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• 作者：Hubert Flenner ; Shulim Kaliman and Mikhail Zaidenberg
• 刊名：Transformation Groups
• 出版年：2008
• 出版时间：June, 2008
• 年：2008
• 卷：13
• 期：2
• 页码：305-354
• 全文大小：760.8 KB

文摘

A Gizatullin surface is a normal affine surface V over \mathbbC \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 \mathbbC * \mathbb{C}^{ * } -actions and \mathbbA\text1 \mathbb{A}^{{\text{1}}} -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with \mathbbCb>\text + b> \mathbb{C}_{{\text{ + }}} -actions on V considered up to a “speed change”. Non-Gizatullin surfaces are known to admit at most one \mathbbA1 \mathbb{A}^{1} -fibration V → S up to an isomorphism of the base S. Moreover, an effective \mathbbC * \mathbb{C}^{ * } -action on them, if it does exist, is unique up to conjugation and inversion t ? \mapsto t −1 of \mathbbC * \mathbb{C}^{ * } . Obviously, uniqueness of \mathbbC * \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 \mathbbC * \mathbb{C}^{ * } -actions and \mathbbA\text1 \mathbb{A}^{{\text{1}}} -fibrations, see, e.g., [FKZ1].