Linearisation of finite abelian subgroups of the Cremona group of the plane (Q1016447)

Let \(\mathrm{Bir}(\mathbb{P}^2)\) be the Cremona group of birational automorphisms of \(\mathbb{P}^2\) and let \(G < \mathrm{Bir}(\mathbb{P}^2)\) be a finite subgroup of it. \(G\) is said to be birationally conjugate to a group of automorphisms of a rational surface if there is a surface \(S\) and a birational map \(\phi : S \dasharrow \mathbb{P}^2\) such that \(G\) is isomorphic to a subgroup of \(\mathrm{Aut}(S)\) under the induced map \(\phi_{\ast} : \mathrm{Aut}(S) \rightarrow \mathrm{Bir}(\mathbb{P}^2)\). The purpose of the paper under review is to find a geometric condition in order \(G\) to be birationally conjugate to a group of automorphisms of a minimal rational surface, i.e., of either \(\mathbb{P}^2\), \(\mathbb{P}^1 \times \mathbb{P}^1\) or \(\mathbb{F}_n\), \(n \geq 2\). If \(G\) is birationally conjugate to a group of automorphisms of a minimal rational surface, then no non-trivial element of \(G\) fixes a curve of positive genus. In this paper the author studies to what extend the converse is true. It is shown that if \(G\) is abelian and no non-trivial element of \(G\) fixes a curve of positive genus, then \(G\) is either conjugate to a subgroup of \(\mathrm{Aut}(\mathbb{P}^2)\), or to a subgroup of \(\mathrm{Aut}(\mathbb{P}^1 \times \mathbb{P}^1)\), or to the subgroup \(Cs_{24}\cong \mathbb{Z}_2 \times \mathbb{Z}_4 \) of \(\mathrm{Bir}(\mathbb{P}^2)\) generated by the birational transformations \((x:y:z)\dasharrow (yz:xy:-xz)\) and \((x:y:z)\dasharrow (yz(y-z):xz(y+z):xy(y+z))\). The last group is not birationally conjugate to a group of automorphisms of any minimal rational surface. In particular, if \(G\) is finite cyclic such that no non-trivial element of \(G\) fixes a curve of positive genus, then \(G\) is birationally conjugate to either a subgroup of \(\mathrm{Aut}(\mathbb{P}^2)\), or to a subgroup of \(\mathrm{Aut}(\mathbb{P}^1 \times \mathbb{P}^1)\).