$\mathrm{\ell }$-away ACM Bundles on Fano Surfaces

We propose the definition of $\mathrm{\ell }$-away ACM bundle on a polarized variety $(X,{\mathcal{O}}_X(h))$. Then we give constructions of $\mathrm{\ell }$-away ACM bundles on $({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1))$, $($\mathbb{P}$^1 \times $\mathbb{P}$^1, $\mathcal{O}${$\mathbb{P}$^1 \times $\mathbb{P}$^1}(1,1))$ and the anticanonically polarized blow up of ${\mathbb{P}}^2$ up to three non collinear points. Also, we give the complete classification of $\mathrm{\ell }$-away ACM bundles $\mathcal{E}$ of rank 2 for values $1\le \mathrm{\ell }\le 2$ on $({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1))$. Similarly, on $($\mathbb{P}$^1 \times $\mathbb{P}$^1, $\mathcal{O}${$\mathbb{P}$^1 \times $\mathbb{P}$^1}(1,1))$, we give such a classification if $$\mathrm{d}\mathrm{e}\mathrm{t}$($\mathcal{E}$) = $\mathcal{O}${$\mathbb{P}$^1 \times $\mathbb{P}$^1}(a,a)$forsome$a \in $\mathbb{Z}$$\mathrm{.}Moreover,weprovethatthecorrespondinggradedmodule$$\mathrm{H}$*^1 ( $\mathcal{E}$) = \underset{{t \in $\mathbb{Z}$ }}{\bigoplus} $\mathrm{H}$^1 ($\mathcal{E}$ (th))$ is connected, extending the similar result for bundles on ${\mathbb{P}}^2$.