Sergei Kovalenko

Gizatullin surfaces or quasi-homogeneous surfaces are normal affine surfaces such that the complement of the big orbit of the automorphism group is finite. If the action of the automorphism group is transitive, the surface is called homogeneous. Examples of non-homogeneous Gizatullin surfaces were constructed in [Ko], but on more restricted conditions. We show that a similar result holds under less constrained assumptions. Moreover, we exhibit examples of smooth affine surfaces V with a non-transitive action of the subgroup of automorphisms, whereas the automorphism group Aut(V) is big. This means that the subgroup Aut(V)_{alg} which is generated by algebraic subgroups of Aut(V) is not generated by a countable set of algebraic subgroups and the quotient Aut(V)/Aut(V)_{alg} contains a free group over an uncountable set of generators.

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