Algebraic group actions and the structure of affine algebraic varieties

代数群作用和仿射代数簇的结构

• 批准号: 15540043
• 负责人: MASUDA Kayo
• 金额: $ 1.73万
• 依托单位: University of Hyogo (2004-2005)Himeji Institute of Technology (2003)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540043/
• 关键词: affine pseudo-plane; Cancellation Problem; Makar-Limanov invariant; Linearization Problem; Jacobian Conjecture; A^1-fibration; algebraic torus action; Cancellation Problem; Makar-Limanov invariant; A^1-fibration; 埋め込み問題; G_m作用; G endonlorphism

1.Linearization ProblemWe have had some results on the Cancellation Problem which is closely related to the Linearization Problem.(1)There exists an infinite dimensional family of affine pseudo-planes without cancellation property. Furthermore, tom Dieck surfaces are characterized as affine pseudo-planes with nontrivial actions of an algebraic torus.(2)The kernel of a triangular derivation with a slice is a polynomial ring.2.Generalized Jacobian Problem(1)We studied the Jacobian Problem for smooth affine surfaces with A^1-fibrations, especially, affine pseudo-planes. We succeeded in making clear of the structure of affine pseudo-planes to some extent with joint work with R.V.Gurjar and P.Russell, but could not resolve the Generalized Jacobian Problem for affine pseudo-planes.(2)Miyanishi showed that the Generalized Jacobian Problem for affine pseudo-planes has an affirmative answer if they satisfy the generalized Sard property.

1.线性化问题我们在与线性化问题密切相关的消除问题上取得了一些成果。(1)存在一个无限维仿射伪平面族，不具有消除性质。此外，Tom Dieck曲面被描述为具有代数环面非平凡作用的仿射伪平面。(2)切片三角导数的核是多项式环。2.广义雅可比问题(1)我们研究了雅可比问题对于具有 A^1 纤维的光滑仿射表面，尤其是仿射伪平面。我们与R.V.Gurjar和P.Russell合作，在一定程度上成功地阐明了仿射伪平面的结构，但未能解决仿射伪平面的广义雅可比问题。(2)Miyanishi证明了广义雅可比问题如果仿射伪平面满足广义 Sard 性质，则它们的问题得到肯定的答案。

项目成果

Equivariant classification of Gorenstein open log del Pezzo surfaces with finite group actions

具有有限群作用的 Gorenstein 开对数 del Pezzo 曲面的等变分类

• 发表时间: 2004
• 期刊: J.Math.Soc.Japan 56
• 作者: M.Miyanishi;De-Qi Zhang
• 通讯作者: De-Qi Zhang

R.V.Gurjar, M.Miyanishi: "Automorphisms of affine surfaces with A^1-fibrations"Michigan Math.J.. (2004)

R.V.Gurjar、M.Miyanishi：“A^1-fibrations 的仿射曲面的自同构”Michigan Math.J.. (2004)

Affine pseudo-converings of algebraic surfaces

代数曲面的仿射伪转换

• 发表时间: 2005
• 期刊: J. Algebra 194
• 作者: T.Kambayashi;M.Miyanishi;M.Miyanishi
• 通讯作者: M.Miyanishi

The additive group actions on Q-homology planes

Q-同调平面上的加性群作用

• 发表时间: 2003
• 期刊: Ann. Inst. Fourier. Grenoble 53
• 作者: K.Masuda;M.Miyanishi
• 通讯作者: M.Miyanishi

Recent developments in affine algebraic geometry

仿射代数几何的最新进展

• 期刊: Affine Algebraic Geometry (in honor of Prof. M. Miyanishi) (in press)
• 作者: R.V.Gurjar;K.Masuda et al.;K.Masuda;M.Miyanishi;M.Miyanishi
• 通讯作者: M.Miyanishi