$\ell$-away ACM line bundles on a nonsingular cubic surface

TL;DR

This paper classifies and constructs $ ext{ell}$-away ACM line bundles on nonsingular cubic surfaces, extending previous work on ACM bundles and providing explicit examples and existence results for various degrees.

Contribution

It completely classifies $ ext{ell}$-away ACM line bundles for $ ext{ell} \, \leq 2$ on cubic surfaces and constructs examples for higher $ ext{ell}$, expanding understanding of these bundles.

Findings

Complete classification of $ ext{ell}$-away ACM line bundles for $ ext{ell} \leq 2$

Construction of examples for $ ext{ell} \geq 3$

Existence of smooth hypersurfaces with $ ext{ell}$-away ACM line bundles

Abstract

Let $X \subset \mathbb P^3$ be a nonsingular cubic hypersurface. Faenzi (\cite{F}) and later Pons-Llopis and Tonini (\cite{PLT}) have completely characterized ACM line bundles over $X$. As a natural continuation of their study in the non-ACM direction, in this paper, we completely classify $\ell$-away ACM line bundles (introduced recently by Gawron and Genc (\cite{GG})) over $X$, when $\ell \leq 2$. For $\ell\geq 3$, we give examples of $\ell$-away ACM line bundles on $X$ and for each $\ell \geq 1$, we establish the existence of smooth hypersurfaces $X^{(d)}$ of degree $d >\ell$ in $\mathbb P^3$ admitting $\ell$-away ACM line bundles.