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摘要翻译：
Gizatullin曲面是一个正规仿射曲面$V$上的$\bf C$,它可以用一个之字形来完成；即由光滑有理曲线组成的线性链。本文讨论了在这类曲面V$上的$\bf C^*$-作用和$\bf a^1$-纤维直到自同构的唯一性问题。后者的纤维与$\bf C_+$-在$V$上的作用一一对应，被认为是“速度变化”。已知非Gizatullin曲面最多允许一个$\bf a^1$-fibration$v\to s$直到一个基$s$的同构。此外，对它们有效的$\bf c^{*}$-作用，如果存在的话，对于共轭和反转$t\mapsto t^{-1}$$\bf c^*$是唯一的。对于仿射曲面，$\bf C^*$-actions的唯一性明显失效；然而，在这种情况下，我们证明了最多有两个共轭类的$\bf a^1$-纤维。还有另一个有趣的非toric Gizatullin曲面族，称为Danilov-Gizatullin曲面，其中通常有几个共轭类:$\bf c^*$-作用和$\bf a^1$-纤维。本文得到了Gizatullin曲面的a^1$-纤维共轭到$V$和基$S$的自同构的一个判据。我们还展示了Gizatullin$\bf C^{*}$-曲面的一个大子类，对于这些曲面，$\bf C^*$-作用本质上是唯一的，对于这些曲面，$\bf a^1$-在$\bf a^1$上最多有两个共轭类$\bf a^1$-纤维。
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英文标题：
《Uniqueness of $\bf C^*$- and $\bf C_+$-actions on Gizatullin surfaces》
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作者：
Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
A Gizatullin surface is a normal affine surface $V$ over $\bf C$, 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 $\bf C^*$-actions and $\bf A^1$-fibrations on such a surface $V$ up to automorphisms. The latter fibrations are in one to one correspondence with $\bf C_+$-actions on $V$ considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one $\bf A^1$-fibration $V\to S$ up to an isomorphism of the base $S$. Moreover an effective $\bf C^{*}$-action on them, if it does exist, is unique up to conjugation and inversion $t\mapsto t^{-1}$ of $\bf C^*$. Obviously uniqueness of $\bf C^*$-actions fails for affine toric surfaces; however we show in this case that there are at most two conjugacy classes of $\bf A^1$-fibrations. There is a further interesting family of non-toric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $\bf C^*$-actions and $\bf A^1$-fibrations. In the present paper we obtain a criterion as to when $\bf A^1$-fibrations of Gizatullin surfaces are conjugate up to an automorphism of $V$ and the base $S$. We exhibit as well a large subclasses of Gizatullin $\bf C^{*}$-surfaces for which a $\bf C^*$-action is essentially unique and for which there are at most two conjugacy classes of $\bf A^1$-fibrations over $\bf A^1$.
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PDF链接：
https://arxiv.org/pdf/0706.2261
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