Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it is birationally conjugate to a group of automorphisms of a minimal surface.
In particular, we prove that a finite cyclic group of birational transformations of the plane is linearisable if and only if none of its non-trivial elements fix a curve of positive genus. For finite Abelian groups, there exists only one surprising exception, a group isomorphic to ℤ/2ℤ × ℤ/4ℤ, whose non-trivial elements do not fix a curve of positive genus but which is not conjugate to a group of automorphisms of a minimal rational surface.
We also give some descriptions of automorphisms (not necessarily of finite order) of del Pezzo surfaces and conic bundles.
Cite this article
Jérémy Blanc, Linearisation of finite Abelian subgroups of the Cremona group of the plane. Groups Geom. Dyn. 3 (2009), no. 2, pp. 215–266DOI 10.4171/GGD/55