Some subgroups of the Cremona groups

Vladimir L. Popov

Published 2011-10-11, updated 2012-07-16Version 4

We explore algebraic subgroups of of the Cremona group $\mathcal C_n$ over an algebraically closed field of characteristic zero. First, we consider some class of algebraic subgroups of $\mathcal C_n$ that we call flattenable. It contains all tori. Linearizability of the natural rational actions of flattenable subgroups on the affine space $\An$ is intimately related to rationality of the invariant fields and, for tori, is equivalent to it. We prove stable linearizability of these actions and show the existence of nonlinearizable actions among them. This is applied to exploring maximal tori in $\mathcal C_n$ and to proving the existence of nonlinearizable, but stably linearizable elements of infinite order in $\mathcal C_n$ for $n\geqslant 6$. Then we consider some subgroups $\mathcal J(x_1,..., x_n)$ of $\mathcal C_n$ that we call the rational de Jonqui\`eres subgroups. We prove that every affine algebraic subgroup of $\mathcal J(x_1,..., x_n)$ is solvable and the group of its connected components is Abelian. We also prove that every reductive algebraic subgroup of $\mathcal J(x_1,..., x_n)$ is diagonalizable. Further, we prove that the natural rational action on $\An$ of any unipotent algebraic subgroup of $\mathcal J(x_1,..., x_n)$ admits a rational cross-section which is an affine subspace of $\An$. We show that in this statement "unipotent" cannot be replaced by "connected solvable".\;This is applied to proving a conjecture of A. Joseph on the existence of "rational slices" for the coadjoint representations of finite-dimensional algebraic Lie algebras $\mathfrak g$ under the assumption that the Levi decomposition of $\mathfrak g$ is a direct product.