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2004 Fiscal Year Final Research Report Summary

Moduli spaces and arithmetic geometry

Research Project

Project/Area Number 13440008
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

MORIWAKI Atushi  Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (70191062)

Co-Investigator(Kenkyū-buntansha) MARUYAMA Masaki  Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (50025459)
UENO Kenji  Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (40011655)
FUKAYA Kenji  Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (30165261)
NAKAJIMA Hiraku  Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (00201666)
KATO Fumiharu  Kyoto University, Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (50294880)
Project Period (FY) 2001 – 2004
KeywordsLogarithmic Geometry / Diophantine Geometry / Rational point / Moduli space
Research Abstract

During this research project, we mainly studied the following three materials :
(1)Picard group of the moduli space of curves
(2)Counting problem of algebraic cycles
(3)Kobayashi-Ochiai's theorem in the category of log schemes
In the following, we explain the details of each material.
(1)We did not know the problem of algebraic cycles on the moduli space of curves in positive characteristic even for the divisor case. We justify this problem. Namely, we showed that the Picard group of the moduli space of stable n-pointed curves is generated by the tautological line bundles and the boundary classes. By this theorem, several results in characteristic zero were generalized to the case of positive characteristic by Gibney-Keel-Morrison and Schroeer. Besides them, we obtained the results concerning the Mori cone.
(2)We estimated the order of growth of the number of algebraic cycles with bounded arithmetic degree on an arithmetic variety. By this, we can introduce a new kind of zeta functions in terms of the number of algebraic cycles. Similarly, we obtained the same result on an algebraic variety over a finite field.
(3)Kabayashi-Ochiai‘s theorem states that the number of dominant rational maps onto a compact complex manifold of general type is finite. From the view-point of Diophantine geometry, this theorem means that the number of rational points of a compact manifold of general type is finite for a big function field. This gives rise to an evidence for Lang's conjecture. Kazuya Kato conjectured a similar result in the category of log schemes. We proved the conjecture by the joint work with Dr.Iwanari. In the proof of this result, the crucial points are the local structure theorem and the rigidity theorem, which were generalized to the case of semistable schemes over a locally noetherian scheme.

  • Research Products

    (9 results)

All 2004 2002 2001

All Journal Article (7 results) Book (2 results)

  • [Journal Article] The number of algebraic cycle with bounded degree2004

    • Author(s)
      Atsushi Moriwaki
    • Journal Title

      J.Math.Kyoto Univ. 44

      Pages: 819-890

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Diophantine geometry viewed from Arakelov geometry2004

    • Author(s)
      Atsushi Moriwaki
    • Journal Title

      Sugaku Expositions 17

      Pages: 219-234

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The number of algebraic cycles with bounded degree2004

    • Author(s)
      Atsushi Moriwaki
    • Journal Title

      J.Math.Kyoto Univ. 44

      Pages: 819-890

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Nef divisors in codimension one on the moduli space of stable curves2002

    • Author(s)
      Atsushi Moriwaki
    • Journal Title

      Compositio Math. 132

      Pages: 191-228

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The Q-Picard group of the moduli space of curves in positive characteristic2001

    • Author(s)
      Atsushi Moriwaki
    • Journal Title

      Internat.J.Math. 12

      Pages: 519-534

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Inequalities for semistable families of arithmetic varieties2001

    • Author(s)
      Atsushi Moriwaki, Shu Kawaguchi
    • Journal Title

      J.Math.Kyoto Univ. 41

      Pages: 97-182

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A generalization of conjectures of Bogomolov and Lang over finitely generated fields2001

    • Author(s)
      Atsushi Moriwaki
    • Journal Title

      Duke Math.J. 107

      Pages: 85-102

    • Description
      「研究成果報告書概要(和文)」より
  • [Book] Algebraic geometry in East Asia2002

    • Author(s)
      Akira Ohbuchi, Kazuhiro Konno, Sampei Usui, Atsushi Moriwaki, Noboru Nakayama
    • Total Pages
      263
    • Publisher
      World Scientific Publishing Co., Inc.
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] Algebraic geometry in East Asia2002

    • Author(s)
      Akira Ohbuchi, Kazuhiro Konno, Sampei Usui, Atsushi Moriwaki, Noboru Nakayama
    • Total Pages
      263
    • Publisher
      World Scientific Publishing Co., Inc
    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2006-07-11  

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