t-stabilities for a weighted projective line

Shiquan Ruan, Xintian Wang

Published 2017-10-03Version 1

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