Tautological Line Bundle coincides with Invertible Sheaf $\mathcal{O}_{\mathbb{P}_n}(-1)$

Question

Let consider the complex projective space in two ways: Topologically by setting $$\mathbb{P}_n =\mathbb{P}(\mathbb{C}^{n+1}) = \mathbb{C}^{n+1}/{\sim}$$

as space of complex lines and via

$$\mathbb{P}^n = \operatorname{Proj}(\mathbb{C}[T_0, T_1, \ldots, T_n])$$

as $\mathbb{C}$-scheme.

We have the (tautological) invertible sheaf $\mathcal{O}_{\mathbb{P}_n}(1)$ defined locally by $\mathcal{O}_{\mathbb{P}_n}(1) \vert _{D_+(T_i)} = \operatorname{Spec}(\mathbb{C}[T_0, T_1, \ldots, T_n]_{(T_i),1})= \operatorname{Spec}(\mathbb{C}[T_0, T_1, \ldots, T_n])$ where $\mathbb{C}[T_0, T_1, \ldots, T_n]_{(T_i),1}$ means the $1$-graded part of canonically graded ring $\bigoplus _i \mathbb{C}[T_0, T_1, \ldots, T_n]_{(T_i),i}$. Let $\mathcal{O}_{\mathbb{P}_n}(-1)$ be the dual sheaf of $\mathcal{O}_{\mathbb{P}_n}(1)$.

It's easy to see that $\{(l,u)\in \mathbb{P}_n \times \mathbb{C}^{n+1} \vert v \in l\}$ is a line bundle over $\mathbb{P}_n$.

My question is why holds

$$\mathcal{O}_{\mathbb{P}_n}(-1) = \{(l,u)\in \mathbb{P}_n \times \mathbb{C}^{n+1} \mid v \in l\}$$

Remark: I know that we can identify locally free sheaves with vector bundles if global sections of the structure sheaf are given as regular functions $U \to \mathbb{C}$, but I don't see how to conclude the desired equation.

Answer 1

Let $V$ be a vector space, and $\Bbb P(V) = X$ be the space of its lines. Write $\mathcal O_X(-1)'$ for the line bundle $\{(l,v) \in X \times V : v \in l\}$.

We will build an isomorphism $\mathcal O_X(-1)' \times \mathcal O_X(1) \to \mathcal O_X$. If $s'\in \Gamma(U,O(-1)')$ we remark that $s'$ is the same as an algebraic function $v : U \to V$ where $v(u) \in u$ for each $u \in U$. A section $s$ $\in \Gamma(U,\mathcal O(1))$ is just a regular section on $U$ and of degree $1$, say $\varphi$.

Warning : it does not define a function, e.g the section $x_0$ has no well defined value at $[1:0:0]$. There are two ways to avoid this problem : first, working with quotients of sections gives a well defined map (but this is not regular anymore), or you can also consider $\varphi$ as a regular function on $W = \{v \in V : [v] \in U\}$.

In particular we obtain an "evaluation map" $$\text{ev} : \mathcal O_X(-1) \otimes \mathcal O_X(1) \to \mathcal O_X, (v, \varphi) \mapsto \varphi(v) : (u \mapsto \varphi(v(u)))$$ which is easily checked to be an isomorphism.