We study linear systems on a smooth complex curve C ⊂ P3 cut out by cones of fixed degree. We develop a systematic study of the families of these systems, considering their limits, their infinitesimal behaviour and some associated geometric structures. As an application, we prove the existence of a pencil of quartics with only one base point and all whose members are irreducible. The work is in collaboration with R. Moschetti and L. Stoppino