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Degree of irrationalities of algebraic surfaces with Kodaira dimension zero

Research Project

Project/Area Number 10640013
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNIIGATA UNIVERSITY

Principal Investigator

YOSHIHARA Hisao  Faculty of Science, NIIGATA UNIVERSITY Professor, 理学部, 教授 (60114807)

Co-Investigator(Kenkyū-buntansha) TAJIMA Shin-ichi  Faculty of Technology, NIIGATA UNIVERSITY Assistant Professor, 工学部, 助教授 (70155076)
AKIYAMA Shigeki  Faculty of Science, NIIGATA UNIVERSITY Assistant Professor, 理学部, 助教授 (60212445)
TAKEUCHI Teruo  Faculty of Science, NIIGATA UNIVERSITY Assistant Professor, 理学部, 助教授 (10018848)
TOKUNAGA Hiro-o  Kochi University, Faculty of Science, Assistant Professor, 理学部, 助教授 (30211395)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 1999: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1998: ¥1,700,000 (Direct Cost: ¥1,700,000)
Keywordsdegree of irrationality / algebraic surface / Kodaira dimension / algebraic curve / Galois point / maximal rational intermediate subfield / 非有理次数 / 極大有理中間体
Research Abstract

1. Let Si (I = 1,2) be smooth projective surfaces and f : SィイD21ィエD2 → SィイD22ィエD2 be the surjective morphism. There has been a problem that whether the inequality dr(SィイD21ィエD2) 【greater than or equal】 dr(SィイD22ィエD2) hold true, where dr is the degree of irrationality. We have found examples which do not satisfy the inequality in the class of hyperelliptic surfaces.
2. Let S be a hyperelliptic surface. We have proved that dr(S) = 2, 3 or 4, and that dr(S) = 2 if and only if 2KィイD2sィエD2 is trivial, where KィイD2sィエD2 is the canonical bundle of S.
3. Let C be a plane curve of degree d and K = k(C) be the rational function field of C. We consider a projection пp of C from P ∈ C to a line l =ィイD4〜ィエD4 PィイD11ィエD1. The projection induces the extension of fields πィイD2PィエD2ィイD1*ィエD1 : k(l) * K. We have studied the structure of this extension from geometrical viewpoint. If the extension is Galois, we call P Galois point. We have determined all the Galois points. If P is not a Galois point, we cosider the minimal splitting field Lp of the extension K/k(l) and the Galois group Gal(Lp/k(l)). Our study have been done in this latter case only for d = 4, the general case must be studied in the furture.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (17 results)

All Other

All Publications (17 results)

  • [Publications] Hisao Yoshihara: "A note on the inequality of degrees of irrationalities of algebraic surfaces"Journal of Algebra. 207. 272-275 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiro-o Tokunaga: "Some examples of Zariski pairs arising from certain elliptic K3 surfaces"Mathematische Zeitschrift. 230. 389-400 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hisao Yoshihara: "Degree of irrationality of Hyperelliptic surfaces"Algebra Colloquium. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kei Miura: "Field theory for function fields of plane quartic curves"Journal of Algebra. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kei Miura: "Field theory for function fields of the quintic Fermat curves"Communications in Algebra. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hisao Yoshihara: "A note on the inequality of degrees of irrationalities of algebraic surfaces"Journal of Algebra. Vol. 207. 272-175 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiro-o Tokunaga: "Some examples of Zariski pairs arising from certain elliptic K3 surfaces"Mathematische Zeitschrift. Vol. 230. 389-400 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hisao Yoshihara: "Degree of irrationality of hyperelliptic surfaces"Algebra Colloquium. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kei Miura: "Field theory for function fields of plane quartic curves"Journal of Algebra. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kei Miura: "Field theory for function field of the quintic Fermat curve"Communications in Algebra. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hisao Yoshihara: "Degree of irrationality of Hyperelliptic surfaces"Algebra Colloquium. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] Kei Miura: "Field theory for function fields of plane quartic curves"Journal of Algebra. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] Kei Miura: "Field theory for function field of the quintic Fermat curve"Communications in Algebra. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] Hisao Yoshihara: "A note on the inequality of degrees of irrationalities of algebraic surface" Journal of Algebra. 207. 272-275 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hiro-o Tokunaga: "Some examples of Zariski pairs orising from certain elliptic K3 surfaces" Mathematische Zeitschrift. 227. 465-477 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hiro-o Tokunaga: "Some examples of Zariski pairs arising from certain elliptic K3 surfaces" Mathematische Zeitschift. 230. 389-400 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Shinichi Tajima: "Perturbed Lame equation and Buslav phase" Ukraine Journal of Mathematics. (in press).

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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