2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101153We investigate the bounded derived category of coherent sheaves on irreducible singular projective curves of arithmetic genus one. A description of the group of exact auto-equivalences and the set of all t-structures of this category is given. We describe the moduli space of stability conditions, obtain a complete classification of all spherical objects in this category and show that the group of exact auto-equivalences acts transitively on them. Harder-Narasimhan filtrations in the sense of Bridgeland are used as our main technical tool.44 pages; minor modifications; to appear in Compositio MathAlgebraic GeometryRepresentation Theory18E30; 14H60; 14H52; 14H20; 14H10text