Modular subvarieties and birational geometry of SUC(r)

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Let C be an algebraic smooth complex genus g>1 curve. The object of this paper is the study of the birational structure of the coarse moduli space UC(r,0) of semi-stable rank r vector bundles on C with degree 0 determinant and of its moduli subspace SUC(r) given by the vector bundles with trivial determinant. Notably we prove that UC(r,0) (resp. SUC(r)) is birational to a fibration over the symmetric product C(rg) (resp. over P⁽ʳ⁻¹⁾ᵍ) whose fibres are GIT quotients (Pʳ⁻¹)rg//PGL(r). In the cases of low rank and genus our construction produces families of classical modular varieties contained in the Coble hypersurfaces.