In this paper all non-splitting rank-two vector bundles E without intermediate cohomology on a general quartic hypersurface X in P4 are classified. In particular, the existence of some curves on a general quartic hypersurface is proved.
We survey some parts of the vast literature on vector bundles on Hirzebruch surfaces, focusing on the rank-two case.
We give a lower bound for the Seshadri constants of ample vector bundles which depends only on the numerical properties of the Chern classes and on a “stability” condition.
We characterize minimal free resolutions of homogeneous bundles on . Besides we study stability and simplicity of homogeneous bundles on by means of their minimal free resolutions; in particular we give a criterion to see when a homogeneous bundle is simple by means of its minimal resolution in the case the first bundle of the resolution is irreducible.