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?