Study on Galois embeddings of algebraic surfaces

代数曲面的伽罗瓦嵌入研究

• 批准号: 17540018
• 负责人: YOSHIHARA Hisao
• 金额: $ 1.98万
• 依托单位: Niigata University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540018/
• 关键词: algebraic surface; algebraic curve; Galois point; Galois embedding; abelian surface; birational transformation; Galois group; ガロワ部分空間; ガロワ群の表現

We have studied the Galois embedding of algebraic curves and surfaces, especially rational curves and abelian surfaces.In the case of abelian surfaces we have obtained all the possible types of the Galois groups which can appear as the covering transformation groups. Moreove we listed a lot of examples of abelian surfaces with given Galois groups of embeddings. In particular we have shown that such abelian surfaces are isogenous to the products of two elliptic curves. On the other hand, we have found the least number N such that abelinan surfaces have the embeddings into PAN. Concernig this study we have studied for singular plane rational curves. We determined all possible type of Galois group, i.e., they are cyclic, dihedral, A_4, S_4 and S_4 and have shown the examples with such Galois groups. Connecting with this research, we have studied if the Galois automorphism can be extended to a birational transformation or not. As a result we have obtained that there are a lot of rational curves such that the autopmorphism cannot be extended to birational transformation, for example we found rational curve with only nodes as singularities and the degree is bigger than 7 with outer Galois point.

我们已经研究了代数曲线和表面，尤其是理性曲线和阿贝尔表面的Galois嵌入。在Abelian表面的情况下，我们获得了所有可能显示为覆盖转换组的Galois基团的类型。 Moreove我们列出了许多带有Galois嵌入组的Abelian表面的例子。特别是我们已经表明，这种Abelian表面对两条椭圆形曲线的产物不同步。另一方面，我们发现了最少的N，因此Abelinan表面将嵌入到PAN中。我们研究了这项研究，我们研究了奇异平面有理曲线。我们确定了Galois组的所有可能类型，即它们是环状，二面的，A_4，S_4和S_4，并显示了此类Galois组的示例。与这项研究相关，我们研究了Galois的自动形态是否可以扩展到异性转变。结果，我们已经获得了许多有理曲线，因此自动晶型不能扩展到异常转化，例如，我们发现只有节点为奇异性的有理曲线，并且该度的程度大于7的galois点。

项目成果

Galois embedding of algebraic variety and its application to abelian suface

代数簇的伽罗瓦嵌入及其在阿贝尔面中的应用

• 发表时间: 2007
• 期刊: Rendiconti Sem.Mat., Universita di Pavia (in press)
• 作者: Tetsuo Fukui;Jiro Sekiguchi;Hisao Yoshihara
• 通讯作者: Hisao Yoshihara

Galois embedding of algebraic variety and its application to abelian surface

代数簇的伽罗瓦嵌入及其在阿贝尔曲面上的应用

• 期刊: Rendiconti Sem. Mat. Universita di Padova (In press)
• 作者: Hisao Yoshihara;Hisao YOSHIHARA;Hisao Yoshihara
• 通讯作者: Hisao Yoshihara

Galois points for plane rational curves

平面有理曲线的伽罗瓦点

• 发表时间: 2007
• 期刊: Far east J. of Math. Sci vol.25
• 作者: 渡部隆夫;吉満隆亮;K. Konno and M. Lopes;T. Takata;Jun-ichi Miyachi;渡部隆夫;K. Sekigawa and A. Yamada;渡部隆夫;渡部隆夫;Hisao Yoshihara;H. Yoshihara
• 通讯作者: H. Yoshihara