On the splitting of Lazarsfeld–Mukai bundles on K3 surfaces

Let X be a K3 surface, let C be a smooth curve on X, and let Z be a base point free pencil on C  . Then, the Lazarsfeld–Mukai bundle EC,ZEC,Z of rank 2 associated with C and Z   is given by an extension of the torsion free sheaf JZ⊗OX(C)JZ⊗OX(C) by OXOX, where JZJZ is the ideal sheaf of Z in X. We can see that if C is very ample as a divisor on X  , EC,ZEC,Z is an ACM bundle with respect to OX(C)OX(C). In this paper, by using this fact, we will characterize a necessary condition for EC,ZEC,Z to be given by an extension of two line bundles on X  , by ACM line bundles with respect to OX(C)OX(C).