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The embedding structure, defining ideals and the projective m-normality of projective varieties

Research Project

Project/Area Number 17K05197
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionYokohama National University

Principal Investigator

Noma Atsushi  横浜国立大学, 大学院環境情報研究院, 教授 (90262401)

Project Period (FY) 2017-04-01 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords射影多様体 / 射影埋め込み / 線形射影 / 定義イデアル / 二重点因子 / 斉次イデアル / 定義方程式 / 超曲面 / m正規性 / カステルヌーボーマンフォード正則数 / カステルヌーボ-マンフォード正則数 / 代数学 / 代数幾何学
Outline of Final Research Achievements

We studied the relation between the embedding structure of projective varieties and their defining ideal. For a projective variety, its double point divisor is the nonisomorphic locus of the variety by the inner projection from the linear subspace spanned by its general (e-1)-points to its image. On the other hand, a nonbirational center of a projective variety is a point from which the variety is projected nonisomorphically. The locus of nonbirational centers off the variety (resp. on its smooth locus) is called outer (resp. Inner) Segre locus of the variety. We get the following two results. The first result is to show that the linear subsystem consisting of double point divisors of a projective variety has the base points in the singular locus or the inner Segre locus of the variety. The second result is to give upper bounds of the number of irreducible components of the Segre locus of a projective variety by its degree, dimension and codimension.

Academic Significance and Societal Importance of the Research Achievements

本研究で得られた結果,射影多様体の定義方程式を線形射影によって与える方法,セグレローカスの構造,2重点因子の豊富性は,射影代数幾何の観点から興味深いのみならず,今後の応用も期待でき,さらには解決の見通しの立っていないregularity予想の状況証拠や解決への糸口としても意義があると考えられる.これらの研究は,計算代数や計算代数幾何などへの応用が今後期待される.

Report

(6 results)
  • 2021 Annual Research Report   Final Research Report ( PDF )
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (9 results)

All 2019 2018

All Journal Article (2 results) (of which Peer Reviewed: 2 results) Presentation (7 results) (of which Invited: 4 results)

  • [Journal Article] Projective varieties with nonbirational linear projections and applications2018

    • Author(s)
      Atsushi Noma
    • Journal Title

      Transactions of the American Mathematical Society

      Volume: 370 Issue: 4 Pages: 2299-2320

    • DOI

      10.1090/tran/7086

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Base-point-freeness of double-point divisors of smooth birational-divisors on conical rational scrolls2018

    • Author(s)
      Atsushi Noma
    • Journal Title

      Journal of Algebra

      Volume: 504 Pages: 39-53

    • DOI

      10.1016/j.jalgebra.2018.01.014

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Presentation] 射影多様体の点射影が像と非双有理となる中心点の集合について2019

    • Author(s)
      野間 淳
    • Organizer
      熊本大学代数幾何セミナー(熊本大学)
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 射影多様体の点射影が像と非双有理となる中心点の集合について2019

    • Author(s)
      野間 淳
    • Organizer
      都の西北 代数幾何学シンポジウム2019 (早稲田大学理工学部)
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 非特異射影曲線の非双有理線形射影中心点の上限について2018

    • Author(s)
      野間 淳
    • Organizer
      特異点セミナー,日本大学文理学部
    • Related Report
      2018 Research-status Report
  • [Presentation] 射影多様体の非双有理中心点集合の既約成分の個数の上限2018

    • Author(s)
      野間 淳
    • Organizer
      特異点セミナー,日本大学文理学部
    • Related Report
      2018 Research-status Report
  • [Presentation] 非特異射影多様体の非双有理射影中心内点の数の上限2018

    • Author(s)
      野間 淳
    • Organizer
      研究集会 射影多様体の幾何とその周辺2018,高知工科大学永国寺キャンパス
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Generic inner projections of projective varieties2018

    • Author(s)
      野間 淳
    • Organizer
      第5回代数幾何学研究集会-宇部-,宇部高等専門学校
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Double-point divisors for generic inner projections of projective varieties2018

    • Author(s)
      野間 淳
    • Organizer
      特異点セミナー,日本大学文理学部
    • Related Report
      2017 Research-status Report

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Published: 2017-04-28   Modified: 2023-01-30  

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