RATIONALITY OF MODULI SPACES OF TORSION-FREE SHEAVES OVER RATIONAL SURFACES

In this paper we show that for rational ruled surfaces many moduli spa ces of torsion free sheaves with given Chern classes are rational. We deal with the case that the first Chern class c(1) satisfies c(1).F = 0 for a fibre F of the ruling. The main tool are priority sheaves intr oduced by Hirschowitz-Laszlo, and Walter, which enable us to reduce th e problem to the construction of a. family of sheaves over a big enoug h rational base.