x^2, y^2, z^2, xy, yz, xz$.
It seems to me (by my calculations), however, that $\mathcal{O}(2)$ is generated by $x^2, y^2$, and $z^2$. Meaning, these 3 global sections generate the stalks at each point of $X$. I'm suspicious, though. Is this true?
David