Birational classification of curves on rational surfaces

Abstract

In this paper we consider the birational classification of pairs $\left(S,\mathcal{L}\right)$, with $S$ a rational surface and $\mathcal{L}$ a linear system on $S$. We give a classification theorem for such pairs, and we determine, for each irreducible plane curve $B$, its Cremona minimal models, that is, those plane curves which are equivalent to $B$ via a Cremona transformation and have minimal degree under this condition.