| The study of birational transformations finds its primary motivation in the classification problem of algebraic varieties. In this thesis, two different approaches are followed.; The first approach is based on finding opportune birational modifications in order to turn birational transformations into automorphisms. This process is called “resolution of indeterminacy of pairs”, and enables us to reduce to the simpler problem of studying automorphisms of varieties. Existence of such a resolution is proven for birational transformations of finite order. The classification of prime order elements in the Cremona group is then given, by means of this method.; The second approach aims to control the singularities of the locus where a given birational transformation is not defined. Important information can be deduced by accurate bounds on the singularities of this locus. A general formula, that extends a known result holding in dimension two to all dimensions, is proven. Applications to the birational rigidity problem are then investigated, and the “method of maximal singularities”, introduced by the Russian school to address this problem, is reviewed. |
