Vladimir L. Popov. Some subgroups of the Cremona groups

Natural Sciences / Mathematics / Algebra

Submitted on: Apr 11, 2012, 07:12:57

Description: We explore algebraic subgroups of of the Cremona group over an algebraically closed field of characteristic zero. First, we consider some class of algebraic subgroups of C_n that we call flattenable. It contains all tori. Linearizability of the natural rational actions of flattenable subgroups on the affine space A^n 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 C_n and to proving the existence of nonlinearizable, but stably linearizable elements of infinite order in C_n for n>=6. Then we consider some subgroups J(x_1..., x_n) of C_n that we call the rational de Jonquieres subgroups. We prove that every affine algebraic subgroup of J(x_1..., x_n) is solvable and the group of its connected components is Abelian.