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Research on algebraic varieties with an algebraic group action

具有代数群作用的代数簇研究

基本信息

批准号：
17540041
负责人：
NAKANO Tetsuo
金额：
$ 2.34万
依托单位：
Tokyo Denki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540041/
关键词：
Algebraic Variety Invariant Theory Moiuli Space Algebraic Group Action Integral Convex Polytope Toric Variety 群作用

项目摘要

(1) Research on the moduli space of pointed algebraic curves of low genus. We have studied the moduli space M(g,1 ; N) of pointed algebraic curves (X,P) of genus g with a given semigroup N at the point P, and have got a theorem saying that, if the genus g is less than or equal to 6 and the number of generators of the semigroup N is less than or equal to 4, then the moduli space M(g,1 ; N) is an irreducible rational variety. This theorem is contained in the paper titled "On the moduli space of pointed algebraic curves of low genus II - rationality -", which is to appear in Tokyo J. Math. 31(2008).(2) Research on the invariant rings of the finite subgroups of the special linear group SL(4,C) of degree 4. We have been computing the generators and relations of the invariant rings of the finite primitive subgroup of SL(4,C), which are 30 of them in all. So far, we have successfully got the generators and relations of about 15 of them. The remaining groups have big order and the direct computation is difficult. We are now seeking for a method for computation for those subgroups with big order.(3) Research on the toric varieties and the combinatorics. We have studied the complete linear systems on the complete toric Gorenstein del Pezzo surfaces, and determined the unique linear system of minimal dimension and degree. As an application, we can compute the defining equations of the image of the minimal dimensional embedding and also determine if a complete toric Gorenstein del Pezzo surface is a global complete intersection or not.

(1)低亏格尖代数曲线的模空间研究。研究了亏格g的点代数曲线（X，P）与给定的半群N在点P处的模空间M（g，1 ; N），得到了一个定理：如果亏格g小于或等于6，且半群N的生成元数小于或等于4，则模空间M（g，1 ; N）是不可约有理簇。该定理包含在题为“On the moduli space of pointed algebraic curves of low genus II - rationality -”的论文中，该论文将发表在东京数学杂志31（2008）上。(2)4次特殊线性群SL（4，C）的有限子群的不变环的研究。本文计算了SL（4，C）的有限本原子群的不变环的生成元和关系，共30个。到目前为止，我们已经成功地得到了其中约15个的生成元及其关系。其余群的阶数较大，直接计算困难。我们正在寻求一种方法来计算那些具有大阶的子群。(3)复曲面簇与组合数学的研究。研究了完备复曲面GorensteindelPezzo曲面上的完备线性系统，确定了唯一的最小维数和最小次数的线性系统。作为应用，我们可以计算最小维嵌入的图像的定义方程，并确定一个完整的复曲面Gorenstein del Pezzo曲面是否是一个整体完全相交。