2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/117923This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We have thus to enumerate all the cases, which gives a long description, and then to show whether two cases are distinct or not, using some conjugacy invariants. For example, we use the non-rational curves fixed by one element of the group, and the action of the whole group on these curves. From this classification, we deduce a sequence of more general results on birational transformations, as for example the existence of infinitely many conjugacy classes of elements of order n, for any even number n, a result false in the odd case. We prove also that a root of some linear transformation of finite order is itself conjugate to a linear transformation.PHD Thesis, 189 pages, 34 figures, original text may be found at http://www.unige.ch/cyberdocuments/theses2006/BlancJ/meta.htmlAlgebraic GeometryGroup Theory14E07; 14J26; 14J50; 14L30text