In this paper we investigate the effect of curvature, in the quasi-one-dimensional, quasi-steady approximation, on detonations where the self-sustaining steady, planar wave is of the pathological (or eigenvalue) type. Planar detonation waves can be of the pathological type when there are endothermic or dissipative effects in the system. Such detonations are self-sustaining, travel faster than the Chapman-Jouguet (CJ) speed, and have an internal frozen sonic point where the thermicity vanishes. We consider a system with two consecutive, irreversible reactions, A --> B --> C, with an Arrhenius form of the reaction rates and the second reaction endothermic. Detonations with slightly curved fronts are also of the pathological type due to the divergence of the flow, even when the planar detonation is CJ. We show that the effect of curvature on the planar, pathological detonations is qualitatively the same as that on CJ detonations. We also discuss the difficulties involved in finding the detonation speed-curvature relation and the curved detonation structures when there is more than one reaction, and describe a numerical method for overcoming these.