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.