Hypersurfaces in the three dimensional projective spaces and its complement
Project/Area Number  02640027 
Research Category 
GrantinAid for General Scientific Research (C)

Allocation Type  Singleyear Grants 
Research Field 
代数学・幾何学

Research Institution  Niigata University 
Principal Investigator 
YOSHIHARA Hisao College of General Education, Assistant Professor, 教養部, 助教授 (60114807)

CoInvestigator(Kenkyūbuntansha) 
TAJIMA Shinichi College of General Education, Lecturer, 教養部, 講師 (70155076)
TAKEUCHI Teruo College of General Education, Assistant Professor, 教養部, 助教授 (10018848)
SERIZAWA Hisamitsu College of General Education, Assistant Professor, 教養部, 助教授 (00042771)

Project Period (FY) 
1990 – 1991

Project Status 
Completed(Fiscal Year 1991)

Budget Amount *help 
¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1991 : ¥200,000 (Direct Cost : ¥200,000)
Fiscal Year 1990 : ¥800,000 (Direct Cost : ¥800,000)

Keywords  Double covering / Quadratic extension / Birational classification / Surface of general type / Ouadratic extention / Degree of irrationality / 2重被覆 / 一般型代数曲面 / 分岐因子 / 有理曲面 / 通常特異点 
Research Abstract 
Let k be an algebraically closed field of characteristic zero. We denote by K a purely transcendental extension of k, and by L a quadratic extension of K, If dim_k K = 1 or equivalently if K = K(x), then L can be expressed as K(y), where the element y satisfies the equation y^2 = (xa_1)・・・(xa_<2n+1>) with distinct a_iE_k. The model of L is a rational, an elliptic or a hyperelliptic curve. In this study we consider the similar subject in the case when dim_kK = 2. Suppose K = k(x, y) and let S be a nonsingular model of L. Then we study the structure of S from the birational viewpoint. First we obtain the following result : THEOREM 1. If L/K is a quadratic extension, then the field L can be written as K(ROO<f(x, y)> where f(x, y) is a reduced polynomial of even degree such that the curve C : f = 0 in P2 has at most ordinary singularities. From the above we conclude the following : THEOREM 2. A belian and hyperelliptic surfaces cannot be birationally equivalent to double coverings of P^2, Except those, every class in the classification can appear.

Report
(3results)
Research Products
(7results)