I would like to know a classification for all holomorphic vector bundles on the second Hirzebruch surface $\mathbb{F}_2$. Is this known? What is known? In particular, I'm looking for holomorphic vector bundles trivial on some fixed line, but I'll take any information I can get.
For example, it is known that on $\mathbb{CP}^2$, all holomorphic vector bundles trivial on some line comes from a monad. Is there a nice description for $\mathbb{F}_2$?
I'll also say that $\mathbb{F}_2$ is a ruled surface. It is also a rational surface (birationally equivalent to $\mathbb{P}^2$) so maybe there's something we can say about the moduli space of holomorphic vector bundles (may or may not be fixed on some line) on $\mathbb{F}_2$ thinking of $\mathbb{F}_2$ as one of these objects?
Thanks for the help.