On the splitting of Lazarsfeld–Mukai bundles on K3 surfaces

Abstract

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,Z of rank 2 associated with C and Z is given by an extension of the torsion free sheaf JZ⊗OX(C) by OX, where JZ is the ideal sheaf of Z in X. We can see that if C is very ample as a divisor on X, EC,Z is an ACM bundle with respect to OX(C). In this paper, by using this fact, we will characterize a necessary condition for EC,Z to be given by an extension of two line bundles on X, by ACM line bundles with respect to OX(C).