Tautological bundle: algebraic geometry vs topology

Question

I'm going to compare the two construction of twisted sheaf/bundle $\mathcal{O}(1)$ from algebraic and topological viewpoint:

1) Algebraic construction (Hartshorne's Algebraic Geometry, p. 117):

Definition. Let $S$ be a graded ring, and let $X = \operatorname{Proj} (S)$. For any $n \in \mathbb{Z}$, we define the sheaf $\mathcal{O}_X(n)$ to be $S(n)^{\sim}$. We call $\mathcal{O}_X(-1)$ the Tautological bundle.

For $X= \mathbb{P}^n$ we can $\mathcal{O}_X(l)$ also characterize by cycle condition: the twisted sheaf $\mathcal O_{P^n}(l)$ is fully determined by it. as $\mathcal O_{P^n}(l)$ are invertible the restrictions $\mathcal O_{P^n}(l) \vert _{U_i}$ to $U_i := D_+(X_i)=Proj(k[X_1,...,X_n])_{(T_i)}= Spec(k[X_1/X_i,...,X_n/X_i])$ are generated by certain regular sections $s_i \in O_{P^n}(l)(U_i)$. The cycle condition is noting but a family of $\phi_{ij} \in O_{P^n}(l)(U_i \cap U_j)^*$ such that $\phi_{ij} s_i = s_j$. for $l \in \mathbb{Z}$ the cycle is given by $\phi_{ij} = (\frac{X_i}{X_j})^l \in O_{P^n}(l)(U_i \cap U_j)^*$. Recall that $O_{P^n}(l)$ is uniqely determined by the the data $(\phi_{ij})_{ij}$ up to glocal section $a \in O_{P^n}(l)(X)^*$, i.e. $(\phi_{ij})_{ij}$ and $(a \cdot \phi_{ij})_{ij}$ determine the same line bundle $O_{P^n}(l)$ for every $a \in O_{P^n}(l)(X)^*$.

2) Topological construction:

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

Q: If we take $S= k[X_1,...,X_n]$ and thus $\operatorname{Proj}(S)= \mathbb{P}_k^n$, how can I connect these both constructions explicitly and understand that the tautological bundle in both constructions in "certain way" coincide with each other.

to be more precise: if we use the cycle condition for description of 1) for $l=1$, how the data $\phi_{ij} = (\frac{X_i}{X_j})^{-1} \in O_{P^n}(-1)(U_i \cap U_j)^*$ is reflected in topological version $L=\{(l,v) \in X \times V : v \in l\}$?

Assume $\operatorname{char}(k)=0$. Then GAGA theorems provide correspondence $\mathcal{F} \to \mathcal{F}^{an}$ that defines an exact functor from the category of sheaves over $ (X,\mathcal{O}_{X})$ to the category of sheaves of $ (X^{an},\mathcal{O}^{an}_{X})$. The bundles of $ (X^{an},\mathcal{O}^{an}_{X})$ are the "topological" bundles and therefore we obtain identification between $Pic(X) = \mathbb{Z}$ and line bundles over $X$. Thus formally we can establish such correspondence.

The motivation of this question is more focused on intuitive approach to understand why $\mathcal{O}_X(-1)$ and $L=\{(l,v) \in X \times V : v \in l\}$ by these correspondence are the "same" . Is there any geometric intuition which makes this identifiction plausible focused on how the cycle condition is "reflected" in topological pendant $L$? I would very thankful if somebody could take some time to explain how one have to think intuitively about this identification.