Submission: 2011, Oct 9
We explore algebraic subgroups of the Cremona group C_n of rank n 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 n-dimensional 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>5. Then we consider some subgroups J(x_1,..., x_n) of C_n that we call the rational de Jonqui\`eres 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. We also prove that every reductive algebraic subgroup of J(x_1,..., x_n) is diagonalizable. Further, we prove that the natural rational action on A^n of any unipotent algebraic subgroup of J(x_1,..., x_n) admits a rational cross-section which is an affine subspace of A^n. 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 a finite-dimensional algebraic Lie algebras under the assumption that their Levi decomposition is a direct product. We then consider some natural overgroup of J(x_1,\ldots, x_n) and prove that every torus in it is linearizable. Finally, we prove the existence of an element g in C_3 of order 2 such that g is not contained in every connected affine algebraic subgroup of C_\infty; in particular, g is not stably linearizable.
2010 Mathematics Subject Classification: 14E07
Keywords and Phrases: Cremona group; linearizability; tori; invariant fields; the rational de Jonqui\`eres subgroups; cross-sections
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