Seminar Algebra and Geometry: Sergei Kovalenko (Ruhr-Universität Bochum)

Transitivity of automorphism groups of Gizatullin surfaces

Gizatullin surfaces are normal affine surfaces completable by a zigzag, i. e. by a linear chain of smooth rational curves. An equivalent characterization of such surfaces V, except for the surface C^∗ × C^∗, is that the automorphism group acts with a big orbit O, i. e. V \ O is finite. Considering some examples of Gizatullin surfaces like the affine plane A² or the Danielewski surfaces V = {xy − P(z) = 0} ⊆ A³ it follows that the big orbit O coincides with the smooth locus V_reg . Gizatullin formulated in his pioneer works the following conjecture:

Conjecture (Gizatullin): The big orbit of a Gizatullin surface V coincides with its smooth locus, i. e. O = V_reg .

We show that the action of the automorphism group of a smooth Gizatullin surface with a distinguished and rigid extended divisor is not transitive in general. Thus such surfaces represent counterexamples to Gizatullin’s conjecture. For such surfaces we give an explicit orbit decomposition of the natural action of the automorphism group. Moreover, the automorphism group of such smooth Gizatullin surfaces can be represented as an amalgamated product of two automorphism subgroups.

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