Stability approach to torsion pairs on abelian categories

TL;DR

This paper introduces a local-refinement method for stability data in abelian categories, classifies finest stability data for specific categories, and applies this to classify torsion pairs, providing new insights and proofs.

Contribution

It presents a novel local-refinement procedure for stability data and classifies all finest stability data for categories of coherent sheaves on certain weighted projective curves.

Findings

Classified all finest stability data for specific categories

Provided a new proof for torsion pairs in tube categories

Connected stability data with torsion pair classification

Abstract

In this paper we introduce a local-refinement procedure to investigate stability data on an abelian category, and provide a sufficient and necessary condition for a stability data to be finest. We classify all the finest stability data for the categories of coherent sheaves over certain weighted projective curves, including the classical projective line, smooth elliptic curves and certain weighted projective lines. As applications, we obtain a classification of torsion pairs for these categories via stability data approach. As a by-product, a new proof for the classification of torsion pairs in any tube category is also provided.