Stalk of $\mathcal{O}(1)_{\mathbb{P}^n}$

Question

Let us consider the projective space $\mathbb{P}^n$ over $\operatorname{Spec}(k)$, being $k$ a field, or even over a general ring $A$. Given a point $p\in\mathbb{P}^n$, what is the stalk of the twisted sheaf $\mathcal{O}_{\mathbb{P}^1}(1)$ at the point $p$? What is the geometric meaning of this stalk? And what happens in the case we consider the sheaf $\mathcal{O}_{\mathbb{P}^1}(-1)$? There exists any relationship between both?

Answer 1

Question: "What is the geometric meaning of this stalk? And what happens in the case we consider the sheaf $\mathcal{O}_{P^1}(−1)$? There exists any relationship between both?"

Answer: If $V:=k\{e_0,e_1\}$ and $V^*:=k\{x_0,x_1\}$ by definition

$$C:=\mathbb{P}^1_k:=Proj(Sym^*_k(V^*)):=Proj(k[x_0,x_1]).$$

There is a sequence of graded $k[x_0,x_1]$-modules

$$\phi^*: k[x_0,x_1]\otimes_k V^* \rightarrow k[x_0,x_1](1)$$

where $\phi(f_0,f_1):=f_0x_0+f_1x_1$. The sheafification gives the surjection

$$(*)\text{ } \phi: \mathcal{O}_C \otimes V^* \rightarrow \mathcal{O}_C(1).$$

The prime ideal $I:=(a_1x_0-a_0x_1) \subseteq S:=k[x_0,x_1]$ corresponds to the $k$-rational point $p:=(a_0:a_1)\in C(k)$. When you take the fiber of $(*)$ at the points $p$ you get the surjection

$$\phi(p): V^* \rightarrow k\{x_0\}:=l(p)\rightarrow 0$$

defined by $\phi(p)(x_0)=x_0, \phi(x_1)=\frac{a_1}{a_0}x_0$. Dualize and you get a line $l(p)^* \subseteq V^{**} \cong V$. This gives a 1-1 correspondence between $k$-rational points $p\in C(k)$ and lines $l(p)^* \subseteq V$. Hence the tautological sequence $(*)$ is related to the fact that $C$ "parametrize lines in $V$".