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

## TL;DR

This paper proves criteria for when finite abelian subgroups of the Cremona group of the plane are conjugate to automorphism groups of minimal surfaces, revealing a unique exception and describing automorphisms of certain rational surfaces.

## Contribution

It provides a method to determine birational conjugacy of finite abelian subgroups to automorphism groups of minimal surfaces and characterizes linearisability conditions.

## Key findings

- A finite cyclic group is linearisable iff no non-trivial element fixes a positive genus curve.
- The only exception among finite abelian groups is Z/2Z x Z/4Z, which is not conjugate to automorphisms of a minimal surface.
- Descriptions of automorphisms of del Pezzo surfaces and conic bundles are included.

## Abstract

This article gives the proof of results announced in [J. Blanc, Finite Abelian 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 rational surfaces. 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/2ZxZ/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.