ACM bundles of rank 2 on quartic hypersurfaces in $${\mathbb {P}}^3$$ and Lazarsfeld-Mukai bundles

Abstract

Let X be a smooth quartic hypersurface in $$\mathbb {P}^3$$ . 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 $$E_{C,Z}$$ 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 $$E_{C,Z}$$ to be indecomposable initialized and aCM, in the case where the class of C in $${{\,\mathrm{Pic}\,}}(X)$$ 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.