Uniqueness of ℂ* - and ℂ₊ -actions on Gizatullin surfaces

A Gizatullin surface is a normal affine surface V over ℂ, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of ℂ* -actions and double-struck A¹ -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with ℂ₊ -actions on V considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one double-struck A ¹ -fibration V → S up to an isomorphism of the base S. Moreover, an effective ℂ* -action on them, if it does exist, is unique up to conjugation and inversion t → ^-1 of ℂ*. Obviously, uniqueness of ℂ* -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of ℂ* -actions and double-struck A¹ -fibrations, see, e.g., [FKZ₁]. In the present paper we obtain a criterion as to when double-struck A¹ -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base S ≅ double-struck A ¹. We exhibit as well large subclasses of Gizatullin ℂ* -surfaces for which a ℂ* -action is essentially unique and for which there are at most two conjugacy classes of double-struck A¹ -fibrations over double-struck A¹.