Transformation Groups ( IF 0.4 ) Pub Date : 2020-11-18 , DOI: 10.1007/s00031-020-09631-y KAYO MASUDA
Let k be an algebraically closed field of characteristic zero and B a factorial affine k-domain equipped with a locally nilpotent derivation δ. We investigate B when there exists an element z ∈ B such that δ(z) = αp for p ≥ 1 and a prime element α of A = Ker δ. One such example of particular interest is the coordinate ring of an affine pseudo-n-space, which is defined as a smooth affine variety X equipped with a faithfully at morphism q : X → 𝔸1 such that q–1(𝔸\( {}_{\ast}^1 \)) ≅ 𝔸\( {}_{\ast}^1 \) × 𝔸 n–1 and q*(0) is irreducible and reduced where 𝔸\( {}_{\ast}^1 \) = 𝔸1 \ {0}. In [9], a criterion was given for an affine pseudo-3-space to be isomorphic to a hypersurface of form xpy – 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 that A/αA is factorial, we describe B in terms of equivariant affine modification developed by Kaliman and Zaidenberg [15] and give a criterion for B to be isomorphic to the residue ring A[Y;Z]/(αpY – g(Z)) for p ≥ 1 and an irreducible polynomial g(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 for n ≥ 3.
中文翻译:
与 DANIELEWSKI 型超表面同构的因子仿射 Gα 变体
令k为特征零的代数闭域,B为阶乘仿射k域,具有局部幂零导数δ。我们研究B当存在一个元素z ∈ B使得δ (z) = α p对于 p ≥ 1 和A的素元素 α = Ker δ。一个特别有趣的例子是仿射伪n空间的坐标环,它被定义为一个光滑的仿射簇X,具有忠实地在态射q : X → 𝔸1使得q –1 (𝔸 \( {}_{\ast}^1 \) ) ≅ 𝔸 \( {}_{\ast}^1 \) × 𝔸 n –1和q * (0) 是不可约的并减少了 𝔸 \( {}_{\ast}^1 \) = 𝔸 1 \ {0}。在 [9] 中,给出了仿射伪 3 空间与形式x p y – g ( x ; t ; z ) = 0 in 𝔸 4的超曲面同构的标准。这种超曲面称为 Danielewski 类型的超曲面,并在 [17]、[16] 中进行了研究。在本文中,在A/α A是阶乘,我们用 Kaliman 和 Zaidenberg [15] 开发的等变仿射修饰来描述B,并给出B与残基环A [ Y同构的标准;Z ]/(α p Y – g ( Z )) 对于p ≥ 1 和不可约多项式g ( Z ) ∈ A [ Z ]\ A。因此,我们获得了一个仿射伪n空间与n ≥ 3的 Danielewski 类型超曲面同构的标准。