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Fano varieties and moduli spaces with emphasis on the Verlinde Formula and the 14^<th> problem of Hilbert

Research Project

Project/Area Number 17340006
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

MUKAI Shigeru  Kyoto University, 数理解析・研究所, 教授 (80115641)

Co-Investigator(Kenkyū-buntansha) MORI Shigefumi  京都大学, 数理解析研究所, 教授 (00093328)
NAKAYAMA Noboru  京都大学, 数理解析研究所, 准教授 (10189079)
ABE Takeshi  京都大学, 数理解析研究所, 助教 (90362409)
NAKAMURA Iku  北海道大学, 大学院・理学研究科, 教授 (50022687)
KURANO Kazuhiko  明治大学, 理工学部, 教授 (90205188)
YOSHIOKA Kota  神戸大学, 大学院・理学研究科, 教授 (40274047)
TAKEUCHI Kiyohiko  椙山女学園大学, 教育学部, 教授 (30236418)
TAKAGI Hiromichi  東京大学, 大学院・数理科学研究科, 准教授 (30322150)
IDE Manabu  常葉学園大学, 教育学部, 講師 (90367582)
Co-Investigator(Renkei-kenkyūsha) MORI Shigefumi  京都大学, 数理解析研究所, 教授 (00093328)
NAKAYAMA Noboru  京都大学, 数理解析研究所, 准教授 (10189079)
ABE Takeshi  京都大学, 数理解析研究所, 助教 (90362409)
NAKAMURA Iku  北海道大学, 大学院・理学研究科, 教授 (50022687)
KURANO Kazuhiko  明治大学, 理工学部, 教授 (90205188)
YOSHIOKA Kota  神戸大学, 大学院・理学研究科, 教授 (40274047)
TAKEUCHI Kiyohiko  椙山女学園大学, 教育学部, 教授 (30236418)
TAKAGI Hiromichi  東京大学, 大学院・数理科学研究科, 准教授 (30322150)
IDE Manabu  常葉学園大学, 教育学部, 講師 (90367582)
Project Period (FY) 2005 – 2008
Project Status Completed (Fiscal Year 2008)
Budget Amount *help
¥8,600,000 (Direct Cost: ¥7,700,000、Indirect Cost: ¥900,000)
Fiscal Year 2008: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2007: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2006: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2005: ¥2,800,000 (Direct Cost: ¥2,800,000)
Keywords代数幾何学 / ヒルベルトの第14問題 / quiver / フェアリンデ公式 / ヒルベルト概型 / エンリケス曲面 / ファノ多様体 / モジュライ / 代数的ベクトル束 / 高次元双有理幾何学 / Verlinde公式 / 代数曲面 / K3曲面 / Enriques曲面 / Hilbert概型 / Brill-Noether理論 / Hom概型 / レベル階数双対性 / Kummer曲面 / Hilbertの第14問題
Research Abstract

代数多様体の中には、曲線、K3曲面とファノ多様体という3つの良い族がある.これらは単独でも興味深いが、互いにモジュライという関係で繋がっていることを観察することによって、より深い理解に到達できると思う.今回の研究課題では、フェアリンデ型公式との関係や不変式環への応用から研究を始めて、エンリケス曲面の位数2のある種の自己同型や偏極K3曲面のモジュライの単有理性問題への応用を研究した.また、Mumfordのpathologyとして有名な現象をよく理解するために、3次元多様体内の曲線の変形に対する障害類が消えないための充分条件についても研究した.

Report

(5 results)
  • 2008 Annual Research Report   Final Research Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (28 results)

All 2009 2008 2007 2006 2005 Other

All Journal Article (14 results) (of which Peer Reviewed: 4 results) Presentation (11 results) Book (2 results) Remarks (1 results)

  • [Journal Article] Obstructions to deforming curves on a 3-fold, I2009

    • Author(s)
      S. Mukai and H. Nasu
    • Journal Title

      Journal of Algebra Geometry (to appear)(掲載確定)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Geometric realization of T-shaped root systems and the Jacobians of del Pezzo surfaces, "Geometry in Osaka", Lecture Notes in Math2008

    • Author(s)
      Shigeru Mukai
    • Journal Title

      Osaka University (vol.9)

      Pages: 134-136

    • Related Report
      2008 Final Research Report
  • [Journal Article] モジュライ入門-数え上げとコンパクト化をめぐって2008

    • Author(s)
      向井 茂
    • Journal Title

      数理科学 3

      Pages: 37-42

    • Related Report
      2007 Annual Research Report
  • [Journal Article] Polarized K3 surfaces of genus thirteen, Adv. Stud2006

    • Author(s)
      Shigeru Mukai
    • Journal Title

      Pure Math. 45

      Pages: 315-326

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Polarized K3 surfaces of genus thirteen2006

    • Author(s)
      S.Mukai
    • Journal Title

      Adv. Stud. Pure Math. 45

      Pages: 315-326

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The cohomology groups of stable quasi-abelian schmes and degenerations associated with the E8-lattice2006

    • Author(s)
      I.Nakamura
    • Journal Title

      Adv. Stud. Pure Math. 45

      Pages: 223-281

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Moduli spaces of twisted sheaves on a projective variety (with appendix by D. Huybrechts and P. Stelari)2006

    • Author(s)
      K.Yoshioka
    • Journal Title

      Adv. Stud. Pure Math. 45

      Pages: 1-42

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The moduli stack of rank-two Gieseker bundles with fixed determinant on a nodal curve2006

    • Author(s)
      T.Abe
    • Journal Title

      Adv. Stud. Pure Math. 45

      Pages: 117-144

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On SL(2)-GL(n) strange duality2006

    • Author(s)
      T.Abe
    • Journal Title

      J. Math. Kyoto Univ. 46

      Pages: 657-692

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Obstructions to deforming space curves and non-reduced components of the Hilbert scheme2006

    • Author(s)
      H.Nasu
    • Journal Title

      Publ. RIMS Kyoto Univ. 42

      Pages: 117-141

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Compact complex surfaces admitting non-trivial surjective endomorphisms2005

    • Author(s)
      N.Nakayama, Yoshio Fujimoto
    • Journal Title

      Tohoku Math.J. 57

      Pages: 395-426

    • NAID

      110001792946

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Obstructions to deforming curves on a 3-fold I - a generalization of Mumford's example and an application to Hom schemes, to appear in J. Alg.

    • Author(s)
      Shigeru Mukai and Hirokazu Nasu
    • Journal Title

      Geometry

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Obstructions to deforming curves on a 3-fold, I --- a generalization of Mumford's example and an application to Horn schemes--

    • Author(s)
      S. Mukai, H. Nasu
    • Journal Title

      J. Alg. Geom. (to appear掲載確定)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Polarized K3 surfaces of genus thirteen

    • Author(s)
      S.Mukai
    • Journal Title

      Adv.Stud.Pure Math. to appear

    • Related Report
      2005 Annual Research Report
  • [Presentation] Polarized K3 surfaces of degree 322008

    • Author(s)
      Shigeru Mukai
    • Organizer
      Moduli Spaces
    • Place of Presentation
      Univ. of Warwick, UK
    • Year and Date
      2008-07-10
    • Related Report
      2008 Final Research Report
  • [Presentation] Cohomologically trivial involutions of Enriques surfaces with application to Shioda-Inose structure2008

    • Author(s)
      Shigeru Mukai
    • Organizer
      Algebraic Geometry
    • Place of Presentation
      Lorentz Center, Leiden,the Netherland
    • Year and Date
      2008-07-02
    • Related Report
      2008 Final Research Report
  • [Presentation] Geometric realization of T-shaped root systems and Jacobians of del Pezzo surfaces2007

    • Author(s)
      S. Mukai
    • Organizer
      Complex geometry in Osaka
    • Place of Presentation
      大阪大学
    • Year and Date
      2007-11-03
    • Related Report
      2007 Annual Research Report
  • [Presentation] Enriques surfaces covered by Kummer's quartics2007

    • Author(s)
      S. Mukai
    • Organizer
      Komplexe Algebraische Geometrie
    • Place of Presentation
      Oberwolfach, 独
    • Year and Date
      2007-10-02
    • Related Report
      2007 Annual Research Report
  • [Presentation] Enriques surfaces associated with bielliptic abelian surfaces2007

    • Author(s)
      S. Mukai
    • Organizer
      Algebraic and arithmetic structures of moduli spaces
    • Place of Presentation
      北海道大学
    • Year and Date
      2007-09-04
    • Related Report
      2007 Annual Research Report
  • [Presentation] Hilbert's 14th problem for the Kronecker quiver2007

    • Author(s)
      Shigeru Mukai
    • Organizer
      "Algebraic geometry in higher dimensions
    • Place of Presentation
      Levico Terme (Trento), Italy
    • Year and Date
      2007-06-08
    • Related Report
      2008 Final Research Report
  • [Presentation] Hilbert's 14th problem for the Kronecker quiver2007

    • Author(s)
      S. Mukai
    • Organizer
      Algebraic geometry in higher dimensions
    • Place of Presentation
      Levico Terme (Trento), 伊
    • Year and Date
      2007-06-08
    • Related Report
      2007 Annual Research Report
  • [Presentation] Cohomologically trivial automorphisms of Enriques surfaces2007

    • Author(s)
      S. Mukai
    • Organizer
      ミラノ大学談話会
    • Place of Presentation
      ミラノ、伊
    • Year and Date
      2007-06-01
    • Related Report
      2007 Annual Research Report
  • [Presentation] Hilbert's 14th problem for the Kronecker quiver2007

    • Author(s)
      S. Mukai
    • Organizer
      Advances in algebra and geometry
    • Place of Presentation
      MSRI, USA
    • Year and Date
      2007-04-30
    • Related Report
      2007 Annual Research Report
  • [Presentation] Hilbert's 14th problem and Verlinde type formulas for rings of invariant polynomials2007

    • Author(s)
      Shigeru Mukai
    • Organizer
      Quwait lecture
    • Place of Presentation
      Cambridge Univ., England
    • Year and Date
      2007-01-30
    • Related Report
      2008 Final Research Report
  • [Presentation] Hilbert's original 14th problem and certain moduli spaces2005

    • Author(s)
      Shigeru Mukai,
    • Organizer
      AMS Summer Institute on Algebraic Geometry
    • Place of Presentation
      Seattle, U.S.A
    • Related Report
      2008 Final Research Report
  • [Book]2008

    • Author(s)
      向井茂
    • Total Pages
      458
    • Publisher
      モジュライ理論I・II(改訂新版)
    • Related Report
      2008 Final Research Report
  • [Book]

    • Author(s)
      岩波書店
    • Related Report
      2008 Final Research Report
  • [Remarks]

    • URL

      http://www.kurims.kyoto-u.ac.jp/~mukai

    • Related Report
      2007 Annual Research Report

URL: 

Published: 2005-04-01   Modified: 2016-04-21  

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