Stability conditions for coherent systems on Integral Curves
Marcos Jardim, Leonardo Roa-Leguizamón, Renato Vidal Martins
TL;DR
The paper develops a Bridgeland-style stability framework for coherent systems on integral curves by constructing a three-parameter family of pre-stability conditions in the derived category through tilting and subsequently identifying parameter regions where genuine stability arises. It provides foundational bounds for global sections, extends standard stability to coherence systems, and introduces standard and tilted stability conditions, including a Bogomolov–Gieseker inequality-empowered tilting theory. The results yield precise descriptions of semistability for tilted torsion and complete systems, establish walls in the gamma-direction, and demonstrate stability in large-gamma limits, enriching moduli-theoretic and geometric invariant theory perspectives for decorated sheaves on singular curves. Overall, the work extends stability notions from coherent sheaves to decorated objects on integral curves, offering new tools for moduli spaces and potential links to Brill–Noether phenomena in singular settings.
Abstract
We present stability conditions for the category of coherent systems on an integral curve. We define a three-parameter family of pre-stability conditions in its derived category using tilting, and we then investigate when these conditions qualify as true stability conditions. Additionally, we examine the semistability of specific objects under these conditions, namely: torsion, free, and complete tilted systems, without relying on the support property.