THE LINE BUNDLES ON THE MODULI OF PARABOLIC G-BUNDLES OVER CURVES ANDTHEIR SECTIONS

Let X be a complex, smooth, complete and connected curve and G be a co mplex simple and simply connected algebraic group. We compute the Pica rd group of the stack of quasi-parabolic G-bundles over X, describe ex plicitly its generators for classical G and G(2) and then identify the corresponding spaces of global sections with the vacua spaces of Tsuc hiya, Ueno and Yamada. The method uses the uniformization theorem whic h describes these stacks as double quotients of certain infinite dimen sional algebraic groups. We describe also the dualizing bundle of the stack of G-bundles and show that it admits a unique square root, which we construct explicitly. If G is not simply connected, the square roo t depends on the choice of a theta-characteristic. These results about stacks allow to recover the Drezet-Narasimhan theorem (for the coarse moduli space) and to show an analogous statement when G = Sp(2r). We prove also that the coarse moduli spaces of semi-stable SOr-bundles ar e not locally factorial for r greater than or equal to 7.