Mappings of Surfaces

In the framework of analytic geometry bimeromorphic maps play the same role as birational maps in algebraic geometry. The first aspect concerning these is the process of desingularization. We have seen (Sect. I.8) that the normalization can be seen as a first step in making the singularities less complicated. So we may assume that the surfaces under consideration are normal. For these there is indeed a desingularization as shown in Sect. 6. The proof is by reduction to the case of Hirzebruch-Jung singularities treated in Sect. 5. The existence of a minimal resolution is very particular for dimension 2 and depends on the fact that one can contract a (−1)-curve, i.e., a smooth rational curve with self intersection − 1, to a smooth point. This and its applications to the existence of minimal models is treated in Sect. 4. Curves contractible to (singular) points are treated more generally in Sect. 2.