Moduli spaces of bundles with fixed determinant on algebraic curves are unirational and very often even rational. For moduli spaces of sheaves on algebraic surfaces the situation differs drastically and, from the point of view of birational geometry, discloses highly interesting features. Once again, the geometry of the surface and of the moduli spaces of sheaves on the surface are intimately related. For example, moduli spaces associated to rational surfaces are expected to be rational and, similarly, moduli spaces associated to minimal surfaces of general type should be of general type. We encountered phenomena of this sort already at various places (cf. Chapter 6).

There are essentially two techniques to obtain information about the birational geometry of moduli spaces. First, one aims for an explicit parametrization of an open subset of the moduli space by means of Serre correspondence, elementary transformation, etc. Second, one may approach the question via the positivity (negativity) of the canonical bundle of the moduli space. The first step was made in Section 8.3. The best result in this direction is due to Li saying that on a minimal surface of general type with a reduced canonical divisor the moduli spaces of rank two sheaves are of general type. This and similar results concerning the Kodaira dimension are presented in Section 11.1. The use of Serre correspondence for a birational description is illustrated by means of two examples in Section 11.3.