Research on Application of Computer Algebra to Algebraic Geometry

代数向量丛的存在性和构造问题与经典的子簇存在性问题密切相关，吸引了许多代数几何学家的注意。在Weil的梦想的刺激下，将向量丛的自守形式一般化，Grothendieck和Atiyah开创了代数向量丛理论。随后Narasimhan，Seshadri，Mumford等人对这一理论进行了深入的研究，并对模空间的构造及其性质进行了研究。由于它们的存在，虽然曲线上的代数向量丛理论还有许多重要的问题有待解决，但它的基础已经奠定。Schwarzenberger首先对代数曲面上的代数向量丛进行了研究，随后该研究项目的主要研究者找到了构造高维簇上代数向量丛的一般方法，然而对于四维以上射影空间上低秩向量丛的存在性和构造问题，我们还没有一个清晰的认识。在目前的情况下，研究Tango丛可能是至关重要的，它本质上是唯一的秩2，5维射影空间上的不可分解向量丛，即使基场的特征为2。在这个项目中，我们设置，因此，我们的主要目标是研究探戈丛使用计算机代数。我们成功地用一个15 × 34的矩阵来表示计算机代数中的Tango丛，该矩阵的元素是6个变量的齐次二次型。通过观察这个矩阵，我们可以确定Tango丛的转移矩阵，并利用计算机代数，我们得到了Tango丛的一个分解，由线丛的直和。然后我们可以计算Tango丛的Chern类。移动Tango丛的第一个Chern类并计算（使用计算机代数）0阶上同调，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.