Elementary transformations and the rationality of the moduli spaces of vector bundles on \(\mathbb{P}^2\) (Q1424384)

Here the authors prove the rationality of the moduli space \(M(r,c_1,c_2)\) of rank \(r\) stable vector bundles on the projective plane with Chern classes \(c_1\) and \(c_2\) for a huge set of triples \((r,c_1,c_2)\). To obtain this result they construct a birational morphism between certain \(M(r,c_1,c_2)\) and \(M(r,d_1,d_2)\) with \((d_1,d_2) \neq (c_1,c_2)\).