Linear systems of rational curves on rational surfaces

Given a curve $C$ on a projective nonsingular rational surface $S$, over an algebraically closed field of characteristic zero, we are interested in the set $\Omega_{C}$ of linear systems $\mathbb{L}$ on $S$ satisfying $C \in \mathbb{L}$, $\dim \mathbb{L} \ge1$, and the general member of $\mathbb{L}$ is a rational curve. The main result of the paper gives a complete description of $\Omega_{C}$ and, in particular, characterizes the curves $C$ for which $\Omega_{C}$ is non empty.