Stability conditions on irreducible projective curves

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.