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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.

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  • $\begingroup$ I waited for two days to see whether someone else understood your question better. What is a monad for you? With the usual definition, any vector bundle on $\mathbb{P}^3$ comes from a monad and in 2-space, from a trivial monad, since they all have a free' resolution of length one, with no hypothesis on restrictions. So, what would be nice' description for you? $\endgroup$ – Mohan Aug 24 '15 at 1:31
  • $\begingroup$ I guess I'm looking for any description. Yeah, so I guess any description of holomorphic vector bundles on $\mathbb{F}_2$. If there exists a description in terms of monads, even better. And then the simpler the pieces of the monads can be, the better. So if the terms are just trivial bundles and the maps between them are nice and simple, that is the best. Does that help? Sorry if I'm not explaining myself well. $\endgroup$ – user46348 Aug 24 '15 at 20:39
  • $\begingroup$ Maybe the article "Monads for framed sheaves on Hirzebruch surfaces" is a good place to start? You can find it at arxiv.org/abs/1205.3613 $\endgroup$ – Bernie Aug 25 '15 at 14:26
  • $\begingroup$ Thanks @Bernie! This is a great resource. $\endgroup$ – user46348 Aug 25 '15 at 17:05

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