Stability conditions on irreducible projective curves
Ziqi Liu
TL;DR
This work classifies stability conditions on the bounded derived category of coherent sheaves for irreducible projective curves, separating smooth from singular cases. For smooth curves of positive genus, it proves $\mathrm{Stab}(\mathbf{D}^b(C))\cong\mathbb{C}\times\mathbb{H}$ and identifies all non-locally-finite stability conditions as $\sigma_{\beta}$ with irrational $\beta$, while describing several boundary compactifications and introducing regular CLSY weak stability to complete the boundary. In the singular setting, it identifies a connected component $\mathrm{Geo}^{\dagger}(C)$ containing geometric stability conditions, shows $\mathrm{Stab}(C)=\mathrm{Geo}^{\dagger}(C)$ for non-rational curves, and proves that locally-finite numerical stability conditions are geometric. Collectively, the results map the stability spaces across smooth and singular curves, clarifying boundary phenomena and providing a framework for regular weak stability conditions to fill gaps in the boundary structure.
Abstract
This note revisits stability conditions on the bounded derived categories of coherent sheaves on irreducible projective curves. In particular, all stability conditions on smooth curves are classified and a connected component of the stability manifold containing all the geometric stability conditions is identified for singular curves. On smooth curves of positive genus, the set of all non-locally-finite stability conditions gives a partial boundary of any known compactification of the stability manifold. To provide a reasonable full boundary, a notion of regular weak stability condition is proposed based on the definition of Collins--Lo--Shi--Yau and is classified for smooth curves of positive genus. On non-rational singular curves, any locally-finite numerical stability condition is shown to be geometric.