ACM bundles of rank 2 on quartic hypersurfaces in P-3 and Lazarsfeld-Mukai bundles

Let X be a smooth quartic hypersurface in P-3. By the Brill-Noether theory of curves on K3 surfaces, if a rank 2 aCM bundle on X is globally generated, then it is the Lazarsfeld-Mukai bundle E-C,E-Z associated with a smooth curve C on X and a base point free pencil Z on C. In this paper, we will focus on the classification of such bundles on X to investigate aCM bundles of rank 2 on X. Concretely, we will give a necessary condition for a rank 2 vector bundle of type E-C,E-Z to be indecomposable initialized and aCM, in the case where the class of C in Pic(X) is contained in the sublattice of rank 2 generated by the hyperplane class of X and a non-trivial initialized aCM line bundle on X.