Semistable vector bundles and Tannaka duality from a computational point of view

TL;DR

This paper introduces a computational algorithm for testing semistability of certain vector bundles on projective spaces, enabling applications in Tannaka duality, stability analysis, and tight closure computations.

Contribution

It develops a novel semistability algorithm for kernel bundles, implements it in CoCoA, and applies it to problems in Tannaka duality, stability of syzygy bundles, and tight closure.

Findings

Successfully computed Tannaka dual groups for stable bundles.

Closed an open case on the stability of syzygy bundles.

Provided a computational approach to tight closure problems.

Abstract

We develop a semistability algorithm for vector bundles which are given as a kernel of a surjective morphism between splitting bundles on the projective space 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 the projective space and on certain smooth complete intersection curves. We also use our algorithm to close an open case left in a recent work of L. Costa, P. Macias Marques and R. M. Miro-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.