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.

中文翻译：

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加权投影线的 t-稳定性

本文重点研究 Gorodentsev、Kuleshov 和 Rudakov 意义上的三角化范畴的 t 稳定性。我们给出了某个三角化类别上最好的 t 稳定性的等效描述，并描述了 $$D^b(\mathrm{coh\,}{{\mathbb {X} }})$$ D b ( coh X ) ，它是加权投影线 $${{\mathbb {X}}}$$ X 的加权类型 (2) 上相干滑轮的有界派生范畴。此外，我们证明了 $$D^b(\mathrm{coh\,}{{\mathbb {X}}})$$ D b ( coh X ) 的 t 异常三元组的存在。作为一个应用，我们获得了 Dimitrov-Katzarkov 的结果，该结果表明 Bridgeland 意义上的每个稳定性条件 $$\sigma $$ σ 承认无环三角箭袋 Q 的 $$\sigma $$ σ - 异常三元组。