Semistable Vector Bundles and Tannaka Duality from a Computational Point of View

Abstract

We develop a semistability algorithm for vector bundles that are given as a kernel of a surjective morphism between splitting bundles on the projective space $\mathbb{P}^N$ over an algebraically closed field $K$ . This class of bundles is a generalization of syzygy bundles. We show how to implement this algorithm in a computer algebra system. Further, we give applications, mainly concerning the computation of Tannaka dual groups of stable vector bundles of degree 0 on $\mathbb{P}^N$ and on certain smooth complete intersection curves. We also use our algorithm to close a case left open in a recent work of L. Costa, P. Macias Marques, and R. M. Miró- Roig regarding the stability of the syzygy bundle of general forms. Finally, we apply our algorithm to provide a computational approach to tight closure. All algorithms are implemented in the computer algebra system CoCoA.