Research on space curve and its Galois line

空间曲线及其伽罗瓦线的研究

• 批准号: 15540016
• 负责人: YOSHIHARA Hisao
• 金额: $ 2.18万
• 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: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540016/
• 关键词: space algebraic curve; Galois line; Galois group; automorphism group; Galois embedding; ramified covering; function field; birational transformation; アーベル曲面

Let C, L and L_0 be a curve and lines in the projective three space P^3 respectively. Consider a projection p_L:P^3--→L_0 with center L, where L and L_0 have no intersection. Restricting p_L to C, we get a morphism p_L|C:C--→L_0 and an extension of fields:k(C)/k(L_0). We have studied the algebraic structure of the extension and the geometric one of C. If this extension is Galois, then we call L a Galois line. In particular we have studied the structure of the Galois group and the number of Galois lines for some special cases, for example, we obtained that the number is at most one if the degree of C is a prime number. After completed the first aims, we started to study the following research : Let V be a smooth projective variety and D be a very ample divisor. Let f:V--→ P^N be the projective embedding associated with |D|. Consider a projection p with a center W such that dim W=N-n-1 and f(V) does not meet W. Ifp f:V--→ P^n induces a Galois extension of function fields, then (V,D) is said to define a Galois embedding. Under this condition we have shown several properties of the Galois group of the covering. After general discussions we study the subject for abelian surfaces in detail.

令C，L和L_0分别为投射三个空间p^3中的曲线和线条。考虑一个投影p_l：p^3--→l_0带中心L，其中l和l_0没有相交。将p_l限制为c，我们得到了态度p_l | c：c - →l_0和字段的扩展：k（c）/k（l_0）。我们已经研究了延伸的代数结构和C的几何形状。如果此扩展为galois，那么我们称之为l galois线。特别是我们研究了Galois组的结构和某些特殊情况的Galois线数量，例如，如果C的程度是质量数，则最多获得数字。完成第一个目标后，我们开始研究以下研究：让V是一个平稳的投射品种，并且D是一个非常足够的除数。令f：v-→p^n是与| d |相关的投射嵌入。考虑一个具有中心w的投影p，使得dim w = n-n-1和f（v）不符合w。在这种情况下，我们显示了覆盖范围的Galois组的几个特性。在一般讨论之后，我们详细研究了阿贝尔表面的主题。

项目成果

Families of Galois closure curves for plane quartic curves

平面四次曲线的伽罗瓦闭合曲线族

• 发表时间: 2003
• 期刊: Journal of Mathematics of Kyoto University 43
• 作者: W.Nakai;T.Nakanishi;Ma.Cristina Lumakin Duyaguit;Ma.Cristina Lumakin Duyaguit;Ma.Cristina Lumakin Duyaguit;Hisao Yoshihara
• 通讯作者: Hisao Yoshihara

Galois lines for normal elliptic space curves

正规椭圆空间曲线的伽罗瓦线

• 发表时间: 2005
• 期刊: Algebra Colloquium 12
• 作者: W.Nakai;T.Nakanishi;Ma.Cristina Lumakin Duyaguit
• 通讯作者: Ma.Cristina Lumakin Duyaguit

Hisao Yoshihara: "Families of Galois closure curves for plane quartic curves"Journal of Mathematics of Kyoto University. (印刷中).

Hisao Yoshihara：“平面四次曲线的伽罗瓦闭合曲线族”京都大学数学杂志（正在出版）。