In this paper, it is proved that certain stable rank-3 vector bundles can be written as extensions of line bundles and stable rank-2 bundles. As an application, we show the rationality of certain moduli spaces of stable rank-3 bundles over the projective plane P 2 .

1 Moduli problems, spaces, and stacks. Vector bundles and K-theory 3 1.1 Some category theory . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Back to moduli spaces . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The way out of the problem . . . . . . . . . . . . . . . . . . . 7 1.4 Algebraic stacks and moduli of vector bundles . . . . . . . . . 7 1.5 K-theory of schemes . . . . . . . . . . ...

We describe the category of equivariant vector bundles on a smooth (partial) group completion of an adjoint simple algebraic group. As a corollary of our description, we prove that every equivariant vector bundle of rank less than or equal to rkG on the canonical (wonderful) completion splits into a direct sum of line bundles.

We describe the category of equivariant vector bundles on a smooth (partial) group completion of an adjoint simple algebraic group. As a corollary of our description, we prove that every equivariant vector bundle of rank less than or equal to rkG on the canonical (wonderful) completion splits into a direct sum of line bundles.

We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of certain moduli space of stable bundles on a rational ruled surface.