Automorphism group of the moduli space of parabolic bundles over a curve

We find the automorphism group of moduli space parabolic bundles on a smooth curve (with fixed determinant and system weights). This is generated by: automorphisms marked curve, tensoring with line bundle, taking dual, Hecke transforms (using filtrations given by structure). A Torelli theorem for arbitrary rank generic weights also obtained. These results are extended to classification birational equivalences which defined over “big” open subsets (3-birational maps, i.e. maps giving an isomorphism between complement codimension at least 3). Finally, analysis stability chambers performed in order determine precisely when two spaces vector different parameters (curve, rank, weights) can be isomorphic.