Uniqueness of $\mathbb{C}^{ * }$- and $\mathbb{C}_{{\text{ + }}}$-actions on Gizatullin Surfaces

Abstract

A Gizatullin surface is a normal affine surface V over $\mathbb{C}$, 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 $\mathbb{C}^{ * }$-actions and $\mathbb{A}^{{\text{1}}}$-fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with $\mathbb{C}_{{\text{ + }}}$-actions on V considered up to a “speed change”.

Non-Gizatullin surfaces are known to admit at most one $\mathbb{A}^{1}$-fibration V → S up to an isomorphism of the base S. Moreover, an effective $\mathbb{C}^{ * }$-action on them, if it does exist, is unique up to conjugation and inversion t $\mapsto$ t ⁻¹ of $\mathbb{C}^{ * }$. Obviously, uniqueness of $\mathbb{C}^{ * }$-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 $\mathbb{C}^{ * }$-actions and $\mathbb{A}^{{\text{1}}}$-fibrations, see, e.g., [FKZ1].

In the present paper we obtain a criterion as to when $\mathbb{A}^{{\text{1}}}$-fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base $S \cong \mathbb{A}^{{\text{1}}}$. We exhibit as well large subclasses of Gizatullin $\mathbb{C}^{ * }$-surfaces for which a $\mathbb{C}^{ * }$-action is essentially unique and for which there are at most two conjugacy classes of $\mathbb{A}^{{\text{1}}}$-fibrations over $\mathbb{A}^{{\text{1}}}$.