Smooth Non-Homogeneous Gizatullin Surfaces

Quasi-homogeneous surfaces, or Gizatullin surfaces, are normal affine surfaces such that there exists an open orbit of the automorphism group with a finite complement. 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 with a non-transitive action of the automorphism group whereas the automorphism group is huge. This means that it is not generated by a countable set of algebraic subgroups and that its quotient by the (normal) subgroup generated by all algebraic subgroups contains a free group over an uncountable set of generators.