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Research on Application of Computer Algebra to Algebraic Geometry

计算机代数在代数几何中的应用研究

基本信息

批准号：
15540024
负责人：
MARUYAMA Masaki
金额：
$ 2.3万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

The existence and the construction problem of algebraic vector bundles has attracted many algebraic geometers, in connection with the classical existence problem of subvarieties. Stimulated by Weil's dream to genralarize the automorphic forms in terms of vector bundles, Grothendieck and Atiyah initiated the theory of algebraic vector bundles. Then Narasimhan, Seshadri, Mumford et al. have deeply studied the theory and have gone to the construction of the moduli spaces and their properties. Thanks to them, the foundation of the theory of algebraic vector bundles on curves has been settled though many serious problems are still remaining to be solved. Schwarzenberger began the study on algebraic vector bundles on algebraic surface and then the head investigator of this research project found a general way to construct algebraic vector bundles on higher dimensional varieties.We have, however, no clear perspective about the existence and construction of low rank vector bundles on the projective spaces of dimension not less than four. In the present situation, it might be crucial to study the Tango bundle, which is essentially unique rank 2, indecomposable vector bundle on 5-dimensional projective space even though the ground field is of characteristic 2. In this project we set, therefore, our main target to study the Tango bundle by using Computer Algebra. We succeeded to represent the Tango bundle on Computer Algebra by a 15 x34 matrix whose entries are homogeneous quadratic forms in 6 variables. Watching this matrix we can determine the transition matrices of the Tango bundle and by using Computer Algebra we get a resolution of the Tango bundle by direct sums of line bundles. Then we can compute the Chern class of the Tango bundle. Shifting the first Chern class of the Tango bundle and computing (using Computer Algebra) the 0-th cohomology, we see that the Tango bundle is stable.

代数向量束的存在性和构造问题与经典的子变量的存在性问题联系在一起，引起了许多代数几何学者的关注。受Weil用向量束概括自同构形式的梦想的启发，Grothendieck和Atiyah开创了代数向量束理论。随后Narasimhan， Seshadri， Mumford等人对该理论进行了深入的研究，探讨了模空间的构造及其性质。他们奠定了曲线上代数向量束理论的基础，但仍有许多重大问题有待解决。施瓦岑贝格开始了对代数曲面上的代数向量束的研究，然后这个研究项目的负责人找到了在高维变量上构造代数向量束的一般方法。然而，对于不小于4维的射影空间上的低秩向量束的存在性和构造，我们还没有明确的认识。在目前的情况下，研究Tango束可能是至关重要的，因为Tango束本质上是5维射影空间上唯一的2阶不可分解向量束，尽管地面场的特征为2。因此，在这个项目中，我们将主要目标设置为使用Computer Algebra来研究Tango包。我们成功地用一个15 × 34矩阵在计算机代数上表示Tango束，该矩阵的项是6变量的齐次二次型。观察这个矩阵，我们可以确定Tango束的转移矩阵，并通过计算机代数通过直线束的直接和得到Tango束的解析。然后我们可以计算Tango包的Chern类。移动Tango包的第一个Chern类并计算（使用Computer Algebra）第0个上同调，我们看到Tango包是稳定的。