On rational functions and singularities of plane algebraic curves

• Project/Area Number: 20540038
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Saitama University
• Principal Investigator: SAKAI Fumio Saitama University, 理工学研究科, 教授 (40036596)
• Co-Investigator(Renkei-kenkyūsha): KOJIMA Hisashi 埼玉大学, 大学院・理工学研究科, 教授 (90146118) ; FUKUI Toshizumi 埼玉大学, 大学院・理工学研究科, 教授 (90218892) ; EBIHARA Madoka 埼玉大学, 大学院・理工学研究科, 講師 (80213578) ; TONO Keita 埼玉大学, 大学院・理工学研究科, 非常勤講師 (30422215)
• Project Period (FY): 2008 – 2010
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 代数幾何 / 平面曲線 / ワイエルシュトラス点 / 分岐点 / 自己同型群 / セクスタクティック点 / 変曲点 / 代数曲線 / 特異点 / 有理関数体 / ゴナリティ

Research Abstract

We studied Weierstrass points on Picard curves, 2-Weierstrass points on Kuribayashi curves and 3-Weierstrass points on genus two curves. We published papers on these topics. We also obtained new results concerning the criterion for the gonality of singular plane curves.

Research Products

• [Journal Article] The 3-Weierstrass points on genus two curves with extra involutions (2011)
  • Author(s): Mohamed Farahat, Fumio Sakai
  • Journal Title: Saitama Math. J Volume: Vol.28 Pages: 1-12
  • Peer Reviewed
• [Journal Article] S4 geometry of Picard curves (2010)
  • Author(s): Weierstrass points, Fumio Sakai
  • Journal Title: School in Algebraic Geometry Volume: Vol.3 Pages: 143-158
• [Journal Article] Hyperelliptic plane curves of type (d, d-2) (2010)
  • Author(s): Fumio Sakai, Mohammad Saleem, Keita. Tono
  • Journal Title: Beitrage zur Algebra und Geometri Volume: Vol.51 Pages: 31-44
  • Peer Reviewed
• [Journal Article] Weierstrass points and S4 geometry of Picard curves (2010)
  • Author(s): Fumio Sakai
  • Journal Title: School in Algebraic Geometry 3 Pages: 143-158
• [Journal Article] Mohammad Saleem and Keita. Tono, Hyperelliptic plane curves of type(d, d-2) (2010)
  • Author(s): Fumio Sakai
  • Journal Title: Beitrage zur Algebra und Geometri 51 Pages: 31-44
  • Peer Reviewed
• [Journal Article] Hyperelliptic plane curves of type (d,d-2) (2010)
  • Author(s): F.Sakai, M.Saleem, K.Tono
  • Journal Title: Beitrage zur Algebra und Geometri Volume: 51 Pages: 67-82
  • Peer Reviewed
• [Journal Article] Kamel and Geometry and computation of 2-Weierstrass points on Kuribayashi quartic curves (2009)
  • Author(s): Alwaleed, Fumio Sakai
  • Journal Title: Saitama Math. J Volume: Vol.26 Pages: 67-82
  • Peer Reviewed
• [Journal Article] Geometry and computation of 2-Weierstrass points on Kuribayashi quartic curves (2009)
  • Author(s): Kamel Alwaleed, Fumio Sakai
  • Journal Title: Saitama Math.J. 26 Pages: 67-82
  • Peer Reviewed
• [Journal Article] Geometry and computation of 2-Weierstrass points on Kuribayashi quartic curves (2009)
  • Author(s): K.Alwaleed, F.Sakai
  • Journal Title: Saitama Math.J. 26 Pages: 67-82
  • Peer Reviewed
• [Journal Article] Geometry of Weierstrass 3 points for 2-parameter families of plane curves (2008)
  • Author(s): Fumio Sakai
  • Journal Title: 第6回代数曲線論シンポジウム報告集 Pages: 94-103
• [Journal Article] Geometry of Weierstrass points for 2-parameter families of plane curves (2008)
  • Author(s): Fumio Sakai
  • Journal Title: 第6回代数曲線論シンポジウム報告集(神奈川大学) Pages: 94-103
• [Journal Article] Geometry of Weierstrass points for 2-parameter families of plane curves (2008)
  • Author(s): F. Sakai
  • Journal Title: 第6回代数曲線論シンポジウム報告集 Pages: 94-103
• [Presentation] 平面曲線族における高次ワイエルシュトラス点の計算 (2011)
  • Author(s): 酒井文雄
  • Organizer: 津山代数幾何シンポジウム
  • Place of Presentation: 津山高専
  • Year and Date: 2011-07-27
• [Presentation] On the gonality of singular plane curves II (2011)
  • Author(s): 酒井文雄
  • Organizer: アフィン代数幾何研究集会
  • Place of Presentation: 関西学院大梅田キャンパス
  • Year and Date: 2011-03-05
• [Presentation] On the gonality of singular plane curves II (2011)
  • Author(s): 酒井文雄
  • Organizer: アフィン代数幾何研究集会
  • Place of Presentation: 関西学院大梅田キャンパス(招待講演)
  • Year and Date: 2011-03-05
• [Presentation] points Kuribayashi quartic curves (2010)
  • Author(s): 酒井文雄
  • Organizer: Tsuda College Mini-Workshop on Number Theory and Physics at the Crossroads
  • Place of Presentation: 津田塾大学
  • Year and Date: 2010-08-29
• [Presentation] 2-Weierstrass points on Kuribayashi quartic curves (2010)
  • Author(s): 酒井文雄
  • Organizer: Tsuda College Mini-Workshop on Number Theory and Physics at the Crossroads
  • Place of Presentation: 招待講演
  • Year and Date: 2010-08-29
• [Presentation] Geometry and computation of Weierstrass points (2010)
  • Author(s): 酒井文雄
  • Organizer: Tsuda College Mini-Workshop on Number Theory and Physics at the Crossroads
  • Place of Presentation: 津田塾大学
  • Year and Date: 2010-08-29
• [Presentation] S4 geometry of 2-Weierstrass points on Kuribayashi quartic curves (2010)
  • Author(s): K.Alwaleed, 酒井文雄
  • Organizer: 日本数学会
  • Place of Presentation: 慶応大学
  • Year and Date: 2010-03-24
• [Presentation] Geometry and computation of Weierstrass points (2009)
  • Author(s): 酒井文雄
  • Place of Presentation: 神奈川大学富士見高原研修所
  • Year and Date: 2009-09-05
• [Presentation] (2009)
  • Author(s): 酒井文雄
  • Organizer: Geometry and computation of Weierstrass points
  • Place of Presentation: 神奈川大学富士見高原研修所(招待講演)
  • Year and Date: 2009-09-05
• [Presentation] Geometry and computation of Weierstrass points (2009)
  • Author(s): 酒井文雄
  • Organizer: Affine algebraic geometry
  • Place of Presentation: 関西学院大学大阪梅田キャンパス
  • Year and Date: 2009-09-05
• [Presentation] Weierstrass points and S4 geometry of Picard curves (2008)
  • Author(s): 酒井文雄
  • Organizer: A. ne Algebraic Geometry
  • Place of Presentation: Bangalore, India
  • Year and Date: 2008-12-27
• [Presentation] Weierstrass points and S4 geometry of Picard curves (2008)
  • Author(s): 酒井文雄
  • Organizer: Affine Algebraic Geometry
  • Place of Presentation: Bangalore, India
  • Year and Date: 2008-12-27
• [Presentation] Geometry of Weierstrass points for 2-paramerter families of plane curves (2008)
  • Author(s): 酒井文雄
  • Organizer: 第6回代数曲線論シンポジウム
  • Place of Presentation: 神奈川工業大学
  • Year and Date: 2008-12-16
• [Presentation] Weierstrass points and S4 geometry of Picard curves (2008)
  • Author(s): 酒井文雄
  • Organizer: Tsuda College Mini-Workshop on Number Theory and Physics at the Crossroads
  • Place of Presentation: 津田塾大学
  • Year and Date: 2008-08-05
• [Presentation] Weierstrass points and S4 geometry of Picard curves (2008)
  • Author(s): 酒井文雄
  • Organizer: Tsuda College Mini-Workshop on Number Theory and Physics at the Crossroads
  • Place of Presentation: 招待講演
  • Year and Date: 2008-08-05
• [Presentation] Weierstrass ponints and S4 geometry of Picard curves (2008)
  • Author(s): 酒井文雄
  • Organizer: Tsuda College Mini-Workshop on Number Theory and Physics at the Crossroads
  • Place of Presentation: 津田塾大学
  • Year and Date: 2008-08-05
• [Presentation] The gonality and the genus of singular plane curves (2008)
  • Author(s): 酒井文雄
  • Organizer: 2nd NIMS School in Algebraic Geometry
  • Place of Presentation: Daejon, Korea
  • Year and Date: 2008-05-12
• [Presentation] The gonality and the genus of singular plane curves (2008)
  • Author(s): 酒井文雄
  • Organizer: 2nd NIMS School in Algebraic Geometry
  • Place of Presentation: Daejon, Korea(招待講演)
  • Year and Date: 2008-05-12
• [Presentation] Weierstrass points and S4 geometry of Picard curves (2008)
  • Author(s): 酒井文雄
  • Organizer: Affine Algebraic Geometry
  • Place of Presentation: Bangalore, India(招待講演)
  • Year and Date: 2008-02-27
• [Presentation] The gonality and the genus of singular, plane curves (2008)
  • Author(s): 酒井文雄
  • Organizer: 2nd NIMS School in Algebraic Geometry
  • Place of Presentation: Daejon, Korea
• [Book] 大学数学の基礎 (2011)
  • Author(s): 酒井文雄
  • Total Pages: 137
  • Publisher: 共立出版
• [Remarks]
  • URL: http://www.rimath.saitama-u.ac.jp/lab.jp/FumioSakai.html