Visualizing the line bundle associated to the sheaf $\mathcal{O}_{\mathbb{P}^1}(2)$ We have that the line bundle associated to the sheaf $\mathcal{O}_{\mathbb{P}_{\mathbb{C}}^1}(1)$ is given by:
$$
\mathbb{P}_{\mathbb{C}}^2\setminus\{(0:0:1)\} \rightarrow \mathbb{P}_{\mathbb{C}}^1, (x_0:x_1:x_2)\mapsto (x_0:x_1).
$$
This corresponds the projection from $(0:0:1)$ to the projective line $\mathbb{P}_{\mathbb{C}}^1 \simeq  V(x_2)\subset \mathbb{P}_{\mathbb{C}}^2$.
I was wondering If we can find such an explicit description for the line bundle associated to $\mathcal{O}_{\mathbb{P}^1_{\mathbb{C}}}(2)$. In other words, 
I would like to find a variety $L$ and a map $\pi:L\rightarrow \mathbb{P}_{\mathbb{C}}^1$ such that $(L,\pi)$ is a line bundle on $\mathbb{P}_{\mathbb{C}}^1$ and its sheaf of sections is $\mathcal{O}_{\mathbb{P}_{\mathbb{C}}^1}(2)$. Does the map $\pi$ have a geometric interpretation like before?
 A: At Nicolas' recommendation, I will open up Huybrechts and expand a bit (this is on p. 91). I will use $\mathcal O$ to denote the total space of the trivial line bundle, and the usual variations to denote the associated tensor bundles. We have the inclusion
$$
\mathcal O(-1) \subset \mathcal O^{\oplus n+1},
$$
where every fiber of the trivial bundle is the $\mathbb C^{n+1}$ of which we get $\mathbb P^n$ as a quotient, and the fiber of $\mathcal O(-1)$ over $[\ell] \in \mathbb P^n$ is $\ell \subset \mathbb C^{n+1}$. Since any inclusion of bundles induces an inclusion of tensor powers, we have
$$
\mathcal O(-k) \subset \mathcal O^{\oplus k(n+1)}.
$$
Now any $F \in \mathbb C[z_1,...,z_{n+1}]_k$ induces a holomorphic map $\tilde{F}: O^{\oplus k(n+1)} \to \mathbb C$ which is linear on all fibers of $\mathcal O^{\oplus k(n+1)} \to \mathbb P^n$ (hence the same thing as an $\mathcal O$-linear map to $\mathcal O$). Restricting $\tilde{F}$ to $\mathcal O(-k)$ thus produces a holomorphic section of $\mathcal O(k)$ by definition: recall that $\mathcal O(k) = \mathcal Hom(\mathcal O(-k),\mathcal O)$. There are a bunch of details I'm leaving out (in particular you have to use power series to prove that the homogeneous holomorphic functions you obtain are in fact algebraic), but this hopefully conveys some of the geometry you were looking for.
