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Obstructions to deforming curves on algebraic varieties and a study on their Hilbert schemes

Research Project

Project/Area Number 17K05210
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionTokai University

Principal Investigator

Nasu Hirokazu  東海大学, 理学部, 准教授 (30535331)

Project Period (FY) 2017-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Keywordsヒルベルトスキーム / 変形 / 障害 / ファノ多様体 / エンリケス・ファノ多様体 / K3曲面 / エンリケス曲面 / 空間曲線 / 変形障害 / 非被約成分 / エンリケス多様体 / ヒルベルト旗スキーム / 退化
Outline of Final Research Achievements

We study some explicit properties of Hilbert schemes, which are one of the most important objects in algebraic geometry. Our main results are the followings:
1. We prove that the Hilbert scheme of smooth connected curves on every prime Fano threefold contains a generically non-reduced component. By this result, an celebrated example due to Mumford (for the projective 3-space) has been extended to the Hilbert scheme of every smooth Fano threefold of Picard rank one.
2. Let X be an Enriques-Fano threefold (called an EF3), that is a singular Fano threefold. We give a sufficient condition for the Hilbert scheme of smooth connected curves on X to have a generically non-reduced component.

Academic Significance and Societal Importance of the Research Achievements

マンフォード(1962)は3次元射影空間内の非特異連結曲線のヒルベルトスキームが生成的に被約でない既約成分を有すことを証明し当時の人々を驚かせた。研究成果1では指数rの3次元ファノ多様体に対しこの例の一般化を行なった。r=4の場合はマンフォードの例により、r=3とr=2の場合は向井・那須の結果(2009)により知られていた。残りのr=1の場合(主ファノ三様体)が、K3曲面とその上の円錐曲線を用いることにより解決した。研究成果2については3次元多様体に穏やかな特異点を許し、エンリケス曲面とその上のハーフペンシルを用いて非被約成分を構成した。以上によりマンフォードの例が広範囲に一般化された。

Report

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

    (23 results)

All 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (1 results) Journal Article (6 results) (of which Peer Reviewed: 5 results,  Open Access: 2 results) Presentation (12 results) (of which Int'l Joint Research: 3 results,  Invited: 7 results) Remarks (3 results) Funded Workshop (1 results)

  • [Int'l Joint Research] University of Oslo/Oslo Metropolitan University(ノルウェー)

    • Related Report
      2019 Research-status Report
  • [Journal Article] Obstructions to deforming curves on an Enriques-Fano 3-fold2021

    • Author(s)
      Nasu Hirokazu
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 225 Issue: 9 Pages: 106677-106677

    • DOI

      10.1016/j.jpaa.2021.106677

    • NAID

      120006796429

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Corrigendum to “Obstructions to deforming curves on a 3-fold, III: Deformations of curves lying on a K3 surface”2020

    • Author(s)
      Nasu Hirokazu
    • Journal Title

      International Journal of Mathematics

      Volume: 31 Issue: 12 Pages: 2092001-2092001

    • DOI

      10.1142/s0129167x20920019

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Obstructions to deforming curves on an Enriques-Fano 3-fold2020

    • Author(s)
      Hirokazu Nasu
    • Journal Title

      「城崎代数幾何学シンポジウム2019」報告集

      Volume: 2019 Pages: 85-94

    • NAID

      120006796429

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] Obstructions to deforming curves on a prime Fano 3‐fold2019

    • Author(s)
      Nasu Hirokazu
    • Journal Title

      Mathematische Nachrichten

      Volume: 292 Issue: 8 Pages: 1777-1790

    • DOI

      10.1002/mana.201800185

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] 書評 R. Hartshorne: Deformation Theory (Grad. Texts in Math., 257)2019

    • Author(s)
      那須 弘和
    • Journal Title

      数学

      Volume: 71

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Obstructions to deforming curves on a 3-fold, III: Deformations of curves lying on a K3 surface2017

    • Author(s)
      Hirokazu Nasu
    • Journal Title

      International Journal of Mathematics

      Volume: 28 Issue: 13 Pages: 1750099-1750099

    • DOI

      10.1142/s0129167x17500999

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Presentation] Obstructions to deforming curves on an Enriques-Fano 3-fold2019

    • Author(s)
      Hirokazu NASU
    • Organizer
      Seminar in Algebra and Algebraic Geometry, University of Oslo
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Obstructions to deforming curves on an Enriques-Fano 3-fold2019

    • Author(s)
      Hirokazu NASU
    • Organizer
      Kinosaki Algebraic Geometry Symposium 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 3次元エンリケス・ファノ多様体上の曲線の変形障害について2019

    • Author(s)
      那須弘和
    • Organizer
      東京電機大学数学講演会, 東京電機大学
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Obstructions to deforming curves on an Enriques-Fano 3-fold2019

    • Author(s)
      Hirokazu Nasu
    • Organizer
      Algebraic Geometry and Moduli Theory ―Conference in Honor of Shigeru Mukai's Retirement―, RIMS, Kyoto University
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Obstructions to deforming curves on a prime Fano threefold2018

    • Author(s)
      那須弘和
    • Organizer
      熊本代数幾何・数論セミナー, 熊本大学
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Obstructions to deforming curves on an Enriques-Fano threefold2018

    • Author(s)
      那須弘和
    • Organizer
      射影多様体の幾何とその周辺, 高知工科大学
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Obstructions to deforming curves on an Enriques-Fano 3-fold2018

    • Author(s)
      那須弘和
    • Organizer
      東工大数論・幾何学セミナー, 東京工業大学
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] On the image of Hilbert-flag schemes and non-reduced components of the Hilbert scheme of Fano 3-folds2018

    • Author(s)
      那須弘和
    • Organizer
      第4回杜の都代数幾何学研究集会@仙台
    • Related Report
      2017 Research-status Report
  • [Presentation] Obstructions to deforming curves on a prime Fano threefold2017

    • Author(s)
      那須弘和
    • Organizer
      第16回代数曲面ワークショップ at 秋葉原
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Obstructions to deforming curves on a prime Fano threefold2017

    • Author(s)
      那須弘和
    • Organizer
      日本数学会2017年度秋季総合分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] Obstructions to deforming curves on a prime Fano threefold2017

    • Author(s)
      那須弘和
    • Organizer
      第5回K3曲面・エンリケス曲面ワークショップ
    • Related Report
      2017 Research-status Report
  • [Presentation] Obstructions to deforming curves on a prime Fano threefold2017

    • Author(s)
      Hirokazu Nasu
    • Organizer
      Hakodate workshop on arithmetic geometry 2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Remarks] 研究者ホームページ

    • URL

      http://fuji.ss.u-tokai.ac.jp/nasu/

    • Related Report
      2020 Annual Research Report 2019 Research-status Report 2018 Research-status Report
  • [Remarks] 東海大学総合理工学研究科 研究事例紹介(所属研究機関ページ)

    • URL

      https://www.u-tokai.ac.jp/uploads/sites/19/2021/03/nasu.pdf

    • Related Report
      2020 Annual Research Report
  • [Remarks] 研究者Webページ

    • URL

      http://fuji.ss.u-tokai.ac.jp/nasu/index.html

    • Related Report
      2017 Research-status Report
  • [Funded Workshop] Algebraic Geometry and Moduli Theory ―Conference in Honor of Shigeru Mukai's Retirement―2019

    • Related Report
      2018 Research-status Report

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Published: 2017-04-28   Modified: 2022-01-27  

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