Study of algebraic varieties with group action
Project/Area Number  10640039 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  Tokyo Denki University 
Principal Investigator 
NAKANO Tetsuo Tokyo Denki University, College of Science and Engineering, Associate professor, 理工学部, 助教授 (00217796)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥2,800,000 (Direct Cost : ¥2,800,000)
Fiscal Year 2000 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥1,800,000 (Direct Cost : ¥1,800,000)

Keywords  3dimensional projective varieties / branched covering over plane / quotient terminal singularity / finite matrix group / 3次元射影代数多様体 / 平面上の分岐被覆 / 商末端特異点 / 有限行列群 / 極小モデル / 群作用 / ガロア被覆 / SL(2)作用 
Research Abstract 
As "Study of algebraic varieties with group action", the author made study of the following four topics during the period 19982000 : (1) The author studied the 3dimensional nonsingular projective algebraic varieties on which SL(2) acts with 2dimensional general orbits. This class of varieties is the last class whose structure remains unknown among the 3dimensional nonsingular projective algebraic varieties on which a simple algebraic group acts nontrivially. The author succeeded in classification of these varieties under the condition that they have no fixed points. This research was published as "Projective threefolds on which SL(2) acts with 2dimensional general orbits" (Transactions of AMS). (2) The author studied with H.Nishikubo the maximal Galois branched covering over affine and projective planes with branch locus x^2=y^q, and clarified the structure of the Galois groups of these coverings and also got the condition for the existence for the maximal Galois covering. This research is to appear as "On some maximal Galois coverings over affine and projective planes II" in Tokyo J.Math. (3) The author studied with H.Takamidori the defining equations and the rigidity of 3dimensional quotient terminal singularities from the viewpoint of computational algebraic geometry. This research has been submitted to a certain journal. (4) The finite primitive subgroups of SL(4) was classified by Blichfeldt to 30 types. The author confirmed with K.Niitsuma that these 30 types of matrix groups are actually primitive, using the computer algebra system MAGMA.Further, we calculated the Hilbert series of these finite matrix groups and determined the candidates among these matrix groups whose invariant rings are complete intersections. This research will be published as a paper "On primitive finite subgroups of SL(4) and their invariant rings".

Report
(5results)
Research Output
(9results)