The author studies the existence of smooth curves passing through the vertex of the cone over a (possibly singular) plane curve \(D\) of degree \(d\). Given a positive integer \(m\), he gives conditions for there to exist such curves that meet each ruling of the cone \(m\) times. In particular, they exist if \(D\) is rational -- in this case, the proof consists in looking at curves on the desingularization of the cone. If \(D\) is not rational, such curves exist under certain conditions on \(m\) and \(D\) as well as on the characteristic of the base field. This proof involves a generalization of Cayley's classical monoid construction. The results can be applied to the problem of finding smooth space curves which are set-theoretic complete intersections of a cone with some other surface.