Study of algebraic varieties with group action

在1998-2000年期间，作者在“群作用代数变体的研究”中，研究了以下四个主题：(1)研究了SL(2)作用于二维一般轨道的三维非奇异射影代数变体。这类变体是简单代数群非平凡作用的三维非奇异射影代数变体中最后一类结构未知的变体。作者在这些品种没有定点的条件下，成功地对它们进行了分类。该研究发表在《SL(2)作用于二维一般轨道的射影三折》（Transactions of AMS）上。(2)与nishikubo一起研究了分支轨迹为x^2=y^q的仿射平面和射影平面上的极大伽罗瓦分支覆盖，澄清了这些覆盖的伽罗瓦群的结构，并得到了极大伽罗瓦覆盖存在的条件。本研究发表于《关于仿射平面和射影平面上的一些极大伽罗瓦覆盖ⅱ》，发表于《东京数学杂志》。(3)从计算代数几何的角度，与takamidori一起研究了三维商端点奇点的定义方程和刚性。这项研究已提交给某杂志。(4) SL(4)的有限原始子群被Blichfeldt划分为30种类型。作者利用计算机代数系统MAGMA与k.n itsuma确认了这30种矩阵群实际上是原始的。进一步，我们计算了这些有限矩阵群的Hilbert级数，并确定了这些矩阵群中不变环为完全相交的候选矩阵群。本研究将以论文《论SL(4)的原始有限子群及其不变环》发表。

As "Study of algebraic varieties with group action", the author made study of the following four topics during the period 1998-2000 :(1) The author studied the 3-dimensional nonsingular projective algebraic varieties on which SL(2) acts with 2-dimensional general orbits. This class of varieties is the last class whose structure remains unknown among the 3-dimensional nonsingular projective algebraic varieties on which a simple algebraic group acts non-trivially. 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 2-dimensional 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 3-dimensional 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".