I am trying to construct line bundles on varieties. I've seen two definitions that work, but I can't see how to connect them. The first is what seems to be the classical definition of a variety $L$ and a regular surjective projection $L\to X$ with an open cover $\{U_i\mid i\in I\}$ and isomorphisms $\varphi_i:\pi^{-1}(U_i)\to U_i\times k$ with the connecting properties that $\varphi(x)=(\pi(x),*)$ and that $\varphi_i:\pi^{-1}(x)\to\{x\}\times k$ is a linear isomorphism.
The second definition is to just consider the cover $\{U_i\mid i\in I\}$ with the transition functions $\rho_{i,j}:U_i\cap U_j\to k[\mathbb A^1]$.
It is easy to see why the second definition is true when the first one is. However, I don't know how to reconstruct the variety $L$ given the cover and the transition functions. In the case of manifolds, it is not that hard to reconstruct. But if I use the same construction I don't know how to ensure that the resulting object is indeed a variety.
Any ideas?
In case it is needed, I am mostly interested in smooth irreducible varieties over an algebraically closed field of characteristic $0$.