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

• Project/Area Number: 17340006
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto 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: ¥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.

Research Products

• [Journal Article] Obstructions to deforming curves on a 3-fold, I (2009)
  • Author(s): S. Mukai and H. Nasu
  • Journal Title: Journal of Algebra Geometry (to appear)(掲載確定)
  • Peer Reviewed
• [Journal Article] Geometric realization of T-shaped root systems and the Jacobians of del Pezzo surfaces, "Geometry in Osaka", Lecture Notes in Math (2008)
  • Author(s): Shigeru Mukai
  • Journal Title: Osaka University (vol.9) Pages: 134-136
• [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
• [Journal Article] Polarized K3 surfaces of genus thirteen, Adv. Stud (2006)
  • Author(s): Shigeru Mukai
  • Journal Title: Pure Math. 45 Pages: 315-326
  • Peer Reviewed
• [Journal Article] Polarized K3 surfaces of genus thirteen (2006)
  • Author(s): S.Mukai
  • Journal Title: Adv. Stud. Pure Math. 45 Pages: 315-326
• [Journal Article] The cohomology groups of stable quasi-abelian schmes and degenerations associated with the E8-lattice (2006)
  • Author(s): I.Nakamura
  • Journal Title: Adv. Stud. Pure Math. 45 Pages: 223-281
• [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
• [Journal Article] The moduli stack of rank-two Gieseker bundles with fixed determinant on a nodal curve (2006)
  • Author(s): T.Abe
  • Journal Title: Adv. Stud. Pure Math. 45 Pages: 117-144
• [Journal Article] On SL(2)-GL(n) strange duality (2006)
  • Author(s): T.Abe
  • Journal Title: J. Math. Kyoto Univ. 46 Pages: 657-692
• [Journal Article] Obstructions to deforming space curves and non-reduced components of the Hilbert scheme (2006)
  • Author(s): H.Nasu
  • Journal Title: Publ. RIMS Kyoto Univ. 42 Pages: 117-141
• [Journal Article] Compact complex surfaces admitting non-trivial surjective endomorphisms (2005)
  • Author(s): N.Nakayama, Yoshio Fujimoto
  • Journal Title: Tohoku Math.J. 57 Pages: 395-426
  • NAID: 110001792946
• [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
  • 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)
  • Peer Reviewed
• [Journal Article] Polarized K3 surfaces of genus thirteen
  • Author(s): S.Mukai
  • Journal Title: Adv.Stud.Pure Math. to appear
• [Presentation] Polarized K3 surfaces of degree 32 (2008)
  • Author(s): Shigeru Mukai
  • Organizer: Moduli Spaces
  • Place of Presentation: Univ. of Warwick, UK
  • Year and Date: 2008-07-10
• [Presentation] Cohomologically trivial involutions of Enriques surfaces with application to Shioda-Inose structure (2008)
  • Author(s): Shigeru Mukai
  • Organizer: Algebraic Geometry
  • Place of Presentation: Lorentz Center, Leiden,the Netherland
  • Year and Date: 2008-07-02
• [Presentation] Geometric realization of T-shaped root systems and Jacobians of del Pezzo surfaces (2007)
  • Author(s): S. Mukai
  • Organizer: Complex geometry in Osaka
  • Place of Presentation: 大阪大学
  • Year and Date: 2007-11-03
• [Presentation] Enriques surfaces covered by Kummer's quartics (2007)
  • Author(s): S. Mukai
  • Organizer: Komplexe Algebraische Geometrie
  • Place of Presentation: Oberwolfach, 独
  • Year and Date: 2007-10-02
• [Presentation] Enriques surfaces associated with bielliptic abelian surfaces (2007)
  • 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
• [Presentation] Hilbert's 14th problem for the Kronecker quiver (2007)
  • Author(s): Shigeru Mukai
  • Organizer: "Algebraic geometry in higher dimensions
  • Place of Presentation: Levico Terme (Trento), Italy
  • Year and Date: 2007-06-08
• [Presentation] Hilbert's 14th problem for the Kronecker quiver (2007)
  • Author(s): S. Mukai
  • Organizer: Algebraic geometry in higher dimensions
  • Place of Presentation: Levico Terme (Trento), 伊
  • Year and Date: 2007-06-08
• [Presentation] Cohomologically trivial automorphisms of Enriques surfaces (2007)
  • 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
• [Presentation] Hilbert's 14th problem for the Kronecker quiver (2007)
  • Author(s): S. Mukai
  • Organizer: Advances in algebra and geometry
  • Place of Presentation: MSRI, USA
  • Year and Date: 2007-04-30
• [Presentation] Hilbert's 14th problem and Verlinde type formulas for rings of invariant polynomials (2007)
  • Author(s): Shigeru Mukai
  • Organizer: Quwait lecture
  • Place of Presentation: Cambridge Univ., England
  • Year and Date: 2007-01-30
• [Presentation] Hilbert's original 14th problem and certain moduli spaces (2005)
  • Author(s): Shigeru Mukai,
  • Organizer: AMS Summer Institute on Algebraic Geometry
  • Place of Presentation: Seattle, U.S.A
• [Book] (2008)
  • Author(s): 向井茂
  • Total Pages: 458
  • Publisher: モジュライ理論I・II(改訂新版)
• [Book]
  • Author(s): 岩波書店
• [Remarks]
  • URL: http://www.kurims.kyoto-u.ac.jp/~mukai