T-stabilities for a weighted projective line

The present paper focuses on the study of t-stabilities on a triangulated category in the sense of Gorodentsev, Kuleshov and Rudakov. We give an equivalent description for the finest t-stability on a piecewise hereditary triangulated category and, describe the semistable subcategories and final HN triangles for (exceptional) coherent sheaves in $D^b(\rm{coh}\mathbb{X})$, which is the bounded derived category of coherent sheaves on the weighted projective line $\mathbb{X}$ of weight type (2). Furthermore, we show the existence of a t-exceptional triple for $D^b(\rm{coh}\mathbb{X})$. As an application, we obtain a result of Dimitrov--Katzarkov which states that each stability condition $\sigma$ in the sense of Bridgeland admits a $\sigma$-exceptional triple for the acyclic triangular quiver $Q$. Note that this implies the connectedness of the space of stability conditions associated to $Q$.