t-stabilities for a weighted projective line (Q2663071)

The authors prove interesting results about the space of Bridgeland stability conditions over the derived category of coherent sheaves \(\mathcal{D}\) of the weighted projective line of type \((2)\). The paper starts with some background material on weighted projective lines and their exceptional collections, closing the first section with an important proposition about the minimal approximation of a line bundle in the derived category. This approximation is later translated into a concrete description of the last triangle in the Harder-Narasimhan filtration for a finest t-stability as defined by Gorodentsev-Kuleshov-Rudakov. To achieve such finest t-stability the authors define an algorithm that refines any t-stability into a finest t-stability and then define a distance function inside of a t-stability, which measures the least distance of two points in the weighted projective line such that the structure sheaf twisted by them respectively is semistable. At the same time, the paper concretely express all the Ext-exceptional triples in \(\mathcal{D}\) which, combined with the distance function, allows for them to prove the existence a class of Ext-exceptional triples adapted to a given effective finest t-stability. Furthermore, they go on to establishing a number of technical results about the stability of the simple sheaves with respect to Bridgeland's stability condition which combined with the existence of Ext-exceptional triple adapted to the t-stability, proves that every Bridgeland stability have a Ext-exceptional tripe associated to it and, as corollary, the connectedness of the space of Bridgeland stability conditions. The appendices of the paper concentrate two different types of results: The first appendix deals with concrete calculations used thoughout the paper involving stable tubes and Auslander-Reiten theory, while the second is used to prove a different characterization of the finest t-stabilities.