Integrated study on problems related to branched Galois covers

分支伽罗瓦覆盖问题的综合研究

• 批准号: 11640034
• 负责人: TOKUNAGA Hiroo
• 金额: $ 2.5万
• 依托单位: Tokyo Metropolitan University (2000)Kochi University (1999)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640034/
• 关键词: Galois cover; Linear representaion; Zariski pair; (2, 3) torus curves; dual curves; elliptic surface; elliptic curve; lattices; 交点数; Fiber sum

1. Construction problem of Galois covers : Galois covers for *_4 and *_4 have been studied by the head investigator. In 1999, he made investigation on such Galois covers based on the Lagrange's method in solving quartic equations. The idea was to understand the Langranges method in terms of linear equivalence of divisors on algebraic varieties. In the first half of 2000, he polished the results in 1999, and made them into more "user-friendly" form. All of these results and their applications are written in his preprint "Galois covers for *_4 and *_4 and their applications, which is submitted to Osaka Math.J.In the second half of 2000, he has started investigation about versal Galois covers. It gaves new daylight in the study of Galois covers with other kinds of finite groups such as *_5.2. Topology of open algebraic varieties : Some new examples of Zariski pairs were found by Tokunaga. Oka made the intensive study on (2, 3) torus sextic curve and the topology of their complement. Shimada gave a new invariant for the fundamental group of the complement to plane curves and applied it to study the Zariski pari given by Namba and Tsuchihashi.3. Singularites : Oka and his PhD Student Pho Duc Tai figured out almost all possible configurations of singularites of non-tame (2, 3) torus sextic curves.4. The fundamental study for Galois covers : Kurano and Ogoma gave some new results in commutative algebra. Tsuchimoto studied non-commutative algebraic geometry.

1. Galois覆盖的构造问题：主要研究者研究了 *_4和 *_4的Galois覆盖。1999年，他在求解四次方程的拉格朗日方法的基础上，对这类Galois覆盖进行了研究。这个想法是为了理解Langranges方法的线性等价因子代数簇。2000年上半年，他对1999年的结果进行了润色，使其成为更“用户友好”的形式。所有这些结果及其应用都写在他提交给Osaka Math.J的预印本“Galois covers for *_4 and *_4 and their applications”中。它为研究有限群的Galois覆盖提供了新的思路。开代数簇的拓扑：德永发现了一些新的Zuraki对的例子。Oka对（2，3）环面六次曲线及其补曲线的拓扑作了深入的研究。Shimada给出了平面曲线的补曲线的基本群的一个新的不变量，并应用它研究了Namba和Tsuchihashi给出的Zakki pari.奇点主义者：Oka和他的博士生Pho Duc Tai计算出了非驯服（2，3）环面六次曲线的奇点的几乎所有可能的配置。伽罗瓦的基础研究包括：Kurano和Ogoma在交换代数中给出了一些新的结果。土本研究非交换代数几何。

项目成果

H.Tokunaga: "Some examples of Zariski pairs arising from certain elliptic K3 surfaces,II"Math.Z. 230. 389-400 (1999)

H.Tokunaga：“由某些椭圆 K3 表面产生的 Zariski 对的一些例子，II”Math.Z。

H.Tokunaga: "Irreducible plane curves with the Albanese dimesion 2"Proc.AMS. 127. 1935-1940 (1999)

H.Tokunaga：“Albanese 维数 2 的不可约平面曲线”Proc.AMS。

M.Oka: "Geometry of cuspidal sextics and their dual curves"Singularities-Sapporo 1998, Advanced Studies in Math.. 29. 245-277 (2000)

M.Oka：“尖六角几何及其对偶曲线”Singularities-Sapporo 1998，数学高级研究.. 29. 245-277 (2000)

H.Tokunaga, (with E.Artal Bartolo): "Zariski pairs of index 19 and the Mordell-Weil groups K3 surfaces"Proc. London Math. Soc.. 80. 127-144 (2000)

H.Tokunaga（与 E.Artal Bartolo）：“索引 19 的 Zariski 对和 Mordell-Weil 群 K3 曲面”Proc。

H.Tokunaga, (E.Artal Bartolo, J.Carmona Ruber and J.I.Cogolludo): "Sextics with singular points in special positions"J.of Knot theory and its ramifications. (to appear).

H.Tokunaga，（E.Artal Bartolo、J.Carmona Ruber 和 J.I.Cogolludo）：“在特殊位置具有奇点的六角学”J. of Knot 理论及其分支。