Rationality of moduli spaces of torsion free sheaves over rational surfaces

In this paper we show that for rational ruled surfaces many moduli spaces of torsion free sheaves with given Chern classes are rational. We deal with the case that the first Chern classc ₁ satisfiesc ₁.F=0 for a fibreF of the ruling. The main tool are priority sheaves introduced by Hirschowitz-Laszlo and Walter, which enable us to reduce the problem to the construction of a family of sheaves over a big enough rational base.