“…over the Hyeon-Morrison's moduli stack M wp g of weakly-pseudo-stable curves). In higher rank, the conjectural first step of the LMMP for the Pandharipande's compactification U r,d,g has been described by Grimes in [22]: using the torsion free approach, he constructs a compactification U ps r,d,g of the moduli space of slope-semistable vector bundles over M ps g . In order to construct birational compact models for the Pandharipande compactification of U r,d,g , it is useful to have an explicit description of its rational Picard group which naturally embeds into the rational Picard group of the moduli stack T F ss r,d,g of slope-semistable torsion free sheaves over stable curves.…”