In this paper we consider the birational classification of pairs , with a rational surface and a linear system on . We give a classification theorem for such pairs, and we determine, for each irreducible plane curve , its Cremona minimal models, that is, those plane curves which are equivalent to via a Cremona transformation and have minimal degree under this condition.
"Birational classification of curves on rational surfaces." Nagoya Math. J. 199 43 - 93, September 2010. https://doi.org/10.1215/00277630-2010-003