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-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 $Q$.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2017
- DOI:
- 10.48550/arXiv.1710.00986
- arXiv:
- arXiv:1710.00986
- Bibcode:
- 2017arXiv171000986R
- Keywords:
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- Mathematics - Representation Theory;
- 18E10;
- 18E30;
- 16G20;
- 14F05;
- 16G70
- E-Print:
- 22 pages