Research on the structures of hypersurfaces and their function fields

超曲面结构及其函数场研究

• 批准号: 13640013
• 负责人: YOSHIHARA Hisao
• 金额: $ 2.24万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640013/
• 关键词: Galois point; hypersurface; projection; function field; Galois group; space curve; Galoi line; 射影空間; 代数多様体

Let V be a smooth hypersurface in P^<n+1>. We consider a projection of V from P ∈ P^<n+1> to a hyperplane H. This projection induces an extension of fields k(V)/k(H), which does not depend on the choice of H. The point P is called a Galois point if the extension is Galois. If, moreover, P ∈ V [resp. P 【not a member of】 V], then we call P an inner [resp. outer] Galois point. We denote by δ(V) [resp. δ(V^c)] the number of inner [resp. outer] Galois points. We have studied the extension K/K_P from geometrical points of view, especially we have considered the following problems:(1) Find all the Galois points. Do there exist any rules for the distribution of the points ?(2) Find the structure of the Galois group G_P at each point P ∈ P^<n+1>.(3) Find the structure of a nonsingular projective model of L_P.As results we have obtained the following; If V is general in the class of hypersurfaces with d 【greater than or equal】 4, then it has no Galois point. If d = 4 and d 【greater than or equal】 5, then δ(V) 【less than or equal】 4([n/2] + 1) and δ(V) 【less than or equal】 [n/2] + 1 respectively. On the other hand we have δ(V^c) 【less than or equal】 n + 2. The equality holds true if and only if V is projectively equivalent to the Fermat variety.

令V为P^<n+1>中的平滑性超表面。我们考虑V从p∈P^<n+1>到超平面H的投影。此投影诱导场k（v）/k（h）的延伸，这不取决于H的选择。如果扩展为Galois，则点P称为Galois点。此外，如果p∈V[resp。 p [不是] v的成员，然后我们称p为内部[resp。外部] galois点。我们用δ（v）表示[resp。 δ（v^c）]内部数量[resp。外部] galois点。我们已经从几何学角度研究了扩展K/K_P，尤其是我们考虑了以下问题：（1）找到所有Galois点。是否有任何分布分布的规则？（2）在每个点p∈P^p^<n+1>>>中找到Galois组G_P的结构。如果V在具有D [大于或相等] 4的Hypersurfaces中是一般的，则它没有Galois点。如果d = 4和d [大于或相等] 5，则δ（v）[小于或等于] 4（[n/2] + 1）和δ（v）[小于或相等] [n/2] + 1。另一方面，我们有δ（v^c）[小于或相等] n + 2。当v在v上与fermat品种相当时，相等性是正确的。

项目成果

Hisao Yoshihara: "Galois points for smooth hypersurfaces"Journal of Algebra. (発表予定).

Hisao Yoshihara：“光滑超曲面的伽罗瓦点”代数杂志（待出版）。

Cristina Duyaguit: "Galois lines for normal elliptic space curves"Algebra Colloquium. (発表予定).

Cristina Duyaguit：“正常椭圆空间曲线的伽罗瓦线”代数讨论会（待提交）。

Hisao Yoshihara: "Galois points for smooth hypersurfaces"Journal of Algebra.

Hisao Yoshihara：“光滑超曲面的伽罗瓦点”代数杂志。

Cristina Duyaguit: "Galois lines for normal elliptic space curves"Algebra Colloquium.

Cristina Duyaguit：“正常椭圆空间曲线的伽罗瓦线”代数讨论会。