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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.
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