Additive group actions on Danielewski varieties and the cancellation problem
Additive group actions on Danielewski varieties and the cancellation problem
复制标题
Danielewski 品种的附加群体行动和取消问题
DOI：
10.1007/s00209-006-0013-3
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发表时间：
2005
影响因子：
0.8
通讯作者：
A. Dubouloz
中科院分区：
文献类型：
--
作者：
A. Dubouloz
给定相同维数的复代数簇X和Y，消去问题问X × Y之间是否存在同构
$$\mathbb{C}$$和Y ×
$$\mathbb{C}$$诱导X和Y之间的同构。Iitaka和Fujita（J. Fac. Sci. 24：123-127，1977）确定了对于任何维度的一大类变种，答案都是肯定的。1989年，Danielewski利用光滑有理仿射曲面构造了一个反例。他的建设是进一步推广的Fieseler（评论。Math. Helvetici 69：5-27，1994）和Wilkens（CR Acad. Sci.巴黎先生I Math.326（9）：1111-1116，1998）来描述更大类的仿射曲面。在这里，我们介绍这些表面的高维类似物。通过研究加法群的代数作用，
$$\mathbb{C}_{+}$$在某些这些品种，我们得到新的反例的取消问题，在每个维d ≥ 2。
AbstractGiven complex algebraic varieties X and Y of the same dimension, the Cancellation Problem asks if an isomorphism between X ×
$$\mathbb{C}$$ and Y ×
$$\mathbb{C}$$ induces an isomorphism between X and Y. Iitaka and Fujita (J. Fac. Sci. Univ. 24:123–127, 1977) established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski constructed a counterexample using smooth rational affine surfaces. His construction was further generalized by Fieseler (Comment. Math. Helvetici 69:5–27, 1994) and Wilkens (C.R. Acad. Sci. Paris Sér. I Math. 326(9):1111–1116, 1998) to describe a larger class of affine surfaces. Here we introduce higher-dimensional analogues of these surfaces. By studying algebraic actions of the additive group
$$\mathbb{C}_{+}$$ on certain of these varieties, we obtain new counterexamples to the Cancellation Problem in every dimension d ≥ 2.