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Hypersurfaces in the three dimensional projective spaces and its complement

Research Project

Project/Area Number 02640027
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionNiigata University

Principal Investigator

YOSHIHARA Hisao  College of General Education, Assistant Professor, 教養部, 助教授 (60114807)

Co-Investigator(Kenkyū-buntansha) 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)
KeywordsDouble 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 = (x-a_1)・・・(x-a_<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

(3 results)
  • 1991 Annual Research Report   Final Research Report Summary
  • 1990 Annual Research Report
  • Research Products

    (7 results)

All Other

All Publications (7 results)

  • [Publications] Hisao YOSHIHARA: "Double Coverings of P^2" Proceedings of the Japan Academy. 66. 233-236 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Hisao YOSHIHARA: "Double Coverings of Rational Surfaces" Manuscripta Mathematica.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Hisao YOSHIHARA: "Double Coverings of P^2" Proceedings of the Japan Academy. 166 no. 8. 233-236

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Hisao YOSHIHARA: "Double Converings of Rational Surfaces" Manuscript Mathematica.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Hisao YOSHIHARA: "Double Coverings of P^2" Proceedings of the Japan Academy. 66. 233-236 (1990)

    • Related Report
      1991 Annual Research Report
  • [Publications] Hisao YOSHIHARA: "Double Coverings of Rational Surfaces" Manuscripta Mathematica.

    • Related Report
      1991 Annual Research Report
  • [Publications] Hisao YOSHIHARA: "Double Coverings of P^2" Proceedings of THE JAPAN ACADEMY,Ser.A. 66. 233-236 (1990)

    • Related Report
      1990 Annual Research Report

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Published: 1990-04-01   Modified: 2016-04-21  

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