ACM line bundles on polarized K3 surfaces

TL;DR

This paper characterizes when certain line bundles on polarized K3 surfaces are arithmetically Cohen-Macaulay and initialized, based on geometric and cohomological conditions.

Contribution

It provides a necessary and sufficient condition for line bundles to be ACM and initialized on polarized K3 surfaces, extending understanding of vector bundles in algebraic geometry.

Findings

Characterization of ACM line bundles on polarized K3 surfaces

Conditions involving divisors with specific intersection properties

Criteria for line bundles with empty linear systems to be ACM

Abstract

An ACM bundle on a polarized algebraic variety is defined as a vector bundle whose intermediate cohomology vanishes. We are interested in ACM bundles of rank one with respect to a very ample line bundle on a K3 surface. In this paper, we give a necessary and sufficient condition for a non-trivial line bundle $\mathcal{O}_X(D)$ on $X$ with $|D|=\emptyset$ and $D^2\geq L^2-6$ to be an ACM and initialized line bundle with respect to $L$, for a given K3 surface $X$ and a very ample line bundle $L$ on $X$.