Rationality and Poincaré Families for Vector Bundles with Extra Structure on a Curve

Iterated Grassmannian bundles over moduli stacks of vector bundles on a curve are shown to be birationally equivalent to an affine space times a moduli stack of degree 0 vector bundles, following the method of King and Schofield. Applications include the birational type of some Brill–Noether loci, of moduli schemes for vector bundles with parabolic structure or with level structure and for A. Schmitt's decorated vector bundles. A further consequence concerns the existence of Poincaré families on finite coverings of the moduli schemes.