Gizatullin surfaces with a Q-trivial canonical class.

By a remarkable result of Bandmann and Makar-Limanov a smooth Gizatullin surface V
with trivial canonical class can be embedded into the ane 3-space with equation xy = P(t) for some
polynomial P(t). More recently this was generalized to normal surfaces by D. Daigle, where again the
same result holds. We give a report about the following result of Kai Ledwig, a PhD student of mine.
Theorem : Let V be a normal Gizatullin surface with a Q-trivial canonical class. Then V is isomorphic to
a locally closed subset of a weighted projective space P of dimension 4. More precisely, V is isomorphic to
a complement DnV , where in suitable weighted homogeneous coordinates (x; y; s; z) the closed subvariety
V has equation xy = P(s; z) and D is the open subset z 6= 0.
The main step in the proof is to show that a normal Gizatullin surface with a Q-trivial canonical class
always admits a hyperbolic C-action. Moreover it turns out that the DPD-presentation of such a surface
is of a special type as studied in a recent paper of Kaliman, Zaidenberg and myself. Applying one of the
main results of that paper the theorem follows.