ℓ -away ACM bundles on Fano surfaces

We propose the definition of ℓ -away ACM bundle on a polarized variety (X, 𝒪_X(h)) . Then we give constructions of ℓ -away ACM bundles on (ℙ^2, 𝒪_ℙ^2(1)) , (ℙ^1×ℙ^1, 𝒪_ℙ^1×ℙ^1(1,1)) and the anticanonically polarized blow up of ℙ^2 up to three non collinear points. Also, we give the complete classification of ℓ -away ACM bundles ℰ of rank 2 for values 1 ≤ℓ≤ 2 on (ℙ^2, 𝒪_ℙ^2(1)) . Similarly, on (ℙ^1×ℙ^1, 𝒪_ℙ^1×ℙ^1(1,1)) , we give such a classification if det(ℰ) = 𝒪_ℙ^1×ℙ^1(a,a) for some a ∈ℤ . Moreover, we prove that the corresponding graded module H_*^1 ( ℰ) = ⊕ _t ∈ℤH^1 (ℰ(th)) is connected, extending the similar result for bundles on ℙ^2 .