Study on Glois embedding of surface of non-general type

• Project/Area Number: 24540036
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Niigata University
• Principal Investigator: Yoshihara Hisao 新潟大学, 自然科学系, フェロー (60114807)
• Co-Investigator(Kenkyū-buntansha): TOKUNAGA Hiroo 首都大学東京, 理工学研究科, 教授 (30211395)
• Co-Investigator(Renkei-kenkyūsha): KONDO Shigeyuki 名古屋大学, 多元数理科学研究科, 教授 (50186847) ; KONNO Kazuhiro 大阪大学, 理学研究科, 教授 (10186869) ; KOJIMA Hideo 新潟大学, 自然科学系, 教授 (90332824)
• 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 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2012: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
• Keywords: ガロワ埋め込み / ガロワ群 / ガロワ直線 / ガロワ閉包多様体 / 代数幾何学 / ガロワ被覆 / Galois line / Galois embedding / Galois group / finite covering / ガロワ点 / 分岐被覆 / 射影多様体

Outline of Final Research Achievements

We have studied Galois points for plane curves and some hypersurfaces, after that we have generalized the concept of it and defined Galois embedding of algebraic varieties. Here we study on the Galois embeddings of conclete algebraic varieties. For elliptic curve we embedd it by complete linear system and study the arrangement of Galois lines. For algebraic surface of non-general type we consider if there exists the Galois embedding, in paticular bi-elliptic surface has no Galois embedding. In case some algebraic variety has no Galois embedding, we consider the Galois closure variety, especially we take such variety as smooth cubic.

Research Products

• [Journal Article] Integral section of elliptic surfaces and degenerated (2,3) torus decompositions of a 3-cuspidal quartic (2015)
  • Author(s): H. Tokunaga and K. Tumenbayar
  • Journal Title: SUT J. of Math. Volume: 51
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Galois group at each point for some self-dual curves (2013)
  • Author(s): Hiroyuki Hayashi and Hisao Yoshihara
  • Journal Title: Geometry Volume: 1
  • Peer Reviewed
• [Journal Article] Sextic variety as Galois closure variety of smooth cubic (2013)
  • Author(s): Hisao Yoshihara
  • Journal Title: Affine Algebraic Geometry-Proceedings Volume: 1
  • Peer Reviewed
• [Presentation] Integral sections of rational elliptic surfaces and contact conics to 3-codal quartics (2015)
  • Author(s): H. Tokunaga
  • Organizer: Various Aspects of Algebraic Geometry
  • Place of Presentation: 国際基督教大学
  • Year and Date: 2015-12-13
  • Int'l Joint Research / Invited
• [Presentation] 代数曲面のガロワ埋め込みの可能性 (2015)
  • Author(s): 吉原久夫
  • Organizer: 第二回代数幾何学研究集会ー宇部ー
  • Place of Presentation: 宇部工業高等専門学校
  • Year and Date: 2015-01-24 – 2015-01-25
  • Invited
• [Presentation] 代数曲面のガロワ埋め込み (2015)
  • Author(s): 吉原久夫
  • Organizer: 代数幾何シンポジウム in 岐阜
  • Place of Presentation: ソフトピアジャパンセンタービル
  • Year and Date: 2015-01-10 – 2015-01-11
  • Invited
• [Presentation] Galois embedding of P^n (2014)
  • Author(s): 吉原久夫
  • Organizer: 射影多様体の幾何とその周辺
  • Place of Presentation: 高知大学理学部
  • Year and Date: 2014-11-01 – 2014-11-03
  • Invited
• [Presentation] Abelian embedding of P^n (2014)
  • Author(s): 吉原久夫
  • Organizer: Work shop on Galois points and related topics
  • Place of Presentation: 滋賀大学大津サテライトプラザ
  • Year and Date: 2014-09-13 – 2014-09-15
  • Invited
• [Presentation] J=1728 の楕円曲線の4 次埋め込みのガロワ直線の配置 (2013)
  • Author(s): 吉原久夫
  • Organizer: Work shop on Galois points and related topics
  • Place of Presentation: 山形大学理学部, 山形市
  • Invited
• [Presentation] Non-existence of Galois covering of smooth rational surface by bi-elliptic surface (2013)
  • Author(s): 吉原久夫
  • Organizer: 代数幾何学小研究集会
  • Place of Presentation: 新潟大学駅南キャンパスときめいと, 新潟市
  • Invited
• [Presentation] Arrangement of Galois lines for space elliptic curves (2013)
  • Author(s): Hisao Yoshihara
  • Organizer: Symposium on Algebraic Curves
  • Place of Presentation: 首都大学東京, 八王子市
  • Invited
• [Presentation] Sextic variety as Galois closure varity of some cubic hypersurface (2012)
  • Author(s): 吉原久夫
  • Organizer: 特異点と多様体の幾何学
  • Place of Presentation: 山形大学理学部
  • Invited
• [Presentation] Galois group at each point for some self-dual curves (2012)
  • Author(s): 吉原久夫
  • Organizer: Galois point and related topics
  • Place of Presentation: 山形大学理学部
  • Invited
• [Remarks] Galois point and related topics
  • URL: http://hyoshihara.web.fc2.com/
• [Remarks] Galois points and related topics
  • URL: http://hyoshihara.web.fc2.com/
• [Remarks] OPEN QUESTIONS
  • URL: http://mathweb.sc.niigata-u.ac.jp/~yoshihara/openquestion.html
• [Remarks] Open Questions
  • URL: http://mathweb.sc.niigata-u.ac.jp/~yosihara/openquestion.html