The curvaton scenario predicts an almost scale-invariant spectrum of perturbations in most inflation models. We consider the possibility that renormalizable 4 or Planck scale-suppressed non-renormalizable curvaton potential terms may result in an observable deviation from scale-invariance. We show that if the curvaton initially has a large amplitude and if the total number of e-foldings of inflation is less than about 300 then a running blue perturbation spectrum with an observable deviation from scale-invariance is likely. D-term inflation is considered as an example with a potentially low total number of e-foldings of inflation. A secondary role for the curvaton, in which it drives a period of chaotic inflation leading to D-term or other flat potential inflation from an initially chaotic state, is suggested.