We continue previous works by various authors and study the birationalgeometry of moduli spaces of stable rank-two vector bundles on surfaces withKodaira dimension $-\infty$. To this end, we express vector bundles as naturalextensions, by using two numerical invariants associated to vector bundles,similar to the invariants defined by Brinzanescu and Stoia in the case ofminimal surfaces. We compute explicitly these natural extensions on blowups ofgeneral points on a minimal surface. In the case of rational surfaces, we provethat any irreducible component of a moduli space is either rational or stablyrational.
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