Geometry of branched Galois covers and number theory

分支伽罗瓦覆盖几何和数论

• 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

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Galois覆盖的构造问题和Galois逆问题：这项工作主要由Tokunaga，Miyake和Tsuhihashi完成。Tokunaga主要研究了二维Verse Galois覆盖。两篇关于二维顶点Galois覆盖和有理椭圆曲面的论文正在出版中。此外，他还证明了对二维顶点Galois覆盖的研究归结为对有限群的Cremona表示的研究。关于这个问题的论文现在正在准备中。Tsuhihashi利用环面几何构造了二面体群和对称群的Veral Galois覆盖。他还研究了普适覆盖为多圆盘的P^2的Galois覆盖。Miyake引入了定义在Q上的两条与某些三次域有关的椭圆曲线，并给出了所谓的Mordell曲线的显式简短形式。2.开代数簇的拓扑和奇点：Nakamura研究了Grothendieck-Teichmuller群。Tokunaga与萨拉戈萨大学的Artal Bartolo对Zariskiκ-Plets进行了研究，并给出了一个新的例子。这一结果正在报道中。Oka完整地研究了平面性疗法。Shimada对Picard数为21的超奇异K3曲面上有理双点的所有可能构形进行了分类。

项目成果

H.Nakamura, H.Tsunogai: "Harmonic and equianharmonic equations in the Grothendieck-Teichmuller group"Form Mathematicum. 15. 877-892 (2003)

H.Nakamura、H.Tsunogai：“Grothendieck-Teichmuller 群中的调和和等调和方程”数学形式。

H.Tokunaga: "2-dimensional versal S_4-covers and rational elliptie surfaces"Seminaire et Congres, Soeiete Mathematique de France. (印刷中).

H. Tokunaga：“二维通用 S_4 覆盖和有理椭圆曲面”Seminaire et Congres，法国数学学会（正在出版）。

H.Tokunaga: "Calois covers for S_4 and A_4 and their applications"Osaka Math. J.. 39. 621-645 (2002)

H.Tokunaga：“Calois 涵盖了 S_4 和 A_4 及其应用”Osaka Math。

H.Tokunaga: "Note on a 2-dimensional versal D_8-cover"Osaka Math.J.. (印刷中).

H.Tokunaga：“关于二维通用 D_8-cover 的注释”Osaka Math.J..（正在印刷中）。

E.Artal Bartolo, H.Tokunaga: "Zariski k-plets of rational curve arrangements and dihedral covers"Topology and its Applications. (印刷中).

E.Artal Bartolo、H.Tokunaga：“有理曲线排列和二面覆盖的 Zariski k-plets”拓扑及其应用（出版中）。