Factorial affine G_a-varieties isomorphic to hypersurfaces of Danielewski type
Factorial affine G_a-varieties isomorphic to hypersurfaces of Danielewski type
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与Danielewski型超曲面同构的阶乘仿射G_a-variety
DOI：
10.1007/s00031-020-09631-y
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发表时间：
2020
影响因子：
0.7
通讯作者：
Kayo Masuda
中科院分区：
文献类型：
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作者：
Y. Sakuhara;H. Shimizu;K. Ito;Kayo Masuda
令k 为特征零和Ba 阶乘仿射域的代数闭域，配备局部幂零导数δ。我们研究 B，当存在一个元素 z∈B 使得 δ(z) = αp for p ≥ 1 且 A 的质元素 α= Kerδ 时。其中一个特别令人感兴趣的例子是仿射伪n空间的坐标环，它被定义为一种光滑的仿射簇X，配备忠实的态射q:X→𝔸1，使得q–1(𝔸)≅𝔸×𝔸n–1和q*(0)是不可约的并且可约化，其中𝔸=𝔸1\ {0}。在[9]中，给出了仿射伪3空间与𝔸4中xpy–g(x;t;z) = 0形式的超曲面同构的标准。这种超曲面被称为 Danielewski 型超曲面，并在[17]、[16]中进行了研究。在本文中，在 A/αA 为阶乘的条件下，我们用 Kaliman 和 Zaidenberg [15] 提出的等变仿射修正来描述 B，并给出了当 p≥ 1 时 B 与残差环 A[Y;Z]/(αpY–g(Z)) 同构的准则，以及一个不可约多项式 g(Z) ∈A[Z]\A。因此，我们获得了仿射伪 n 空间与 Danielewski 型超曲面同构的准则，其中 n≥ 3。
Letkbe an algebraically closed field of characteristic zero andBa factorial affinek-domain equipped with a locally nilpotent derivationδ. We investigateBwhen there exists an elementz∈Bsuch thatδ(z) = αpfor p ≥ 1 and a prime element α ofA= Kerδ. One such example of particular interest is the coordinate ring of an affine pseudo-n-space, which is defined as a smooth affine varietyXequipped with a faithfully at morphismq:X→ 𝔸1such thatq–1(𝔸) ≅ 𝔸× 𝔸n–1andq*(0) is irreducible and reduced where 𝔸= 𝔸1\ {0}. In [9], a criterion was given for an affine pseudo-3-space to be isomorphic to a hypersurface of formxpy–g(x;t;z) = 0 in 𝔸4. Such a hypersurface is called a hypersurface of Danielewski type and studied in [17], [16]. In this article, under the condition thatA/αAis factorial, we describeBin terms of equivariant affine modification developed by Kaliman and Zaidenberg [15] and give a criterion forBto be isomorphic to the residue ringA[Y;Z]/(αpY–g(Z)) forp≥ 1 and an irreducible polynomialg(Z) ∈A[Z]\A. As a consequence, we obtain a criterion for an affine pseudo-n-space to be isomorphic to a hypersurface of Danielewski type forn≥ 3.