摘要翻译：
本文证明了[J.Blanc,平面上Cremona群的有限Abel子群，C.R.Acad.Sci.Paris,S\'er.I344(2007),21-26.]中的结果以及有理曲面自同构的一些描述。给出了平面上Cremona群的有限Abel子群，给出了它是否与极小曲面的自同构群双偶共轭的一种判定方法。特别地，我们证明了平面的双分变换的有限循环群是线性的当且仅当它的非平凡元中没有一个固定正亏格的曲线。对于有限阿贝尔群，只有一个惊人的例外，即同构于Z/2ZxZ/4Z的群，它的非平凡元素不固定正亏格曲线，但不共轭于极小有理曲面的自同构群。我们也给出了del Pezzo曲面和二次丛的自同构（不一定是有限阶）的一些描述。
---
英文标题：
《Linearisation of finite abelian subgroups of the Cremona group of the
  plane》
---
作者：
J\'er\'emy Blanc
---
最新提交年份：
2008
---
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
--

---
英文摘要：
  This article gives the proof of results announced in [J. Blanc, Finite Abelian subgroups of the Cremona group of the plane, C.R. Acad. Sci. Paris, S\'er. I 344 (2007), 21-26.] and some description of automorphisms of rational surfaces.   Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it is birationally conjugate to a group of automorphisms of a minimal surface.   In particular, we prove that a finite cyclic group of birational transformations of the plane is linearisable if and only if none of its non-trivial elements fix a curve of positive genus. For finite Abelian groups, there exists only one surprising exception, a group isomorphic to Z/2ZxZ/4Z, whose non-trivial elements do not fix a curve of positive genus but which is not conjugate to a group of automorphisms of a minimal rational surface.   We also give some descriptions of automorphisms (not necessarily of finite order) of del Pezzo surfaces and conic bundles.
---
PDF链接：
https://arxiv.org/pdf/0704.0537