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

TL;DR

This paper classifies rank 2 arithmetically Cohen-Macaulay bundles on smooth quartic hypersurfaces in projective 3-space, focusing on Lazarsfeld-Mukai bundles associated with curves and line bundles on these surfaces.

Contribution

It provides a necessary condition for certain Lazarsfeld-Mukai bundles to be indecomposable, initialized, and aCM on quartic hypersurfaces, advancing the classification of such bundles.

Findings

Identifies conditions for indecomposability of rank 2 aCM bundles

Connects bundle properties to classes in Picard group

Advances understanding of Lazarsfeld-Mukai bundles on quartic hypersurfaces

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 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 $X$.