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New developments of computer-aided research in algebraic geometry

Research Project

Project/Area Number 23K20209
Project/Area Number (Other) 20H01798 (2020-2023)
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeMulti-year Fund (2024)
Single-year Grants (2020-2023)
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionHiroshima University

Principal Investigator

島田 伊知朗  広島大学, 先進理工系科学研究科(理), 教授 (10235616)

Co-Investigator(Kenkyū-buntansha) 木村 俊一  広島大学, 先進理工系科学研究科(理), 教授 (10284150)
金銅 誠之  名古屋大学, 多元数理科学研究科, 名誉教授 (50186847)
高橋 宣能  広島大学, 先進理工系科学研究科(理), 准教授 (60301298)
Project Period (FY) 2020-04-01 – 2025-03-31
Project Status Granted (Fiscal Year 2024)
Budget Amount *help
¥10,790,000 (Direct Cost: ¥8,300,000、Indirect Cost: ¥2,490,000)
Fiscal Year 2024: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2023: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2022: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2021: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2020: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
KeywordsK3曲面 / エンリケス曲面 / 計算機 / 格子理論 / Leech 格子 / Niemeier格子 / 対数的カラビヤウ曲面 / モノドロミー / ザリスキ多重対 / 自己同型 / モーデル・ヴェイユ格子 / ネロン・セヴェリ格子 / 自己同型群 / 数値的ネロン・セヴェリ格子 / 楕円ファイブレーション / コブル曲面 / 双曲格子 / BPS numbers
Outline of Research at the Start

K3曲面やエンリケス曲面をはじめとする代数曲面の詳細な幾何学的構造を計算機を用いて明らかにすることを目標としている.具体的な曲面に対して例えばその自己同型群,ネフ錐などの計算を実行し,代数幾何学に新しい知見をもたらすと同時に,新しくて有用な汎用的アルゴリズムを開発する,あるいは既存のアルゴリズムを高速化するなどの活動を通して,代数幾何学における計算機の応用範囲を拡大することを目指す.

Outline of Annual Research Achievements

研究代表者の島田は以下の成果を得た.切断を持つ楕円K3曲面のモーデル・ヴェイユ群のネロン・セヴェリ格子への作用を計算するアルゴリズムをピカール数が26の「仮想的な,実在しない」K3曲面に適用することにより,Niemeier格子からLeech格子を構成する新しい方法を考案し,論文にまとめた.これは Conway-Sloane によって与えられていた Leech格子の構成に,幾何学的な意味づけを与えるものである.また5本または6本の実アフィン直線の配置で分岐する平面の二重被覆から得られるK3曲面のトポロジーを調べ,中間次元のホモロジー群の生成元を明示的に記述した.さらに,いくつかの異なる文脈においていろいろな観点から調べられてきた Apery-Fermi のK3曲面の自己同型群の研究を開始し,その準備段階としてネロン・セヴェリ格子を詳しく調べ,いくつかの幾何学的対合を発見した.
研究分担者の金銅はKummer 4 次曲面の bitangent lines に関する Kummer の古典的結果の標数2における類似をIgor Dolgachev 氏と求めた.また Klaus Hulek 氏および前田洋太氏とともに,射影直線の12点のモジュライ空間の研究を行い論文にまとめた.
研究分担者の木村は組合せゲーム理論について研究を進め,いくつかの結果を得た.
研究分担者の高橋はLie-Yamaguti代数や無限小s多様体のうち,付随するLie代数が半単純なものの表現について,様々な例の構成を含むより詳しい研究を行なった.また,対数的カラビヤウ曲面上の一次元層のモジュライの部分特異点解消について調べた.

Current Status of Research Progress
Current Status of Research Progress

3: Progress in research has been slightly delayed.

Reason

2022年度に行った,2次曲線上にある6個の通常尖点を持つ射影平面6次曲線で分岐する射影平面の2重被覆の最小特異点解消として得られるK3曲面の自己同型群の決定(プレプリント Mordell-Weil groups and automorphism groups of elliptic K3 surfaces, To appear in Revista Matematica Iberoamericana, に結果を記載)以後,新しいK3曲面もしくはエンリケス曲面の自己同型群の計算の仕事を行なっていない.そのため進捗がやや遅れていると判断した.
その理由として,2023年度は,K3曲面と共形場理論の関係に関して,バックグランドとなる物理学の知識を吸収するために時間を取られたこと,および 2023年6月にスペインで開かれた特異点の研究集会で着想を得た,実数体上定義された直線配置の実構造を用いた複素代数曲面のトポロジーの研究に時間を取られたこと,が考えられる.特に後者は,証明自体は初等的であるが,証明の記述が長く複雑でありまた研究代表者のこれまでの研究課題とは異なる分野の研究であるため,論文の執筆に多くの時間を要することとなった.

Strategy for Future Research Activity

昨年度に引き続き,できるだけ多くの具体的なK3曲面およびエンリケス曲面の自己同型群とネフ錐について計算を続け,知見を蓄積する.さらに計算対象を,(正標数の場合を含む)Coble 曲面,さらには高次元の正則シンプレクティック多様体へと広げることを目指す.
また,K3曲面や正則シンプレクティック多様体のさまざまな射影モデルに対して,そのホモロジー群の基底となる位相的サイクルを明示的に記述するという研究も開始する.現在の Borcherds 法においては,その準備段階の作業として, K3曲面ごとその K3 曲面の特別な性質を用いた ad hoc な方法で「ネロン・セヴェリ格子の基底とグラム行列の決定」という仕事をおこなわなくてはいけない.近年,多くの研究者が研究している数値的代数幾何の方法を用いてこの部分の作業を統一的に行うための手法を確立することを目標としている.さらに周期の数値的計算に必要な道具(単純な数値的な多重積分による方法と Picard-Fuchs の微分方程式による解析接続の方法)を整備し,いくつかの例について予備的な計算をおこない,幾何学的なデータを抽出することを試みる.この方法は研究代表者にとって全く新しい方法であるので,何人かの研究者と研究連絡をとり議論を行う.

Report

(4 results)
  • 2023 Annual Research Report
  • 2022 Annual Research Report
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • Research Products

    (42 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (3 results) Journal Article (18 results) (of which Int'l Joint Research: 8 results,  Peer Reviewed: 18 results,  Open Access: 13 results) Presentation (17 results) (of which Int'l Joint Research: 16 results,  Invited: 17 results) Remarks (4 results)

  • [Int'l Joint Research] Saarland University(ドイツ)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Vietnam National University of Science/Hanoi National University of Education(ベトナム)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Saarland University(ドイツ)

    • Related Report
      2020 Annual Research Report
  • [Journal Article] A note on construction of the Leech lattice2024

    • Author(s)
      I. Shimada
    • Journal Title

      Discrete Math.

      Volume: 347 Issue: 3 Pages: 113832-113832

    • DOI

      10.1016/j.disc.2023.113832

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Mordell-Weil groups and automorphism groups of elliptic $K3$ surfaces2023

    • Author(s)
      Shimada I.
    • Journal Title

      Revista Matematica Iberoamericana

      Volume: - Issue: 4 Pages: 1469-1503

    • DOI

      10.4171/rmi/1449

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] On Characteristic Polynomials of Automorphisms of Enriques Surfaces2023

    • Author(s)
      Simon Brandhorst, Slawomir Rams, Ichiro Shimada
    • Journal Title

      Publications of the Research Institute for Mathematical Sciences

      Volume: 59 Issue: 3 Pages: 633-656

    • DOI

      10.4171/prims/59-3-7

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Coble surfaces in characteristic two2023

    • Author(s)
      Toshiyuki Katsura and Shigeyuki Kondo
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 75 Issue: 4 Pages: 1287-1337

    • DOI

      10.2969/jmsj/87568756

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A note on Quebbemann’s extremal lattices of rank 642023

    • Author(s)
      Shimada Ichiro
    • Journal Title

      Journal de theorie des nombres de Bordeaux

      Volume: 34 Issue: 3 Pages: 813-826

    • DOI

      10.5802/jtnb.1229

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Zariski multiples associated with quartic curves2022

    • Author(s)
      Shimada Ichiro
    • Journal Title

      Journal of Singularities

      Volume: 24 Pages: 169-189

    • DOI

      10.5427/jsing.2022.24g

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Borcherds’ Method for Enriques Surfaces2022

    • Author(s)
      Brandhorst Simon、Shimada Ichiro
    • Journal Title

      Michigan Mathematical Journal

      Volume: 71 Issue: 1 Pages: 3-18

    • DOI

      10.1307/mmj/20195769

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Coble surfaces with finite automorphism group2022

    • Author(s)
      Shigeyuki Kondo
    • Journal Title

      Rend. Circ. Mat. Palermo

      Volume: 71 Issue: 2 Pages: 829-864

    • DOI

      10.1007/s12215-021-00646-2

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Automorphism Groups of Certain Enriques Surfaces2021

    • Author(s)
      Brandhorst Simon、Shimada Ichiro
    • Journal Title

      Foundations of Computational Mathematics

      Volume: -- Issue: 5 Pages: 1463-1512

    • DOI

      10.1007/s10208-021-09530-y

    • Related Report
      2021 Annual Research Report 2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Rational double points on Enriques surfaces2021

    • Author(s)
      Ichiro Shimada
    • Journal Title

      Sci. China Math.

      Volume: 64 Issue: 4 Pages: 665-690

    • DOI

      10.1007/s11425-019-1796-x

    • Related Report
      2021 Annual Research Report 2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Classification of Enriques surfaces covered by the supersingular 𝐾3 surface with Artin invariant 1 in characteristic 22021

    • Author(s)
      Shigeyuki Kondo
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 73 Issue: 1 Pages: 301-328

    • DOI

      10.2969/jmsj/81778177

    • NAID

      130007973746

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2021 Annual Research Report 2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Sheaves of maximal intersection and multiplicities of stable log maps2021

    • Author(s)
      Choi Jinwon、van Garrel Michel、Katz Sheldon、Takahashi Nobuyoshi
    • Journal Title

      Selecta Mathematica

      Volume: 27 Issue: 4

    • DOI

      10.1007/s00029-021-00671-0

    • Related Report
      2021 Annual Research Report 2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Modules over geometric quandles and representations of Lie-Yamaguti algebras2021

    • Author(s)
      Takahashi Nobuyoshi
    • Journal Title

      Journal of Lie Theory

      Volume: 31

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Kummer surfaces associated with group schemes2021

    • Author(s)
      Shigeyuki Kondo, Stefan Schroer
    • Journal Title

      manuscripta mathematica

      Volume: 166 Issue: 3-4 Pages: 323-342

    • DOI

      10.1007/s00229-020-01257-4

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Modules over geometric quandles and representations of Lie-Yamaguti algebras2021

    • Author(s)
      Takahashi, Nobuyoshi
    • Journal Title

      J. Lie Theory

      Volume: 31

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Log BPS numbers of log Calabi-Yau surfaces2021

    • Author(s)
      Jinwon Choi, Michel van Garrel, Sheldon Katz, Nobuyoshi Takahashi
    • Journal Title

      Transactions of the American Mathematical Society

      Volume: 374 Issue: 1 Pages: 687-732

    • DOI

      10.1090/tran/8234

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Enriques involutions on singular K3 surfaces of small discriminants.2020

    • Author(s)
      Shimada, Ichiro, Veniani, Davide Cesare
    • Journal Title

      Ann. Sc. Norm. Super. Pisa

      Volume: 21 Pages: 1667-1701

    • DOI

      10.2422/2036-2145.201902_004

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Classification of Enriques surfaces with finite automorphism group in characteristic 22020

    • Author(s)
      Toshiyuki Katsura, Shigeyuki Kondo and Gebhard Martin
    • Journal Title

      Algebraic Geometry

      Volume: 7 Pages: 390-459

    • DOI

      10.14231/ag-2020-012

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Enforce operation of the disjunctive sum and the Continued Conjunctive Sum2024

    • Author(s)
      Shunichi Kimura
    • Organizer
      Game at Mumbai
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Vanishing cycles of a double plane branching along a real line arrangement2023

    • Author(s)
      Ichiro Shimada
    • Organizer
      Workshop "Topology of Singularities and Related Topics"
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Conway theory for algebraic geometers2023

    • Author(s)
      Ichiro Shimada
    • Organizer
      Working Workshop on Calabi--Yau Varieties and Related Topics 2023
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Kummer surfaces and quadric line complexes in characteristic two2023

    • Author(s)
      Shigeyuki Kondo
    • Organizer
      Aspect of Algebraic Geometry
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] あるK3曲面の自己同型群について2023

    • Author(s)
      Ichiro Shimada
    • Organizer
      K3, Enriques Surfaces, and Related Topics
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Mordell-Weil groups of a certain K3 surface2022

    • Author(s)
      Ichiro Shimada
    • Organizer
      Recent Development in Algebraic Geometry
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The automorphism group of a K3 surface birational to a double plane2022

    • Author(s)
      Ichiro Shimada
    • Organizer
      Real Aspects of Geometry
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Automorphism groups of Enriques surfaces (joint work with Simon Brandhorst)2022

    • Author(s)
      Ichiro Shimada
    • Organizer
      Japanese-European Symposium on Symplectic Varieties and Moduli Spaces -- Sixth Edition
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Coble surfaces with finite automorphism group2022

    • Author(s)
      Shigeyuki Kondo
    • Organizer
      Algebraic Geometry Seminar, Universitat des Saarlandes
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Computation of automorphism groups of Enriques surfaces (joint work with Simon Brandhorst)2021

    • Author(s)
      Ichiro Shimada
    • Organizer
      ODTU-Bilkent Algebraic Geometry Seminar
    • Related Report
      2021 Annual Research Report 2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Computation of the nef cone and the automorphism group of an Enriques surface (joint work with Simon Brandhorst)2021

    • Author(s)
      Ichiro Shimada
    • Organizer
      Online Workshop on Calabi-Yau Varieties and Related Topics
    • Related Report
      2021 Annual Research Report 2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Log BPS numbers and contributions of degenerate log map2021

    • Author(s)
      Nobuyoshi Takahashi
    • Organizer
      Online Workshop on Mirror symmetry and Related Topics
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 有限自己同型群を持つ Coble 曲面の分類2021

    • Author(s)
      Shigeyuki Kondo
    • Organizer
      城崎代数幾何シンポジウム2021
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 有限自己同型群を持つ Coble 曲面2021

    • Author(s)
      Shigeyuki Kondo
    • Organizer
      森先生古希講演会
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Enriques surfaces and Leech lattice2020

    • Author(s)
      Shigeyuki Kondo
    • Organizer
      Algebraic geometry and arithmetic: a conference on the occasion of V.V. Nikulin 70th birthday
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Enriques surfaces and Leech lattice2020

    • Author(s)
      Shigeyuki Kondo
    • Organizer
      Japanese-European symposium on symplectic varieties and moduli spaces
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Enriques surfaces and the Leech lattice (after Brandhorst and Shimada)2020

    • Author(s)
      Shigeyuki Kondo
    • Organizer
      第15回代数・解析・幾何学セミナー
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Remarks] Computation Data

    • URL

      https://home.hiroshima-u.ac.jp/ichiro-shimada/ComputationData.html

    • Related Report
      2023 Annual Research Report
  • [Remarks] Computational data

    • URL

      https://home.hiroshima-u.ac.jp/ichiro-shimada/ComputationData.html

    • Related Report
      2022 Annual Research Report
  • [Remarks] Computational data of K3 and Enriques surfaces

    • URL

      http://www.math.sci.hiroshima-u.ac.jp/shimada/K3andEnriques.html

    • Related Report
      2021 Annual Research Report 2020 Annual Research Report
  • [Remarks] Computational data on lattices

    • URL

      http://www.math.sci.hiroshima-u.ac.jp/shimada/lattice.html

    • Related Report
      2021 Annual Research Report

URL: 

Published: 2020-04-28   Modified: 2024-12-25  

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