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## Infinite transitivity and special automorphisms

It is known that if the special automorphism group SAut(X) of a quasiaffine variety X of dimension at least 2 acts transitively on X, then this action is infinitely transitive. In this paper we question whether this is the only possibility for the automorphism group Aut(X) to act infinitely transitively on X. We show that this is the case, provided X admits a nontrivial G_a- or G_m-action. Moreover, 2-transitivity of the automorphism group implies infinite transitivity.

We say that a group G acts infinitely transitively on a set X if for every m ε N the induced diagonal action of G is transitive on the cartesian mth power X m\δ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of normal affine cones over flag varieties, the second of nondegenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups. Bibliography: 42 titles.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Ths is Proceedings of a Conference on Affine Algebraic Geometry that was held at Osaka Umeda Campus of Kwansei Gakuin University during the period 3--6 March, 2011 on the occasion of the seventieth birthday of Professor Masayaoshi Miyanishi.

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.

This volume is dedicated to Professor M. Miyanishi on the occasion of his 70th birthday.

We explore algebraic subgroups of the Cremona group Cn over an algebraically closed field of characteristic zero. First, we consider some class of algebraic subgroups of Cn that we call flattenable. It contains all tori. Linearizability of the natural rational actions of flattenable subgroups on 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 Cn and to proving the existence of nonlinearizable, but stably linearizable elements of infinite order in Cn for n ? 5. Then we consider some subgroups J (x1, . . . ,xn) of Cn that we call the rational de Jonquie`res subgroups. We prove that every affine algebraic subgroup of J (x1, . . . ,xn) is solvable and the group of its connected components is Abelian. We also prove that every reductive algebraic subgroup of J (x1, . . . ,xn) is diagonalizable. Further, we prove that the natural rational action on An of any unipotent algebraic subgroup of J (x1, . . . ,xn) 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 g under the assumption that the Levi decomposition of g is a direct product. We then consider some overgroup J^ (x1, . . . ,xn) of J (x1, . . . ,xn) and prove that every torus in J^ (x1, . . . ,xn) is linearizable. Finally, we prove the existence of an element g ? C3 of order 2 such that g does not lie in every connected affine algebraic subgroup G of C?; in particular, g is not stably linearizable.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.