Research on the structures of hypersurfaces and their function fields
Project/Area Number 
13640013

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  NIIGATA UNIVERSITY 
Principal Investigator 
YOSHIHARA Hisao NIIGATA UNIVERSITY Faculty of Science, Professor, 理学部, 教授 (60114807)

CoInvestigator(Kenkyūbuntansha) 
KONNO Kazuhiro Osaka University Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (10186869)
OHBUCHI Akira Tokushima University Faculty of Integrated Arts and Sciences, Professor, 総合科学部, 教授 (10211111)
TAJIMA Shinichi Faculty of Engineering, Professor, 工学部, 教授 (70155076)
TOKUNAGA Hiroo Tokyo Metropolitan University Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (30211395)
AKIYAMA Shigeki Faculty of Science, Associate Professor, 理学部, 助教授 (60212445)

Project Period (FY) 
2001 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2002: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2001: ¥1,900,000 (Direct Cost: ¥1,900,000)

Keywords  Galois point / hypersurface / projection / function field / Galois group / space curve / Galoi line / 射影空間 / 代数多様体 
Research Abstract 
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.

Report
(3 results)
Research Products
(6 results)