T-Stabilities for a weighted projective line

Abstract

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-stabilities on certain triangulated category and, describe the semistable subcategories and last HN-triangles for coherent sheaves in $$D^b(\mathrm{coh\,}{{\mathbb {X}}})$$ D b ( coh X ) , which is the bounded derived category of coherent sheaves on the weighted projective line $${{\mathbb {X}}}$$ X of weight type (2). Furthermore, we show the existence of a t-exceptional triple for $$D^b(\mathrm{coh\,}{{\mathbb {X}}})$$ D b ( coh X ) . As an application, we obtain a result of Dimitrov–Katzarkov which states that each stability condition $$\sigma $$ σ in the sense of Bridgeland admits a $$\sigma $$ σ -exceptional triple for the acyclic triangular quiver Q. This result plays an important role in the proof of Dimitrov–Katzarkov that the space of stability conditions associated to Q is connected and contractible.