Research on Application of Computer Algebra to Algebraic Geometry

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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.