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

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 $\ell \le 2$. For $\ell \ge 3$, we give examples of $\ell$-away ACM line bundles on $X$ and for each $\ell \ge 1$, we establish the existence of smooth hypersurfaces $X^{(d)}$ of degree $d>\ell$ in ${\mathbb{P}}^3$ admitting $\ell$-away ACM line bundles.

33 pages. Comments are welcome