Toward the resolution of problems on algebraic varieties in positive characteristic

• Project/Area Number: 24540051
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo University of Science
• Principal Investigator: Ito Hiroyuki 東京理科大学, 理工学部, 教授 (60232469)
• Co-Investigator(Renkei-kenkyūsha): HIROKADO MASAYUKI 広島市立大学, 大学院情報科学研究科, 講師 (40316138) ; SAITOU NATSUO 広島市立大学, 大学院情報科学研究科, 講師 (70382372)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 代数幾何学 / 特異点 / 正標数 / 楕円曲面 / K3曲面 / 群スキーム / 疑似乱数生成 / 非ケーラー幾何 / 代数多様体 / 導分 / Calabi-Yau多様体 / 正標数特異点 / 野生的群作用 / Frobenius写像 / Artin-Schreier拡大 / Enriques曲面

Outline of Final Research Achievements

1) We classified elliptic K3 surfaces whose Mordell-Weil group have p-torsion. As a corollary, we found the strong relationship between the moduli space of such surfaces and deformation spaces of rational double points. 2) We solved the problems on the dual graph of wild quotient singularities. 3) By introducing the notion of pseudo-derivations, we consider quotient singularities not only by finite groups but by group schemes, which enable the unified treatment of various quotient singularities in positive characteristic. 4) We continued the evaluation of AST pseudo-random number generators. We also collect and consider the phenomena which give the evidences of the analogy between non-Kaeler geometry and positive characteristic algebraic geometry.

Research Products

• [Int'l Joint Research] Heinrich Heine University/Munic Technical University(Germany)
• [Journal Article] Wild quotient surface singularities whose dual graphs are not star-shaped (2015)
  • Author(s): Hiroyuki Ito, Stefan Schroeer
  • Journal Title: Asian Journal of Mathematics Volume: 19 Issue: 5 Pages: 951-986
  • DOI: 10.4310/ajm.2015.v19.n5.a7
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Elliptic K3 surfaces with p^n-torsion sections (2013)
  • Author(s): Hiroyuki Ito, Christian Liedtke
  • Journal Title: Journal of Algebraic Geometry Volume: 22 Issue: 1 Pages: 105-139
  • DOI: 10.1090/s1056-3911-2012-00584-0
  • Peer Reviewed
• [Journal Article] Three dimensional canonical singularities in codimension two in positive characteristic (2013)
  • Author(s): Masayuki Hirokado, Hiroyuki Ito, Natsuo Saito
  • Journal Title: Journal of Algebra Volume: 373 Pages: 207-222
  • DOI: 10.1016/j.jalgebra.2012.10.007
  • Peer Reviewed
• [Journal Article] Wildly ramified actions and surfaces of general type arising from Artin-Schreier curves (2012)
  • Author(s): Hiroyuki Ito, Stefan Schroeer
  • Journal Title: Geometry and Arithmetic, EMS Volume: 1
  • Peer Reviewed
• [Presentation] Pseudo-derivations and wild group scheme quotient singularities (2016)
  • Author(s): Hiroyuki Ito
  • Organizer: The 14th Affine Algebraic Geometry Meeting
  • Place of Presentation: Kwansei Gakuin University, Osaka
  • Year and Date: 2016-03-05
  • Int'l Joint Research / Invited
• [Presentation] Pseudo-derivations and wild group scheme quotient singularities (2015)
  • Author(s): Hiroyuki Ito
  • Organizer: Conference on K3 surfaces and related topics
  • Place of Presentation: KIAS, Seoul, Korea
  • Year and Date: 2015-11-17
  • Int'l Joint Research / Invited
• [Presentation] 正標数有理二重点は商特異点か? (2015)
  • Author(s): Hiroyuki Ito
  • Organizer: 射影多様体の幾何とその周辺 2015
  • Place of Presentation: 高知大学理学部
  • Year and Date: 2015-11-01
  • Invited
• [Presentation] Are rational double points quotient singularities even in positive characteristic ? (2015)
  • Author(s): Hiroyuki Ito
  • Organizer: TSUDA COLLEGE MINI-WORKSHOP ON CALABIYAU VARIETIES :ARITHMETIC, GEOMETRY AND PHYSICS
  • Place of Presentation: Tsuda Colledge, Tokyo
  • Year and Date: 2015-08-05
  • Int'l Joint Research / Invited
• [Presentation] On the quotient singularities in positive characteristic (2014)
  • Author(s): Hiroyuki Ito
  • Organizer: TSUDA COLLEGE MINI-WORKSHOP ON CALABIYAU VARIETIES :ARITHMETIC, GEOMETRY AND PHYSICS
  • Place of Presentation: Tsuda College
  • Year and Date: 2014-08-07
  • Invited
• [Presentation] 正標数の$2$次元商特異点について (2014)
  • Author(s): Hiroyuki Ito
  • Organizer: 函館数論幾何ワークショップ
  • Place of Presentation: 函館中央図書館
  • Year and Date: 2014-05-27
  • Invited
• [Presentation] Two-dimsnsional wild quotient singularities in positive characteristic, Birational Geometry and Singularities in Positive Characteristic
  • Author(s): Hiroyuki Ito
  • Organizer: Birational Geometry and Singularities in Positive Characteristic
  • Place of Presentation: University of Tokyo
  • Invited
• [Presentation] 正標数の商特異点について
  • Author(s): 伊藤浩行
  • Organizer: 第5回代数曲面ワークショップ at 秋葉原
  • Place of Presentation: 首都大学東京、秋葉原サテライトキャンパス
  • Invited
• [Presentation] On tautness of surface singularities in positive characteristic
  • Author(s): Hiroyuki Ito
  • Organizer: The 12th Affine Algebraic Geometry Meeting
  • Place of Presentation: Kwansei Gakuin University
  • Invited
• [Presentation] On wild quotient singularities
  • Author(s): 伊藤浩行
  • Organizer: 2013年度代数幾何学城崎シンポジウム
  • Place of Presentation: 兵庫県城崎大会議場
  • Invited
• [Remarks]
  • URL: http://www.ma.noda.tus.ac.jp/u/hi/math/index.html