Slope semistability of rank 2 Lazarsfeld-Mukai bundles on K3 surfaces and ACM line bundles

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Previously, many people have studied a stability of vector bundles of given rank and Chern classes on algebraic varieties. Recently, we are interested in the slope stability of the rank 2 Lazarsfeld-Mukai bundle EC,Z on a K3 surface X associated to a very ample smooth curve C on X and a base point free pencil Z on C with respect to Oₓ(C). In this paper, we will give a sufficient condition for such a Lazarsfeld-Mukai bundle EC,Z to be Oₓ(C)-slope semistable by ACM line bundles with respect to Oₓ(C).