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

### Jérémy Blanc

Universität Basel, Switzerland

## Abstract

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. 215–266

DOI 10.4171/GGD/55