Study of algebraic varieties with group action

群作用的代数簇研究

• 批准号: 10640039
• 负责人: NAKANO Tetsuo
• 金额: $ 1.79万
• 依托单位: Tokyo Denki University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640039/
• 关键词: 3-dimensional projective varieties; branched covering over plane; quotient terminal singularity; finite matrix group; 極小モデル; 群作用; ガロア被覆; SL(2)-作用

As "Study of algebraic varieties with group action", the author made study of the following four topics during the period 1998-2000 :(1) The author studied the 3-dimensional nonsingular projective algebraic varieties on which SL(2) acts with 2-dimensional general orbits. This class of varieties is the last class whose structure remains unknown among the 3-dimensional nonsingular projective algebraic varieties on which a simple algebraic group acts non-trivially. The author succeeded in classification of these varieties under the condition that they have no fixed points. This research was published as "Projective threefolds on which SL(2) acts with 2-dimensional general orbits" (Transactions of AMS).(2) The author studied with H.Nishikubo the maximal Galois branched covering over affine and projective planes with branch locus x^2=y^q, and clarified the structure of the Galois groups of these coverings and also got the condition for the existence for the maximal Galois covering. This research is to appear as "On some maximal Galois coverings over affine and projective planes II" in Tokyo J.Math.(3) The author studied with H.Takamidori the defining equations and the rigidity of 3-dimensional quotient terminal singularities from the viewpoint of computational algebraic geometry. This research has been submitted to a certain journal.(4) The finite primitive subgroups of SL(4) was classified by Blichfeldt to 30 types. The author confirmed with K.Niitsuma that these 30 types of matrix groups are actually primitive, using the computer algebra system MAGMA.Further, we calculated the Hilbert series of these finite matrix groups and determined the candidates among these matrix groups whose invariant rings are complete intersections. This research will be published as a paper "On primitive finite subgroups of SL(4) and their invariant rings".

1998-2000年期间，作者以“群作用代数簇的研究”为题，主要研究了以下四个方面的问题：（1）研究了SL（2）作用于其上的具有二维一般轨道的三维非奇异射影代数簇。这一类的品种是最后一类的结构仍然未知的3维非奇异投影代数簇上的一个简单的代数群的行为非平凡。作者在这些变种没有固定点的情况下，成功地将它们分类。这项研究发表于“SL（2）作用于其上的投影三重与2维一般轨道”（《AMS学报》）。(2)作者与H.Nishikubo研究了仿射平面和射影平面上具有分支轨迹x^2=y^q的极大Galois分支覆盖，阐明了这些覆盖的Galois群的结构，并得到了极大Galois覆盖存在的条件.本文的研究成果将作为“关于仿射和射影平面上的极大伽罗瓦覆盖II”发表在东京数学杂志上。(3)作者与H.Takamidori一起从计算代数几何的观点研究了三维商终端奇点的定义方程和刚性。这项研究已提交给某杂志。(4)Blichfeldt将SL（4）的有限本原子群分为30类。作者与K.Niitsuma一起用计算机代数系统MAGMA证明了这30类矩阵群实际上是本原的，并计算了这些有限矩阵群的Hilbert级数，确定了这些矩阵群中不变环是完全交的候选矩阵群。本文的研究成果将作为论文《关于SL（4）的本原有限子群及其不变环》发表。

项目成果

Tetsuo Nakano: "Projective threefolds on which SL(2) acts with 2-dimensional general orbits"Transactions of the AMS. 350. 2903-2924 (1998)

Tetsuo Nakano：“SL(2) 与二维一般轨道作用的投影三重”AMS 的交易。

Tetsuo Nakano: "Projective threefolds on which SL(2) acts with 2-dimensional general orbits" Transactions of the American Math.Society. 350・7. 2903-2924 (1998)

Tetsuo Nakano：“SL(2) 作用于二维一般轨道的投影三重”美国数学会汇刊 350・7 (1998)。

Tetsuo Nakano: "Projective threefolds on which Si(2) acts with 2-dimensional general orbits"Transactions of the AMS. 350. 2903-2924 (1998)

Tetsuo Nakano：“Si(2) 在二维一般轨道上作用的投影三重”AMS 的交易。

Tetsuo Nakano: "Hiroyasu Nishikabo, On some maximal Galois coverings over offine and projective planes II"Tokyo J.Math. 23. 295-310 (2000)

Tetsuo Nakano：“Hiroyasu Nishikabo，关于离线和射影平面上的一些最大伽罗瓦覆盖 II”Tokyo J.Math。

Tetsao Nakano: "Projective threefolds on which SL(2) acts with 2-dimensional general orbirs"Transactions of the AMS. 350. 2903-2924 (1998)

Tetsao Nakano：“SL(2) 与二维一般轨道作用的投影三重”AMS 的交易。