I'm reading Ravi Vakil's Notes on Foundations of Algebraic Geometry. I'm confused by the remark following 14.1.B on page 198 in Nov1817 edition.
In particular, he mentioned that $\mathscr{O}(n)$ on $\mathbb{P}_k^m$ does not depend on affine cover and then asked questions: where does the section $x_0^3 - x_0 x_1^2$ of $\mathscr{O}(3)$ vanish? Which section do you get when multiplynig $x_0 + x_1$ in $\mathscr{O}(1)$ with $x_0^2$ of $\mathscr{O}(2)$?
To these questions my answer was: (1) It vanishes at $[0: 1], [1: 1], [-1: 1] \in \mathbb{P}^1$; (2) We get the section $x_0^3 + x_1 x_0^2$. But I'm not sure why this demonstrate that the definition of $\mathscr{O}(n)$ does not depend on affine cover.
Thanks!