Research on space curve and its Galois line
Project/Area Number 
15540016

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) 
HOMMA Masaaki Kanagawa University, Faculty of Engineering, Professor, 工学部, 教授 (80145523)
OHBUCHI Akira The University of Tokushima, Faculty of Integrated Arts and Sciences, Professor, 総合科学部, 教授 (10211111)
AKIYAMA Shigeki Niigata University, Faculty of Science, Associate Professor, 理学部, 助教授 (60212445)
TOKUNAGA Hiroo Tokyo Metropolitan University, Graduate School and Faculty of Science, Associate Professor, 大学院・理学研究科, 助教授 (30211395)
KOJIMA Hideo Niigata University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (90332824)

Project Period (FY) 
2003 – 2004

Project Status 
Completed (Fiscal Year 2004)

Budget Amount *help 
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2004: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2003: ¥1,900,000 (Direct Cost: ¥1,900,000)

Keywords  space algebraic curve / Galois line / Galois group / automorphism group / Galois embedding / ramified covering / function field / birational transformation / アーベル曲面 
Research Abstract 
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_LC: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=Nn1 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.

Report
(3 results)
Research Products
(6 results)