Let's assume that we have three affine lines in general position. Every algebraic vector bundle on an affine line is trivial. It is easy to show that every vector bundle on two intersecting lines are also trivial. But when it comes to three lines jumping around lines in a loop can give us a non-trivial element of $GL_n(k[t])$ which happens to be also in $GL_n(k)$ (Because every two lines intersect at a point). Here $n$ is the rank of vector bundle and $k$ is the base field. So the category of vector bundles on three lines seems to be isomorphic to the category of vector bundles on an affine line with an automorphism which is in $GL_n(k)$. The triviality of vector bundle is equivalent for the automorphism to be identity.
Now consider three 2 dimensional planes in the general position (similar to xy,xz and yz planes). Is it true every algebraic vector bundle on this is trivial? Topologically this is true but I cannot figure out it for the algebraic case. It seems we can have non-trivial automorphisms when we jump between planes and I cannot see how such a thing can be trivial.
Consider four 2-dimensional plane in general position (so they form a pyramid). What are the algebraic vector bundles on this?
Is it possible to generalize this to higher dimensions?