Study of algebraic varieties with group action

作为“具有群作用的代数簇的研究”，作者在1998-2000年间进行了以下四个方面的研究：(1)研究了SL(2)具有2维一般轨道的三维非奇异射影代数簇。这类簇是简单代数群非平凡作用的三维非奇异射影代数簇中最后一类结构未知的簇。在没有固定点的情况下，作者成功地对这些品种进行了分类。(2)与H.Nishikubo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Galois分支覆盖，阐明了这些覆盖的Galois群的结构，并得到了存在最大Galois覆盖的条件。这项研究将在东京数学杂志上发表为《关于仿射和射影平面上的极大Galois覆盖II》。(3)作者与H.Takamidori从计算代数几何的观点研究了三维商终端奇点的定义方程和刚性。将SL(4)的有限本原子群按BlichFeldt分类为30种类型。利用计算机代数系统MAGMA，与K.NiitSuma证明了这30类矩阵群实际上是本原的，进而计算了这些有限矩阵群的Hilbert级数，并确定了这些不变环是完全交的矩阵群中的候选矩阵。这项研究将作为论文《关于SL(4)的本原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".