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This article gives the proof of results announced in [J. Blanc, FiniteAbelian subgroups of the Cremona group of the plane, C.R. Acad. Sci. Paris,S\'er. I 344 (2007), 21-26.] and some description of automorphisms of rationalsurfaces.  Given a finite Abelian subgroup of the Cremona group of the plane, we providea way to decide whether it is birationally conjugate to a group ofautomorphisms of a minimal surface.  In particular, we prove that a finite cyclic group of birationaltransformations of the plane is linearisable if and only if none of itsnon-trivial elements fix a curve of positive genus. For finite Abelian groups,there exists only one surprising exception, a group isomorphic to Z/2ZxZ/4Z,whose non-trivial elements do not fix a curve of positive genus but which isnot conjugate to a group of automorphisms of a minimal rational surface.  We also give some descriptions of automorphisms (not necessarily of finiteorder) of del Pezzo surfaces and conic bundles.

Linearisation of finite abelian subgroups of the Cremona group of the plane