What does the notation $\mathcal{O}_{\mathbb{P}^n}(1)$ mean?

Question

I have tried looking at my sheaves notes but couldn't find anything.

Answer 1

For details a reference is the book of Hartshorne "Algebraic Geometry", page 116. If $S$ is a graded ring, then a graded $S$-module $M$ gives a sheaf $\tilde{M}$ on $Proj(S)$. On the basic open, $\tilde{M}(D(f)) \cong \tilde{M_f}$.

Now, it's only pure algebra.

There is a general construction with $S$ a graded ring, and $M$ a graded $S-$module. We can define the "twisted module" $M(k)$ as $M$ with modified grading : $M(k)_n = M_{n+k}$.

Now, you can define $\mathcal O_{\mathbb P^n}(1) = \tilde{S(1)}$.

For more details about the sheaves $\mathcal O(n)$, you can take a look this other question.