Torsion Pairs and t-Structures in \(\textrm{D}^{b}(\text {coh}(\mathbb {X}))\)

We present a bijection between torsion pairs in \(\text {coh}(\mathbb )\) with corresponding t-structures in \(\textrm{D}^{b}(\text {coh}(\mathbb {X}))\) where \(\mathbb {X}\) represents a weighted projective line. When focusing on the split case, we derive a bijection between this class and corresponding torsion pairs in the module category of a concealed canonical algebra. Additionally, we demonstrate that if the aisle of a split t-structure in the derived category of a hereditary category contains an Ext-projective object, then it admits a tilting complex. Finally, we use the structure of the Auslander-Reiten quiver of \(\textrm{D}^{b}(\text {coh}(\mathbb {X}))\) in order to classify split t-structures in \(\textrm{D}^{b}(\text {coh}(\mathbb {X}))\).