Research on Application of Computer Algebra to Algebraic Geometry

代数向量丛的存在和构造问题与经典的子簇存在问题一起吸引了许多代数几何学家。受韦尔用向量丛概括自守形式的梦想的刺激，格洛腾迪克和阿提亚开创了代数向量丛理论。然后是 Narasimhan、Seshadri、Mumford 等人。深入研究了该理论并构建了模空间及其性质。感谢他们，曲线上的代数向量丛理论的基础已经解决，尽管许多严重的问题仍有待解决。施瓦岑贝格开始了代数曲面上的代数向量丛的研究，随后该研究项目的首席研究员找到了在更高维簇上构造代数向量丛的通用方法。然而，对于维度不小于四的射影空间上低秩向量丛的存在和构造，我们还没有明确的观点。在目前的情况下，研究 Tango 丛可能是至关重要的，尽管地面场具有特征 2，但它本质上是 5 维射影空间上唯一的 2 阶不可分解向量丛。因此，在本项目中，我们设定了使用计算机代数研究 Tango 丛的主要目标。我们成功地用一个 15 x 34 矩阵来表示计算机代数上的 Tango 束，该矩阵的条目是 6 个变量的齐次二次形式。观察这个矩阵，我们可以确定 Tango 丛的转移矩阵，并且通过使用计算机代数，我们可以通过线丛的直接求和来获得 Tango 丛的分辨率。然后我们可以计算 Tango 包的 Chern 类。移动 Tango 丛的第一个 Chern 类并计算（使用计算机代数）第 0 次上同调，我们看到 Tango 丛是稳定的。

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.