# Groups, Geometry, and Dynamics

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**Volume 3, Issue 2, 2009, pp. 215–266**

**DOI: 10.4171/GGD/55**

Published online: 2009-06-30

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

Jérémy Blanc^{[1]}(1) Universität Basel, Switzerland

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.

*Keywords: *Birational transformations, fixed curves, linearisation, minimal surfaces

Blanc Jérémy: Linearisation of finite Abelian subgroups of the Cremona group of the plane. *Groups Geom. Dyn.* 3 (2009), 215-266. doi: 10.4171/GGD/55