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Study on integral convex polytopes and toric varieties

Research Project

Project/Area Number 16540004
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionTohoku University

Principal Investigator

OGATA Shoetsu  Tohoku University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (90177113)

Co-Investigator(Kenkyū-buntansha) ISHIDA Masanori  Tohoku University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (30124548)
HARA Nobuo  Tohoku University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (90298167)
KAJIWARA Takeshi  Yokohama National University, Faculty of Enginieering, Associate Professor, 大学院工学研究院, 助教授 (00250663)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywordsconvex polytope / toric variety / algebraic geometry / 孤立特異点 / トロピカル幾何 / アーベル多様体
Research Abstract

Ogata studied about ideals defining projective toric varieties. We say the ideal satisfies the property (Np) if its free resolution is the simplest up to the degree p part. We obtained a best possible estimate of this p with respect to the dimension of the variety and the times of tensor product of the same ample line bundle. We also obtained an algebro-geometric proof of the Theorem of Fakhruddin, which states that any global section of Ample bundle multiplied by a nef bundle is a nultiple of sections of the ample bundle and the nef bundle. Next we proved that an ample line bundle on a nonsingular toric 3-fold is normally generated if the bundle after added with the twice of the canonical bundle has no global sections. As a consequence, we showed that the anti-canonical bundle on nonsingular toric Fano 4-fold is normally generated.
Ishida studied about completion of real fans by using the notion of Zariski-Riemann spaces. We also determined the moduli number of CCI surfaces which are defined by Hirotaka Ishida.
Hara studied about isolated singularities of dimension two in positive characteristic. We define the F-pure threshold ftp(X, D) for a pair of a surface and an effective divisor which is an analog of the logarithmic canonical threshold. And we showed that ftp(X, D) has values in rational numbers by using the method of p-fractal.
Kajiwara studied abuot the relation between tropical geometry and toric geometry. We characterized degenerations of projective toric varieties defined from polyhedral decompositions which are determined naturally by tropical hypersurfaces.We also reconstructed tropical toric varieties in terms of algebraic geometry. As a consequence, we obtained
We also reconstructed tropical toric varieties in terms of algebraic geometry. As a consequence, we obtained an intersection theory on tropical nonsingular toric surfaces.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (24 results)

All 2006 2005 2004

All Journal Article (22 results) Book (2 results)

  • [Journal Article] On higher syzygies of projective toric varieties2006

    • Author(s)
      尾形 庄悦
    • Journal Title

      Mathematical Journal of Okayama University 48

      Pages: 47-55

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Multiplication maps of complete linear systems on projective toric surfaces2006

    • Author(s)
      尾形 庄悦
    • Journal Title

      Interdisciplinary Information Sciences 12

      Pages: 93-107

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Completion of real funs and Zariski-Riemann spaces2006

    • Author(s)
      石田 正典
    • Journal Title

      Tohoku Mathematical Journal 58

      Pages: 189-218

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] F-pure thresholds and F-jumping exponents in dimension two2006

    • Author(s)
      原 伸生
    • Journal Title

      Mathematical Research letter 13

      Pages: 747-760

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Abelian surfaces in projective 4-folds2006

    • Author(s)
      梶原 健
    • Journal Title

      Archiv der Mathematik 86

      Pages: 43-49

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] On higher syzygies of projective toric varieties2006

    • Author(s)
      S.Ogata
    • Journal Title

      Math. J. Okayama Univ. 48

      Pages: 47-55

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Multiplication maps of complete linear systems on projective toric surfaces2006

    • Author(s)
      S.Ogata
    • Journal Title

      Interdisciplinary Information Sciences 12

      Pages: 93-107

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Completion of real fans and Zariski-Riemann spaces2006

    • Author(s)
      M.Ishida
    • Journal Title

      Tohoku Math. J. 58

      Pages: 189-218

    • NAID

      110004669750

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] F-pure thresholds and F-jumping exponents in dimension two2006

    • Author(s)
      N.Hara
    • Journal Title

      Math. Research Letter 13

      Pages: 747-760

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Abelian surfaces in projective 4-folds2006

    • Author(s)
      T.Kajiwara
    • Journal Title

      Archiv der Math. 86

      Pages: 43-49

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Completion of real fans and Zariski-Riemann spaces2006

    • Author(s)
      石田 正典
    • Journal Title

      Tohoku Mathematical Journal 58

      Pages: 189-218

    • NAID

      110004669750

    • Related Report
      2006 Annual Research Report
  • [Journal Article] F-pure thresholds and F-jumping exponents in dimension two2006

    • Author(s)
      原 伸生
    • Journal Title

      Mathematical Research Letter 13

      Pages: 747-760

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Completion of real fans and Zariski-Riemann spaces2006

    • Author(s)
      石田正典
    • Journal Title

      Tohoku Mathematical Journal (未定)

    • NAID

      110004669750

    • Related Report
      2005 Annual Research Report
  • [Journal Article] k-normakity of weighted projective spaces2005

    • Author(s)
      尾形 庄悦
    • Journal Title

      Kodai Mathematical Journal 28

      Pages: 519-524

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A characteristic p analog of multiplier ideals and applications2005

    • Author(s)
      原 伸生
    • Journal Title

      Communications in Algebra 33

      Pages: 3375-3388

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2005 Annual Research Report
  • [Journal Article] k-normality of weighted projective spaces2005

    • Author(s)
      S.Ogata
    • Journal Title

      Kodai Math. J. 28

      Pages: 519-524

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A characteristic p analog of multiplier ideals and applications2005

    • Author(s)
      N.Hara
    • Journal Title

      Communications in Algebra 33

      Pages: 3375-3388

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] k-normality of weighted projective spaces2005

    • Author(s)
      尾形庄悦
    • Journal Title

      Kodai Mathematical Journal 28・3

      Pages: 519-524

    • Related Report
      2005 Annual Research Report
  • [Journal Article] K-normality of weighted progective spaces2005

    • Author(s)
      尾形庄悦
    • Journal Title

      Kodai Mathematical Journal (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A characteristic p analog of multiplier ideals and applications2005

    • Author(s)
      原 伸生
    • Journal Title

      Communication in Algelora 33(印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a generalization of test ideals2004

    • Author(s)
      原 伸生
    • Journal Title

      Nagoya Mathematical Journal 175

      Pages: 59-74

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] On a generalization of test ideals2004

    • Author(s)
      N.Hara
    • Journal Title

      Nagoya Math. J. 175

      Pages: 59-74

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] 代数曲線入門-はじめての代数幾何2004

    • Author(s)
      梶原 健
    • Total Pages
      323
    • Publisher
      日本評論社
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Book] Nihonhyouronsha2004

    • Author(s)
      T.Kajiwara
    • Total Pages
      323
    • Publisher
      Introduction to Algebraic curves-A first course to Algebraic Geometry
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary

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Published: 2004-04-01   Modified: 2016-04-21  

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