Geometry of branched Galois covers and number theory

分支伽罗瓦覆盖几何和数论

基本信息

批准号：
14340015
负责人：
TOKUNAGA Hiroo
金额：
$ 2.88万
依托单位：
Tokyo Metropolitan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14340015/
关键词：
Galois cover versal cover fundahnental group cubic fields Zariski κ-plet super singular K3 surfaces rational double points Mordell curves Zariski Κ-plet 超特異κ3曲面 Zariski pair Alexander多項式 Groebner基底

项目摘要

1.Construction problem of branched Galois covers and the inverse Galois problem : This project were manly carried out by Tokunaga, Miyake and Tsuchihashi. Tokunaga mainly studied on 2-dimensional versal Galois covers. Two papers on 2-dimensional versal Galois covers and rational elliptic surfaces are in press. Also he show that the study on 2-dimensional versal Galois covers is reduced to that of Cremona representations of finite groups. The paper on this subject is now in preparation. Tsuchihashi constructed versal Galois covers for dihedral groups and the symmetric group by using toric geometry. He also studied Galois covers of P^2 whose universal cover is polydisc. Miyake introduced two ellitpic curves defined over Q, which is related to certain cubic fields, and gave explicit short forms so-called "Mordell Cures."2.Topology of open algebraic varieties and singularities : Nakamura made investigation on Grothendieck-Teichmuller group. Tokunaga made study on Zariski κ-plets with Artal Bartolo of Universidad Zaragoza, and gave a new example. This result is in press. Oka intesively studied plane sextic cures. Shimada classified all possible configurations of rational double points on super singular K3 surfaces with Picard number 21.

1.分支伽罗瓦覆盖的构造问题和逆伽罗瓦问题：该项目由德永、三宅和土桥共同完成，主要研究了二维通用伽罗瓦覆盖和有理椭圆的两篇论文。他还表明，二维通用伽罗瓦覆盖的研究被简化为有限克雷莫纳表示的研究。关于这个主题的论文正在准备中。土桥利用复曲面几何构造了二面体群和对称群的伽罗瓦覆盖，其通用覆盖是 Miyake 引入了两个椭圆曲线。 Q 与某些三次域有关，并给出了所谓的“Mordell Cures”的明确简形式。2. 开代数簇和奇点的拓扑：Nakamura 制作Grothendieck-Teichmuller 小组与 Zaragoza 大学的 Artal Bartolo 一起对 Zariski κ-plets 进行了研究，并给出了一个新的例子，Oka 深入研究了平面性治疗的所有可能配置。在皮卡德数为 21 的超奇异 K3 表面上。