I'm not really entirely sure how to think about Serre's twisting sheaves $\mathscr{O}(i)$ - on any $\text{Proj}$ construction, really, but let's stick to something like $\mathbb{P}_2$ for now for simplicity. I know how to construct their sections and transitions functions on the usual open cover, and their global sections as homogenous polynomials, I have a hazy sense of the geometric intuition behind them via the Picard group structure and the very ample $\mathscr{O}(1)$-pullback condition, but I feel like I don't really understand them at all, to be honest. For example, $\mathscr{O}(1)$ is isomorphic as a line bundle hyperplane divisors (which I'm assuming means $\mathscr{O}(H)$ for some hyperplane $H$), according to Wikipedia, but is there any relationship between $\mathscr{O}(i)$ and other divisors? Does a divisor of degree $i$ have line bundle isomorphic to $\mathscr{O}(i)$?

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    $\begingroup$ What do you mean by divisor sheaves? $\mathscr{O}(i)$ is the collection of degree $i$ hypersurfaces from the point of view of divisors. $\endgroup$ – user149792 Jun 3 '15 at 11:07
  • $\begingroup$ This is true for projective space, since $\text{Pic}(\mathbb{P}_n)\cong\mathbb{Z}$, but it's not clear that it holds over a general $\text{Proj}$ construction. I guess you could identify the "$i$-times hyperplane divisor" canonically with $\mathscr{O}(i)$, but I'm unclear what the class represented by this would look like if $S$ were more general object - the prior assumption I think doesn't even hold for coordinate rings of projective varieties over an algebraically closed field in general, let alone on a more general graded ring, or a coherent sheaf of graded $\mathscr{O}_X$-algebras. $\endgroup$ – Peter Xu Jun 3 '15 at 14:42
  • $\begingroup$ For example, if we have an inclusion of varieties $\phi: C_1\to C_2$ and hence a quotient of graded coordinate rings $\phi^*:k_2\to k_1$, is $\mathscr{O}_{C_1}(i)$ the pullback of $\mathscr{O}_{C_2}(i)$ under $\phi$? This would solve the problem in the first case (for projective varieties over algebraically closed fields), and would imply that the class of every very ample divisor over an abstract algebraic variety could be seen as $\mathscr{O}(1)$ with the right coordinatization. What would this construction with the Veronese embedding look like? $\endgroup$ – Peter Xu Jun 3 '15 at 15:23

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