$\newcommand{\Pe}{\mathbb{P}} \newcommand{\oh}{\mathcal{O}} \newcommand{\F}{\mathcal{F}} \newcommand{\ra}{\rightarrow} $I know that for a global section $\sigma \in \Gamma(\Pe^n, \oh(-1)) $ this is equivalent to giving a map $\sigma : \oh \ra \oh(-1)$ (all structure sheaves here are $\oh := \oh_{\Pe^n}$). My first question is why is this? I know in general for a sheaf $\F$ of, say, abelian groups that $s \in \F(U)$ is an element of the abelian group $\F(U)$. But then why is $\sigma \in \oh(-1)(\Pe^n)$ (notice the notation change here for global sections) a map? Is it perhaps something to do with the fact that
$$ \oh(-1) = \oh(1)^\vee = \mathcal{H}om(\oh(1), \oh) $$
where $\mathcal{H}om$ is the sheaf Hom? Then I suppose
$$ \oh(-1)(\Pe^n) = \mathcal{H}om(\oh(1), \oh)(\Pe^n) = \mathcal{H}om(\oh(1)|_{\Pe^n}, \oh|_{\Pe^n}) $$
by the definition of the sheaf Hom, so an element of $\oh(-1)(\Pe^n)$ (i.e. a section $\sigma$) is a morphism $\oh(1)|_{\Pe^n} \ra \oh|_{\Pe^n}$ and dualising we get a morphism $\oh|_{\Pe^n} \ra \oh(-1)|_{\Pe^n}$ i.e. a morphism $\oh \ra \oh(-1)$, our $\sigma$ from the beginning of the question. Is this construction correct?
I also know that $\oh(n)$ is ample, so it's generated by global sections $x_0, \dots, x_r \in \Gamma(\Pe^r, \oh(n))$ where $\Gamma(\Pe^r, \oh(n))$ is a finitely generated $\mathbb{Z}$-module (?) and where $\Pe^r$ is the object in $\mathsf{Sch}$ that satisfies the representability of the functor
$$ \underline{\Pe}^r : \mathsf{Sch} \ra \mathsf{Set} $$
given by
$$ X \mapsto ( \phi : \bigoplus_{i=0}^r \oh_X \twoheadrightarrow \mathcal{L}) $$
where $\mathcal{L}$ is a line bundle (or $\Pe^r = \text{Proj} (\mathbb{Z}[x_0, \dots, x_r])$ via the Proj construction if you prefer; the reason why I outlined the functorial construction is because this section-morphism correspondence rather nicely relates to the definition of $\underline{\Pe}^r$ in my opinion). But then picking a $\sigma \in \Gamma(\Pe^r, \oh(n))$ is the same as specifying a morphism $\oh \ra \oh(n)$ since elements in modules are determined by where the identity (which I'm assuming corresponds to $\oh$ here) is sent, or something along those lines?
My final question would be whether this section-morphism correspondence holds for all sheaves $\F$ as opposed to just twists of the structure sheaf $\oh(n)$? I suspect it does; I can't see anything in my above reasoning (in the part before I talk about representability at least, if it's correct) which wouldn't work if $\F$ is arbitrary. Thank you.
Best Answer
Let $F$ be an $\mathcal{O}_X$ module. Given a global section $s\in F(X)$, one can define $\alpha: \mathcal{O}_X\to F$ as follows: \begin{equation} \alpha(U): g\mapsto g.s|_U \end{equation} Here I mean by $g.$ the action of $\mathcal{O}_X$ on $F$.
Conversely given a homomorphism $\alpha: \mathcal{O}_X\to F$, define \begin{equation} s=\alpha(X)(1)\in F(X) \end{equation} where $1\in \mathcal{O}_X(X)$.
You can check that these two constructions are inverse to each other. Note that $X$ does not need to be a projective space.