Examples of rank two aCM bundles on smooth quartic surfaces in $\mathbb{P}^3$

TL;DR

This paper classifies certain rank two vector bundles on smooth quartic surfaces in projective three-space, focusing on their Chern classes, cohomological properties, and stability, contributing to the understanding of algebraic vector bundles on K3 surfaces.

Contribution

It provides a classification of rank two aCM bundles on smooth quartic surfaces, including their stability properties, which is a new detailed analysis in this context.

Findings

Classification of rank two aCM bundles with specific Chern classes

Identification of stability conditions for these bundles

Explicit examples illustrating the classification

Abstract

Let $F\subseteq\mathbb{P}^3$ be a smooth quartic surface and let $\mathcal{O}_F(h):=\mathcal{O}_{\mathbb{P}^3}(1)\otimes\mathcal{O}_F$. In the present paper we classify locally free sheaves $\mathcal{E}$ of rank $2$ on $F$ such that $c_1(\mathcal{E})=\mathcal{O}_F(2h)$, $c_2(\mathcal{E})=8$ and $h^1\big(F,\mathcal{E}(th)\big)=0$ for $t\in\mathbb{Z}$. We also deal with their stability.