Local trees in the theory of affine plane curves (Q1191304)

Let \(S\) be a complete non singular algebraic surface and \(D\in\text{Div}(S)\) with normal crossing. Then the pair \((S,D)\) determines a weighted graph which carries some information about the surface \(S\backslash D\). If \(D\) does not have normal crossings then one has to blow-up \(S\) at the ``bad'' points of \(D\). In this paper it is developed a graph theory which relates the desingularization process to graph-theoretic devices called ``local trees''. Application of the methods to the classification of birational morphisms of the affine plane with one or two fundamental points is given.