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Noncommutative resolution of singularities in positive characteristic and its applications

Research Project

Project/Area Number 22740020
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Algebra
Research InstitutionOsaka University (2012-2013)
Kagoshima University (2010-2011)

Principal Investigator

YASUDA Takehiko  大阪大学, 理学(系)研究科(研究院), 准教授 (30507166)

Project Period (FY) 2010-04-01 – 2014-03-31
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords特異点 / 特異点解消 / 非可換代数幾何 / 正標数 / モチーフ積分 / 野生商特異点 / 野生McKay対応 / 点のHilbertスキーム / Bhargavaの量公式 / 局所Galois表現 / Deligne-Mumfordスタック / マッカイ対応 / 野生的商特異点 / 悲観特異点解消 / F爆発 / フロベニウス射 / 非可換特異点解消
Research Abstract

In a joint work with Nobuo Hara and Tadakazu Sawada, I studied F-blowups of rational double points and simple elliptic singularities in detail. As a byproduct of studies of Frobenius functors of noncommutative resolutions, I proved that concerning a wild finite group action which is free in codimension one, the associated quotient singularity is not strongly F-regular. Roughly, this means that under a mild condition, wild quotient singularities are always bad. Also I started a research on a generalization of the McKay correspondence to positive characteristics by using motivic integration. It has turned out that singularities and the number theory are closely related.

Report

(5 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Annual Research Report
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • Research Products

    (47 results)

All 2014 2013 2012 2011 2010 Other

All Journal Article (8 results) (of which Peer Reviewed: 8 results) Presentation (39 results) (of which Invited: 8 results)

  • [Journal Article] The p-cyclic McKay correspondence via motivic integration2014

    • Author(s)
      Takehiko Yasuda
    • Journal Title

      Compositio Mathematica

      Volume: 未定

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Toward motivic integration over wild Deligne-Mumford stacks2014

    • Author(s)
      Takehiko Yasuda
    • Journal Title

      the proceedings of the conference " Higher Dimensional Algebraic Geometry - in honour of Professor Yujiro Kawamata's sixtieth birthday"

      Volume: 未定

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] F-blowups of normal surface singularities2013

    • Author(s)
      Nobuo Hara, Tadakazu Swada and Takehiko Yasuda
    • Journal Title

      Algebra & Number Theory

      Volume: 7-3 Pages: 733-763

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms2012

    • Author(s)
      Takehiko Yasuda
    • Journal Title

      Journal of Algebra

      Volume: 370 Pages: 15-31

    • Related Report
      2013 Final Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Frobenius morphisms of noncommutative blowups2012

    • Author(s)
      Takehiko Yasuda
    • Journal Title

      IRMA Lectures in Mathematics and Theoretical Physics

      Volume: 20 Pages: 269-283

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Universal flattening of Frobenius2012

    • Author(s)
      Takehiko Yasuda
    • Journal Title

      American Journal of Mathematics

      Volume: 134 Issue: 2 Pages: 349-378

    • DOI

      10.1353/ajm.2012.0014

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Toward motivic integration over wild Deligne-Mumford stacks

    • Author(s)
      Takehiko Yasuda
    • Journal Title

      研究集会"Higher Dimensional Algebraic Geometry"

      Volume: (掲載決定)

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] The p-cyclic McKay correspondence via motivic integration

    • Author(s)
      Takehiko Yasuda
    • Journal Title

      Compositio Mathematica

      Volume: (掲載決定)

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Presentation] 野生McKay対応の展望2014

    • Author(s)
      Takehiko Yasuda
    • Organizer
      日本数学会年会
    • Place of Presentation
      学習院大学
    • Year and Date
      2014-03-15
    • Related Report
      2013 Final Research Report
  • [Presentation] A brief guide to the wild McKay correspondence2014

    • Author(s)
      Takehiko Yasuda
    • Organizer
      九州代数的整数論2014
    • Place of Presentation
      九州大学
    • Year and Date
      2014-02-06
    • Related Report
      2013 Final Research Report
  • [Presentation] A brief guide to the wild McKay correspondence2014

    • Author(s)
      Takehiko Yasuda
    • Organizer
      九州代数的整数論 2014
    • Place of Presentation
      九州大学
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Quotient singularities and local Galois representations2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Motives in Tokyo 2013
    • Place of Presentation
      東京大学
    • Year and Date
      2013-11-26
    • Related Report
      2013 Final Research Report
  • [Presentation] Motivic integration and wild ramification2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      1st Franco-Japanese-Vietnamese Symposium on Singularities
    • Place of Presentation
      ニース(フランス)
    • Year and Date
      2013-09-16
    • Related Report
      2013 Final Research Report
  • [Presentation] A survey on wild quotient singularities2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Workshop on Algebraic Geometry in Sapporo
    • Place of Presentation
      北海道大学
    • Year and Date
      2013-08-27
    • Related Report
      2013 Final Research Report
  • [Presentation] An invitation to the wild McKay correspondence2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Noncommutative Algebraic Geometry and Related Topics
    • Place of Presentation
      京都大学
    • Year and Date
      2013-07-01
    • Related Report
      2013 Final Research Report
  • [Presentation] The Hilbert Scheme of Points and Extensions of Local Fields2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      PRIMA 2013
    • Place of Presentation
      上海(中国)
    • Year and Date
      2013-06-27
    • Related Report
      2013 Final Research Report
  • [Presentation] Bhargava's formula and the Hilbert scheme of points2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Complex Algebraic Geometry
    • Place of Presentation
      Oberwolfach(ドイツ)
    • Year and Date
      2013-05-30
    • Related Report
      2013 Final Research Report
  • [Presentation] Stringy invariants of p-cyclic quotient singularities2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Arithmetic and Algebraic Geometry 2013
    • Place of Presentation
      東京大学
    • Year and Date
      2013-01-28
    • Related Report
      2013 Final Research Report 2012 Annual Research Report
  • [Presentation] The wild McKay correspondence2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Higher Dimensional Algebraic Geometry
    • Place of Presentation
      東京大学
    • Year and Date
      2013-01-08
    • Related Report
      2013 Final Research Report 2012 Annual Research Report
  • [Presentation] Bhargava's formula and the Hilbert scheme of points2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Complex Algebraic Geometry
    • Place of Presentation
      Oberwolfach, ドイツ
    • Related Report
      2013 Annual Research Report
  • [Presentation] The Hilbert scheme of points and extensions of local fields2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      The 2nd Pacific Rim Mathematical Association COngress
    • Place of Presentation
      上海交通大学、中国
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] An invitation to the wild McKay correspondence2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Noncommutative Algebraic Geometry and Related Topics
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] A survey on wild quotient singularities2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Workshop in Algebraic Geometry in Sapporo
    • Place of Presentation
      北海道大学
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Motivic integration and wild ramification2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      1st Franco-Japanese-Vietnamese Symposium on Singularities
    • Place of Presentation
      ニース、フランス
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Quotient singularities and local Galois representations2013

    • Author(s)
      Takehiko Yasuda
    • Organizer
      International Workshop on motives in Tokyo, 2013
    • Place of Presentation
      東京大学
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] p-cyclic quotient singularities and motivic integration2012

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Motives in Tokyo
    • Place of Presentation
      東京大学
    • Year and Date
      2012-12-11
    • Related Report
      2013 Final Research Report 2012 Annual Research Report
  • [Presentation] p巡回McKay対応2012

    • Author(s)
      Takehiko Yasuda
    • Organizer
      城崎代数幾何学シンポジウム2012
    • Place of Presentation
      城崎
    • Year and Date
      2012-10-23
    • Related Report
      2013 Final Research Report 2012 Annual Research Report
  • [Presentation] Computing F-blowups2012

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Computational Workshop on Frobenius singularities and invariants
    • Place of Presentation
      Ann Arbor(アメリカ)
    • Year and Date
      2012-06-01
    • Related Report
      2013 Final Research Report
  • [Presentation] Motivic integration and modular representations of a cyclic group of prime order2012

    • Author(s)
      Takehiko Yasuda
    • Organizer
      第7回代数解析幾何学セミナー
    • Place of Presentation
      鹿児島大学
    • Year and Date
      2012-02-15
    • Related Report
      2013 Final Research Report
  • [Presentation] Motivic integration and modular representations of a cyclic group of prime order2012

    • Author(s)
      Takehiko Yasuda
    • Organizer
      第7回代数・解析・幾何学セミナー
    • Place of Presentation
      鹿児島大学(招待講演)
    • Year and Date
      2012-02-15
    • Related Report
      2011 Annual Research Report
  • [Presentation] Higher Nash blowups and F-blowups2012

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Clay Mathematics Institute Summer School 2012
    • Place of Presentation
      Obergurgl(オーストリア)
    • Related Report
      2013 Final Research Report
  • [Presentation] Computing F-blowups2012

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Computational workshop on Frobenius singularities and invariants
    • Place of Presentation
      Ann Arbor, アメリカ
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Higher Nash Blowups and F-Blowups2012

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Clay Mathematics Institute Summer School 2012
    • Place of Presentation
      Obergurgl, オーストリア
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Higher Nash blowups2011

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Artin approximation and arcs
    • Place of Presentation
      シュレーディンガー研究所(オーストリア)
    • Year and Date
      2011-11-23
    • Related Report
      2013 Final Research Report
  • [Presentation] Higher Nash blowups2011

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Artin approximation and arcs
    • Place of Presentation
      シュレーディンガー研究所、オーストリア
    • Year and Date
      2011-11-23
    • Related Report
      2011 Annual Research Report
  • [Presentation] Quotient-like singularities in positive characteristic and noncommutative rings2011

    • Author(s)
      Takehiko Yasuda
    • Organizer
      第6回日仏特異点シンポジウム
    • Place of Presentation
      九州大学西陣プラザ
    • Year and Date
      2011-09-07
    • Related Report
      2013 Final Research Report
  • [Presentation] Quotient-like singularities in positive characteristic and noncommutativerings2011

    • Author(s)
      Takehiko Yasuda
    • Organizer
      第6回日仏特異点シンポジウム
    • Place of Presentation
      九州大学西陣プラザ(招待講演)
    • Year and Date
      2011-09-07
    • Related Report
      2011 Annual Research Report
  • [Presentation] 2次元特異点のF 爆発2011

    • Author(s)
      Takehiko Yasuda
    • Organizer
      南九州代数系集会
    • Place of Presentation
      鹿児島大学
    • Year and Date
      2011-08-28
    • Related Report
      2013 Final Research Report
  • [Presentation] 2次元特異点のF爆発2011

    • Author(s)
      安田健彦
    • Organizer
      南九州代数系集会
    • Place of Presentation
      鹿児島大学(招待講演)
    • Year and Date
      2011-08-28
    • Related Report
      2011 Annual Research Report
  • [Presentation] F-blowups in dimension two2011

    • Author(s)
      Takehiko Yasuda
    • Organizer
      特異点とそのひろがり
    • Place of Presentation
      京都大学
    • Year and Date
      2011-08-23
    • Related Report
      2013 Final Research Report
  • [Presentation] F-blowups in dimension two2011

    • Author(s)
      安田健彦
    • Organizer
      特異点論とそのひろがり
    • Place of Presentation
      京都大学(招待講演)
    • Year and Date
      2011-08-23
    • Related Report
      2011 Annual Research Report
  • [Presentation] Noncommutative resolution of pure subrings and Frobenius morphisms2011

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Workshop on non-commutative geometry and the McKay correspondence
    • Place of Presentation
      名古屋大学
    • Year and Date
      2011-03-15
    • Related Report
      2013 Final Research Report
  • [Presentation] Noncommutative resolution of pure subrings and Frobenius morphisma2011

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Workshop on non-commutaive geometry and the McKay corespondence
    • Place of Presentation
      名古屋大学(名古屋市)
    • Year and Date
      2011-03-15
    • Related Report
      2010 Annual Research Report
  • [Presentation] Introduction to F-blowups2010

    • Author(s)
      Takehiko Yasuda
    • Organizer
      特異点と多様体の幾何学
    • Place of Presentation
      山形大学
    • Year and Date
      2010-09-16
    • Related Report
      2013 Final Research Report
  • [Presentation] Introduction to F-blowups2010

    • Author(s)
      Takehiko Yasuda
    • Organizer
      特異点と多様体の幾何学
    • Place of Presentation
      山形大学(山形市)
    • Year and Date
      2010-09-16
    • Related Report
      2010 Annual Research Report
  • [Presentation] Commutative and noncommutative desingularizations via Frobenius morphisms2010

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Workshop on Higher Dimensional Algebraic Geometry 2010
    • Place of Presentation
      National Taiwan University, 台北(台湾)
    • Year and Date
      2010-04-01
    • Related Report
      2013 Final Research Report
  • [Presentation] Commutative and noncommutative desingularizationas via Frobenius morphisms2010

    • Author(s)
      Takehiko Yasuda
    • Organizer
      Workshop on Higher Dimensional Algebraic Geometry, 2010
    • Place of Presentation
      National Taiwan University,台北
    • Year and Date
      2010-04-01
    • Related Report
      2010 Annual Research Report

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Published: 2010-08-23   Modified: 2019-07-29  

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