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.线性化问题与线性化问题密切相关的消去问题已经有了一些结果。（1）存在无限维仿射伪平面族，且不具有消去性质。此外，汤姆Dieck曲面的特点是仿射伪平面的代数环面的非平凡的行动。（2）带切片的三角导子的核是多项式环。2.广义雅可比问题（1）研究了具有A^1-纤维化的光滑仿射曲面，特别是仿射伪平面的雅可比问题。我们与R. V. Gurjar和P.Russell合作，在一定程度上弄清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证明了仿射伪平面的广义雅可比问题在满足广义Sard性质时有肯定解。

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

Affine pseudoplanes and cancellation problem

仿射伪平面和取消问题

• 发表时间: 2005
• 期刊: TransAmer.Math.Soc. 357・12
• 作者: M.Miyanishi;R.V.Gurjar;T.Mabuchi;M.Miyanishi;M.Miyanishi
• 通讯作者: M.Miyanishi

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

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