$\ell$-away ACM Bundles on Fano Surfaces

We propose the definition of $\ell$-away ACM bundle on a polarized variety $(X, \mathcal{O}_{X}(h))$. Then we give constructions of $\ell$-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 $\ell$-away ACM bundles $\mathcal{E}$ of rank 2 for values $1 \leq \ell \leq 2$ 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 $\mathrm{det}(\mathcal{E}) = \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,a)$ for some $a \in \mathbb{Z}$. Moreover, we prove that the corresponding graded module $\mathrm{H}_*^1 ( \mathcal{E}) = \underset{{t \in \mathbb{Z} }}{\bigoplus} \mathrm{H}^1 (\mathcal{E} (th))$ is connected, extending the similar result for bundles on $\mathbb{P}^2$.


Introduction
A vector bundle E on a smooth polarized variety (X, O X (h)) is called arithmetically Cohen-Macaulay (ACM) if its cohomology satisfies H i (X, E(th)) = 0 for all t ∈ Z and 0 < i < dim(X).A well-known result of Horrocks states that a vector bundle E on P n splits into line bundles if and only if E is ACM.Although a decomposition result like that of Horrocks is not known for vector bundles on arbitrary polarized varieties, ACM bundles on such varieties are regarded as simpler in comparison.Consequently, ACM bundles on polarized varieties have received considerable attention among algebraic geometers, leading to a significant body of literature dedicated to their study..
The natural next step is to explore the properties of non-ACM bundles.
Definition 1.1.A non-ACM vector bundle E on a polarized variety (X, O X (h)) is called ℓ-away ACM if there are exactly ℓ pairs (i, t) ∈ Z 2 with 0 < i < dim(X) such that h i (E(th)) = 0.
This article aims to investigate the properties of such bundles for any positive integer ℓ.The main problems are their existence for each ℓ and the classification of them for a given ℓ.Also, the configuration of non zero intermediate cohomology groups is interesting to explore.
The existence of these bundles is significant for existing literature, particularly in the context of weakly Ulrich and supernatural bundles (see Definition 3.4 and Definition 3.7).These types of bundles can have non-vanishing intermediate cohomology groups, making them relevant to the study of ℓ-away ACM bundles.The article presents several results like corollaries 3.8, 3.11, 4.9 and 4.11 in this direction.
In this work, we show the existence of ℓ-away ACM bundles and give their classification on the Fano surfaces (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.Definition 1.2.A vector bundle E on (X, O X (h)) is called special if det(E) = ch for some integer c.
In Section 2, we determine the common properties of special ℓ-away ACM bundles of rank 2 on Fano surfaces.Also, we give some basic definitions and results concerning the theory of bounded derived category of coherent sheaves on a smooth variety.We finish the section with some properties of quivers and their representations.
In Section 3, we give the classification of 1-away and 2-away ACM bundles of rank 2 on P 2 .Then, we point out that they are weakly Ulrich.Also, we show the existence of µ-stable ℓ-away ACM bundle of rank 2 on P 2 for any ℓ > 1 and that this bundle corresponds to a smooth point in the moduli space of µ-stable sheaves lying in a single component of dimension ℓ 2 + 2ℓ − 3. We finish this section by showing the existence of simple ℓ-away ACM bundle of any even rank by using representations of quivers.As a corollary, we are able to show the existence of weakly Ulrich bundles of any even rank on P 2 .
In Section 4, we analyse ℓ-away ACM bundles on P 1 × P 1 .We give the complete classification of ℓ-away ACM line bundles for any ℓ and the complete classification of special 1-away and 2-away ACM bundles of rank 2.Moreover, we show that the module H 1 * (E) is connected for any special rank 2 bundle, where the connectedness can be defined as follows: Definition 1.3.Let E be vector bundle on a polarized variety (X, O X (h)).We say that the module H 1 * (E) is connected if h 1 (E(kh)) = 0 and h 1 (E(mh)) = 0 with k < m implies h 1 (E(th)) = 0 for k < t < m.
The connectedness result mentioned above extends to P 1 × P 1 where the similar result was proved in [3, Proposition 3.1] for P 2 .Additionally, for any ℓ, we show the existence of simple, special ℓ-away ACM bundles of rank 2, representing smooth points of the moduli space of simple sheaves lying in a single component of dimension 2ℓ 2 + 2ℓ − 3.Moreover, we show the existence of special ℓ-away ACM bundle of any even rank.We cap off this section with the existence of weakly Ulrich bundles of any even rank on P 1 × P 1 .
Lastly, in Section 5, we give the complete classification of ℓ-away ACM line bundles on the blow up of P 2 at a single point.Then using these line bundles, we show the existence of ℓ-away ACM bundles of rank 2. We finish the section by showing the existence of ℓ-away ACM bundles of rank 1 and rank 2 on the blow ups of P 2 at two distinct points or three non collinear points.Unfortunately, for these Fano surfaces, we have no complete classification like the case P 2 or P 1 × P 1 .The problem of connectedness of H 1 * (E) for a special rank 2 bundle is open even on the blow up of P 2 at a single point.
1.1.Notations and Conventions.We always work over the field C of complex numbers.
If E is a vector bundle on a variety X, we denote by h i (X, E), hom(X, E) and ext i (X, E) the dimension of the complex vector spaces H i (X, E), Hom(X, E) and Ext i (X, E), respectively.

Preliminaries
Note that, for an ℓ-away ACM bundle E on a polarized variety (X, O X (h)), E(t 0 h) is again an ℓ-away ACM bundle for each integer t 0 .
So, when we classify ℓ-away ACM bundles on a polarized variety, we can restrict to initialized ones.Definition 2.2.A smooth projective variety X is called a Fano variety if its anticanonical divisor −K X is ample.The index i X of a Fano variety is the greatest integer such that K X ∼ = O X (−i X h) for some ample line bundle O X (h) ∈ Pic(X) and the ample divisor h is called the fundamental divisor.
Throughout the paper, when we call a variety Fano surface, we mean a smooth Fano surface polarized with the fundamental divisor.In this case, we have three cases with respect to i X : If i X = 3, then our Fano surface is (P 2 , O P 2 (1)).
If i X = 2, then our Fano surface is (P )).More explicitly, let π i : P 1 ×P 1 → P 1 be the projection map onto the ith factor for i = 1, 2. Write h i for the pull-back of a point on P 1 via the map π i .Then A(P If i X = 1, then our Fano surface is isomorphic to a blow-up of P 2 at a set of points {p 1 , p 2 , . . ., p r } with r ≤ 8 where no three of them lie on a line and no six of them lie on a conic.The fundamental divisor is h = −K X .
Notice that, by the Serre duality, we have for a rank 2 special, initialized bundle on a Fano surface X.
Remark 2.3.Note that E ⊕m is an ℓ-away ACM bundle of rank rm if E is ℓ-away ACM bundle of rank r on a polarized variety (X, h).
When (X, h) is a surface, we define the numbers Notice that we always have s ≥ l − 1.
Proposition 2.4.Let E be a special initialized vector bundle of rank 2 on a Fano surface X with det(E) = ch.If there exists an integer t 0 such that −1 − c ≥ t 0 ≥ 1 − i X and h 1 (E(t 0 h)) = 0 then E can be obtained as an extension of two line bundles Proof.Let s ∈ H 0 (E) and let its zero locus be D ∪ Z where D is the union of the components of pure dimension 1 and Z a scheme of dimension 0.Then, we have . Then, by tensoring 2.2 with O X (t 0 h) and considering its cohomology, we have By assumption h 1 (E(t 0 h)) = 0. Since we assumed On the other hand, by tensoring the short exact sequence with O X ((t 0 + c)h − D) and considering its cohomology, we have Since t 0 + c ≤ −1 by assumption, we have h 0 (O X ((t 0 + c)h − D) = 0. Also, we showed that h 1 (I Z ((t 0 + c)h − D)) = 0. Therefore, h 0 (O Z ) = 0.So Z is empty.
Theorem 2.5.Let E be a special ℓ-away ACM bundle of rank 2 on a Fano surface X.Then the following items hold.
Proof.(i): First, note that Then, by combining this with item (i), we have Notice that h 1 (E((k 0 − 1)h)) = 0 by definition.Also, k 0 − 1 ≥ 1 − i X by the assumption and −1 − c ≥ k 0 − 1 by part (i).Then, by Proposition 2.4, E is an extension of two line bundles This is a contradiction for the case i X = 3, because there is no indecomposable rank 2 vector bundle obtained by extension of two line bundles.
Therefore, the assumption is false and k 0 ≤ −i X + 1 for i X > 1.
Remark 2.6.Notice that, in the item (iii) of Theorem 2.5, we have obtained the result for i X > 1 by showing that there is no non trivial extension in the form 2.4.However, this is not possible in the case of i X = 1, because, in principle, we may have non nef divisor D with very negative intersection number with an exceptional curve.
Lemma 2.7.Let X be a Fano surface.If E is a simple vector bundle on X, then Ext 2 (E, E) = 0.
Proof.One can prove the statement by mimicking the proof of [2, Lemma 2.2].Now, in this part of the paper, we will mention some details about the bounded derived category of coherent sheaves on a smooth varieties.We refer the reader to the paper [9] for more details.Let D b (X) denotes the bounded derived category of coherent sheaves on a smooth variety X.For each sheaf E and integer k we denote by E[k] the trivial complex defined as with the trivial differentials: we will often omit [0] in the notation.A coherent sheaf E is exceptional as trivial complex in An ordered set of exceptional objects (E 0 , . . ., E s ) is an exceptional collection if Ext k X E i , E j = 0 when i > j and k ∈ Z.An exceptional collection is full if it generates D b (X) and strong if Ext k X E i , E j = 0 when i < j and k ∈ Z \ { 0 }.
) is full, then there exists a unique collection (F 0 , . . ., F s ) satisfying Finally, let us review the definition and basic properties of quivers.For an introduction to the topic, see [6].
where Q 0 and Q 1 are sets, whose elements are called vertices and arrows respectively, and two maps s, t : Q 1 → Q 0 that sends each arrow to its source and target respectively.A quiver Q is called finite if Q 0 and Q 1 are finite sets.It is called acyclic if it does not contain any oriented cycle.Definition 2.9.A representation V of the quiver Q is a set of finite-dimensional vector spaces V i for every i ∈ Q 0 and linear maps V a : V i → V j for every arrow a : i → j.V is called trivial if all V i = 0; in this case, we just write V = 0. Given two representations V and W , a morphism φ : V → W is a collection of linear maps φ i : V i → W i such that for every arrow a : i → j, the diagram We can define a category Rep(Q) where the objects are representations of Q and the morphisms are as defined in Definition 2.9.The direct sum of quiver representations, monomorphisms, epimorphisms, isomorphisms, and kernels and cokernels can be defined in a straightforward manner (see [6,Chapter 1.2]).
Let V be a representation of a quiver Q.We can introduce the extension group Ext 1 (V, −) as the derived functor of Hom(V, −).
Lemma 2.10.For any two quiver representations V and W , Ext i (V, W ) = 0 for i ≥ 2.
Proof.See [6,Lemma 2.4.3].This allows us to define the Euler characteristic as follows.
Definition 2.11.Given two quiver representations V and W , the Euler characteristic is defined by where hom(V, W ) and ext(V, W ) stand for dimensions of Hom(V, W ) and Ext 1 (V, W ). Definition 2.12.If the quiver Q is finite and the V i were assumed to be finitedimensional, the dimensions of V i can be arranged in a tuple of integers d = d(V ) ∈ Z Q0 , called dimension vector.Definition 2.13.Given two dimension vectors α, β ∈ Z Q0 , the Euler form is defined by The Euler characteristic and the Euler form are related by the following result.
Proposition 2.14.Given two quiver representations V and W , we have Let us finish this section with the following: Theorem 2.15.Suppose there exists a full strong exceptional collection Then there exists a quiver with relations (Q; J), and a triangulated equivalence Proof.See [13, Theorem 2.6.3].

ℓ-away ACM Bundles on P 2
In this section, we give the complete classification of initialized 1-away and 2away ACM bundles of rank 2 on Fano surface P 2 .Also, we show the existence of simple initialized ℓ-away ACM bundles of any even rank on P 2 for any ℓ > 1.Notice that the fundamental divisor h of P 2 is determined by the line bundle O P 2 (1).
Since, Pic(P 2 ) ∼ = Z < h >, in this section we will use the notation E(t) for E(th) for an integer t.
The Riemann-Roch theorem for a rank 2 vector bundle E on (P 2 , O P 2 (1)) with det(E) = c for some integer c gives First, let us state the standard Beilinson theorem for P 2 .This proposition will be useful for the classification results.Proposition 3.1.Let E be a vector bundle on P 2 .Then E is the cohomology in degree 0 of a complex C • with r th -module Now, we are ready to give the complete classifications of 1-away and 2-away ACM bundles of rank 2 on P 2 .Theorem 3.2.E is an initialized 1-away ACM bundle of rank 2 on P 2 if and only if E ∼ = Ω P 2 (2).
Proof.First, assume that, E is initialized 1-away ACM of rank 2.Then, by Theorem 2.5, Then, again by Equation (3.1), we have The converse is an easy application of Bott's Theorem.
Theorem 3.3.Let E be an initialized rank 2 vector bundle on P 2 .Then, E is a 2-away ACM if and only if E fits into one of the following sequences: Then, by Equation (3.1), we have the following Table 1 Table 1: The values of h q P 2 , E(ph) for P 2 If we apply Proposition 3.1 to E(−2) then E can be represented as Similarly, if we apply Proposition 3.1 to E(−1) then E can be represented as Conversely, by applying Bott's Theorem, one can see that kernel bundles in items (i) and (ii) are 2-away ACM bundles of rank 2 on P 2 .
In the theory of vector bundles, there are particular bundles which are called weakly Ulrich.Definition 3.4.Let E be a vector bundle on a smooth, polarized variety (X, O X (h)) of dimension n.Then, E is weakly Ulrich if For the interested reader, the main references are [5] and [7].Now, we have the complete list of indecomposable weakly Ulrich bundles of rank 2 on P 2 .Corollary 3.5.All indecomposable weakly Ulrich bundles of rank 2 on P 2 are 1away and 2-away ACM bundles on P 2 up to a shift.
Proof.Notice that a non-ACM weakly Ulrich bundle on a surface is ℓ-away ACM for ℓ ≤ 2. Using the classification of rank two ℓ-away ACM bundles on P 2 with ℓ ≤ 2 in Theorems 3.2 and 3.3, it is easy to check that they are weakly Ulrich up to a shift.When ℓ ≥ 3, it gets complicated to give the complete classification of ℓ-away ACM bundles of rank 2 on P 2 .In the meantime, there always exists µ-stable ℓ-away ACM bundle of rank 2 on P 2 for all ℓ.
Theorem 3.6.The bundle E, fitting in the displayed sequence, is a µ-stable ℓ-away ACM bundle of rank 2 on P 2 .Also, the point corresponding to E in the moduli space of µ-stable sheaves on P 2 is smooth and lies in a component of dimension ℓ 2 + 2ℓ − 3.
Proof.Since the general morphism Now, let us show that E is initialized.The cohomology of sequence 3.2 returns Then, by the Serre duality, h 1 E(t) = 0 for t ≥ −1 too.Hence, E is a ℓ-away ACM bundle of rank 2.
It is µ-stable by Hoppe's Criterion (see [11,Corollary 4]), since it is initialized and it has positive first Chern class.
Finally, since it is µ-stable, the moduli space at the corresponding point is smooth by simplicity of E. Therefore, the dimension of moduli space can be computed by the equation If we tensor the sequence 3.2 by E ∨ ≃ E(−ℓ) and use the additivity of the Euler characteristic under exact sequences, the equation 3.3 will be Then, by applying the equation 3.1 for E and E(1), we get h The bundle constructed in Theorem 3.6 is supernatural and important in the Boij-Söderberg theory.Definition 3.7.A bundle E on a polarized variety (X, O X (h)) of dimension n is called supernatural if for each j ∈ Z there is at most one i such that h i (E(jh)) = 0 and the Hilbert polynomial of E has n distinct integral roots.
Corollary 3.8.The bundle E constructed in Theorem 3.6 is a supernatural bundle and its Hilbert polynomial has integral roots −ℓ − 2 and −1.
Proof.We have shown in the proof of Theorem 3.6 that Lemma 3.9.Let E be a supernatural ℓ-away ACM bundle on a polarized variety (X, O X (h)) of dimension n and represented in a flat family.Then there exists an open subset in this family whose points corresponds to supernatural ℓ-away ACM bundles.
Proof.Since the dimension of cohomology is an upper semi-continuous function on a flat family, then there exists an open subset O in the family around the point corresponding to E where for any bundle F ∈ O we have h i (F (jh)) ≤ h i (E(jh)) for each i, j.So, if h i (E(jh)) = 0 is then h i (F (jh)) = 0. Hence, the Hilbert polynomial of F has the same integral roots as E. So, the Hilbert polynomial of F has at least n integral roots.
If h i0 (E(j 0 h)) = 0 then h i0 (F (j 0 h)) = 0.If h i0 (F (j 0 h)) were 0 then F would have one more integral root at j 0 which is a contradiction.Because, the degree of the Hilbert polynomial is n and it may have at most n integral root.Therefore, F is supernatural, ℓ-away ACM.
Theorem 3.10.There exist simple ℓ-away ACM bundles F on P 2 of any even rank r = 2m and for any ℓ > 1.Also, the point corresponding to F in the moduli space of simple sheaves on P 2 is smooth and lies in a component of dimension m 2 (ℓ 2 + 2ℓ − 4) + 1.
Proof.Let E be the bundle in the short exact sequence 3.2.Then by Theorem 2.15 (see also [13,Section 5.1]), E is mapped to the complex of quiver representation The right vertex disappears because Ext i O P 2 (1), E = H i E(−1) = 0 for all i by Corollary 3.8.Similarly, by applying Corollary 3.8, we have that the graded vector space at the left vertex is concentrated in degree 1.Also, the cohomology of sequence (3.2) tensored with Ω(−ℓ) returns h i (E ⊗ Ω(−ℓ)) = 0 for i = 0, 2 and , the middle vertex is also concentrated in degree 1.
Therefore, we have the quiver with two vertices and three arrows (all have the same source and the same target) Then, by Proposition 2.14, we have Now, first notice that E ⊕m is a supernatural ℓ-away bundle, since E is so.Secondly, E ⊕m corresponds to a quiver representation with dimension vector md via the map Φ in Theorem 2.15.Also, since the general element in Rep(md) is indecomposable, Φ −1 can not map a generic element in Rep(md) to a decomposable sheaf.Therefore, there exists an indecomposable ℓ-away ACM bundle F of rank 2m, since the condition of being supernatural ℓ-away ACM is open by Lemma 3.9.
Lastly, notice that ext 2 (F , F ) = 0 by Lemma 2.7 since F is simple.So, the point corresponding to F in the moduli space of simple sheaves on P 2 is smooth and its dimension can be computed by considering the Euler form as Corollary 3.11.There exist indecomposable weakly Ulrich bundle of any even rank on P 2 .
Proof.Let F be a 2-away ACM bundle of rank r = 2m constructed in Theorem 3.10.Then F (−1) is weakly Ulrich.

ℓ-away ACM Bundles on P 1 × P 1
In this section, we will give similar results for Fano surface P 1 × P 1 .Recall that h = h 1 + h 2 is the fundamental divisor.
In this part of the paper, we will give the complete classification of special initialized 1-away and 2-away ACM bundles of rank 2 on P 1 × P 1 .For this purpose, we need the following proposition.Proposition 4.2.Let E be a vector bundle on X = P 1 × P 1 .Then E is the cohomology in degree 0 of a complex C • with r th -module where Proof.It follows from [9, Section 2.7.3]. Theorem 4.3.Let E be an initialized, indecomposable rank 2 vector bundle on X = P 1 × P 1 .Then, E is special 1-away ACM if and only if E is one of the following: Proof.First, assume that E is special, initialized 1-away ACM bundle of rank 2 on X.Then we have s = 0 and i X = 2. So, by Theorem 2.
Table 3: The values of h q−c E(ah 1 + bh 2 ) for P 1 × P 1 Then, by applying Proposition 4.2 to E(−2h), we have Then c = 0 and h 2 (E(th)) = 0 for t ≥ −1.By following exactly the same steps as in the case k 0 = −2, we have the following Table 4 1 Table 4: The values of h q−c E(ah 1 + bh 2 ) for P 1 × P 1 Then, by applying Proposition 4.2 to E(−h), we have Conversely, by applying Künneth's theorem and considering its cohomology, one can check that kernels in items (i) and (ii) are special 1-away ACM bundles of rank 2 on P 1 × P 1 .Corollary 4.4.If E is a special initialized 1-away ACM bundle of rank 2 on P 1 ×P 1 , then it is weakly Ulrich.
Proof.It is a direct consequence of Theorem 4.3 in the light of Definition 3.4.Theorem 4.5.Let E be an indecomposable, initialized, special ℓ-away ACM bundle of rank 2 on X = P 1 × P 1 .Then the module H 1 * (E) is connected (hence s = l − 1).Proof.Assume on the contrary that s ≥ l.Thanks to the Serre duality we can assume that there is a natural number m ≥ s 2 such that h Combining the inequality 4.3 with the item (i) of Theorem 2.5 we have So, by Proposition 2.4, E fits into Therefore, E is a special, initialized ℓ-away ACM bundle of rank 2 on P 1 × P 1 .Now, to show µ-semistability of E, we apply [11,Corollary 4]: we need to show that h 0 E(ah 1 + bh 2 ) = 0 for all a + b < −ℓ − 1.When we tensor the short exact sequence 4.5 with O X (ah 1 + bh 2 ) and then consider the cohomology sequence, one has The right hand side is zero because of Künneth's theorem and inequality a + b < −ℓ − 1.Therefore, E is µ-semistable.
If E were decomposable, E = O X (D)⊕O X ((ℓ+1)h−D) where D = ah 1 +bh 2 for some integers a and b.Since E is initialized, we have both a − 1 < 0 and ℓ − a < 0, which is a contradiction.So E is indecomposable. Since ) is a simple vector bundle.Hence, E is simple.
Proof.We have shown in the proof of Theorem 4.8 that Finally, we can give a construction for ℓ-away ACM bundles of higher rank on P 1 × P 1 for any positive integer ℓ.Theorem 4.10.There exist simple ℓ-away ACM bundle F on P 1 × P 1 of any even rank r = 2m and for any positive number ℓ.Also, the point corresponding to F in the moduli space of simple sheaves on P 2 is smooth and lies in a component of dimension m 2 (2ℓ 2 + 4ℓ − 2) + 1.
Proof.Let E be the bundle in the short exact sequence 4.5.We will follow the same steps as in Theorem 3.10.Then by Theorem 2.15 (see also [13,Section 6]), E is mapped to the complex of quiver representation .

The right vertex vanishes because Ext
Then, by Proposition 2.14, we have Then, one can follow the same steps in Theorem 3.10 and see that the general element in Rep(md) corresponds to an indecomposable ℓ-away ACM bundle.
Lastly, notice that ext 2 (F , F ) = 0 by Lemma 2.7 since F is simple.So, the point corresponding to F in the moduli space of simple sheaves on P 1 × P 1 is smooth and its dimension can be computed by considering the Euler form as Corollary 4.11.There exist weakly Ulrich bundles of any even rank on P 1 × P 1 .
Proof.Let F is a 2-away ACM bundle of rank r = 2m constructed in Theorem 4.10.Then F (−1) is weakly Ulrich.

ℓ-away ACM Bundles on Blow up of P 2 up to three non collinear points
Let X be the blow up of P 2 up to three non collinear points with polarization h = −K X .In this section, we will study ℓ-away ACM line bundles on X.Then, we will use these line bundles to construct ℓ-away ACM bundles of rank 2.
Let us call m for the pull-back of a line in P 2 via blow up map, and e i for exceptional divisors on X.The canonical bundle K X is −3m + i e i = −h.For a line bundle L = O(am + b i e i ) on X, we have where Proposition 5.1.Let X be the blow up of P 2 up to three non collinear points p 1 , p 2 , p 3 and L = O(am + b i e i ) be a line bundle on X.Then, for a ≥ −2 we have where A is the dimension of the space of forms of degree a in three variables and with zeroes of order at least d i at the points p i .
Proof.By the Serre duality, we have If a ≥ −2 then the latter dimension is zero because (−a − 3)m − ((b i − 1)e i )m < 0 and O X (m) is base point free.The assertion for h 0 follows from the definition of X because if b i > 0, it is immediate to check that b i e i is a fixed component.At this point the computation of h 1 is trivial.
By choosing the proper coordinates, we can assume that p 1 = [1 : 0 : 0], p 2 = [0 : 1 : 0] and p 3 = [0 : 0 : 1].Therefore, to calculate A, we can focus on monomials of order a as they form a basis of the space of degree a forms.Also, a form f has order at least d i at p i if and only if every monomial that contributes to f has at least order d i .So we obtain: Proposition 5.2.Let X be the blow-up P 2 up to 1 ≤ n ≤ 3 non collinear points and L = O X (am + b i e i ) be a line bundle on X.If a ≥ −2, then we have A = A n where Proof.Let x, y, z be coordinates in P 2 .
Case n = 1: The monomial f does not have a zero of order d 1 in p 1 if and only if f = gx a−d1+1 .We can choose g arbitrary and we have deg(g)+2 2 = d1+1 2 many ways to do so.Since the number of all monomials is a+2 2 , we obtain our formula.Case n = 2: As we showed earlier, we have d1+1 2 monomials that vanish at p 1 with order smaller than d 1 .This is the same for p 2 .But, there can happen that some monomials lie in both of these bases.Such a case occurs for monomials of the form x a−d1+1 y a−d2+1 g.Since we have d1+d2−a 2 many of them, we obtain Case n = 3: An analogous reasoning as in the previous case gives the desired formula.Now we can characterize ℓ-away ACM line bundles on the blow up of P 2 at a single point.Moreover, h 1 (L(th)) > 0 if and only if t ∈ I where I is as follows: We precede the proof with a couple of technical lemmas.In the beginning, we will establish the conditions for the vanishing of intermediate cohomology of a fixed line bundle M.

2
. Therefore For = χ(M).Thus h 1 (M) = 0 identically.So, h 1 (M) = 0 if and only if d 1 ≤ a + 1 or equivalently b 1 ≥ −(a + 1).Joining this with the inequality from the previous case we obtain Case a ≤ −2: This part follows from the previous cases after applying the Serre duality.
Let us continue and find conditions on t to have h 1 (M(th)) = 0. Lemma 5.5.For a line bundle M = O X (am + b 1 e 1 ) the following holds As a result, H 1 * (M) is connected for any line bundle on X. Proof.Recall that h = 3m − e 1 , thus M(th) = O X ((a + 3t)m + (b 1 − t)e 1 ).Then, h 1 (M(th)) = 0 if and only if inequalities from Lemma 5.4 do not hold for M(th).

So we have two cases:
Case a + 3t > −2: First inequality from Lemma 5.4 must be invalidated, so Transforming this, we obtain Joining this with a + 3t > −2, we get Case a + 3t ≤ −2: Analogous reasoning leads to − a 3 and the analysis of this case is complete.
Putting these two cases together, we obtain We will check below that we can get rid of the fraction −2−a 3 in the middle and write Indeed, it's obvious that I is a subset of the prior union of open intervals.On the other hand, if, for an integer x, we have which gives us a contradiction.Analogously, for the other component of the union of intervals, we have a similar result.Having this we can finally summarize our calculations as From the previous lemma, we can see that M is ℓ-away if and only if the interval I contains ℓ many integers.We will deal below with the restrictions on a and b 1 such that |Z ∩ I| = ℓ.Lemma 5.6.Let M = O X (am + b 1 e 1 ) be a line bundle on X.Then M is ℓ-away ACM if and only if |a + 3b 1 | ∈ {2ℓ + 2, 2ℓ + 3}.Also, M is ACM if and only if |a + 3b 1 | < 4.
Proof.Again we have two cases depending on the form of I (I defined as in the previous lemma).

2
, b 1 − 1 : The second part of the statement is the direct result of the first part.In fact, ACM bundles on anticanonically polarized Fano surfaces have been studied in [15].
Finally, we can prove our theorem.Using Lemma 5.6 the only thing left is to deal with the restrictions on a and b 1 such that L is initialized.
Proof of Theorem 5.3.L is initialized if and only if and this is nonzero only when a ≥ 0 and We obtain two cases: Then |a + 3b 1 | = 4, and so L = O X (2m − 2e 1 ) is an initialized 1-away ACM line bundle.This is the only possibility for L to be initialized ℓ-away ACM with a + 3b 1 < 0. When a + 3b 1 > 0, we have a + 3b 1 = 2ℓ + 2 or a + 3b 1 = 2ℓ + 3.
Since 0 ≤ a ≤ 2, we have two solutions: In the second case 5.2, we have This gives us two solutions: . The minus sign is here because a ≥ 0. Therefore, we have a = ℓ + 1 and b 1 = −(ℓ + 1).
Similarly, for the second solution, we have Inserting the calculated values of a and b 1 into the formula for I we get the "moreover" part.
We can now use the Theorem 5.3 to construct ℓ-away ACM bundles of rank 2.
Theorem 5.7.Let X be the blow up of P 2 at a single point.Then, there exists an indecomposable, initialized ℓ-away ACM vector bundle of rank 2 on X.
Proof.We will prove this theorem separately for ℓ = 1 and ℓ ≥ 2.
For ℓ = 1, consider the following exact sequence: Applying Proposition 5.2 with a = −2 and b 1 = 3 (thus so, there exists a non split extension.From Theorem 5.3 we get that L is initialized and furthermore h 1 (L(th)) = 0 only for t = −1.On the other hand, M is initialized because it satisfies Inequality (5.1) and it is ACM by Corollary ??.Since L and M are initialized, E is also initialized.Because h 1 (L(−h)) = 0 and h 2 (M(−h)) = 0 thus h 1 (E(−h)) = 0 and so E is initialized 1-away ACM.
For ℓ ≥ 2 we take: Denote O X (ℓm − ℓe 1 ) by M and O X (2e 1 ) by L. We have: Applying Proposition 5.2 with a = ℓ and b 1 = −(ℓ + 2) (thus < 0 hence by Proposition 5.1 h 1 (O X (ℓm − (ℓ + 2)e 1 )) > 0 and so there exist a non split extension.From Theorem 5.3 we have that M is initialized and h Finally, let us prove the indecomposability of constructed bundles.For ℓ ≥ 2, assume on the contrary that E = O X (am + be 1 ) ⊕ O X (a ′ m + b ′ e 1 ).Since E is initialized, at least one of these bundles must be initialized too.We can assume without losing the generality that it is the first one.Therefore, from previous calculations, we obtain a ∈ {0, 1, 2} or b = −a or b = −a + 1. (5.5) Comparing Chern classes we must have We have five cases to check by 5.5.If we try to solve equation 5.6 for each one of these five cases, we will have a contradiction.By a similar analysis, we will have the same result for the case ℓ = 1.Therefore, the constructed bundles are indecomposable.
To show the existence of ℓ-away ACM bundles of rank 2 on the blow ups at two distinct and three non collinear points we use the following construction.
Before proving this theorem, we will show the conditions for the vanishing of the middle cohomology of an arbitrary line bundle.Lemma 5.9.Let X be the blow up of P 2 at three non collinear points.Therefore all binomial coefficients in the expansion of Z vanish and this implies that d i + d j < a + 2 for every pair i = j.In the other direction, it's clear that these three inequalities imply Z = 0.If we would show that A = Y + Z then it would be the end since then h 1 (L) = Z = 0. To get A = Y + Z it is enough to show that A > 0. Recall that A was defined as the dimension of the space of forms of degree a in three variables and with zeroes of order at least d i at the points p i .By the proper choice of coordinates, we can assume that p 1 = [1 : 0 : 0], p 2 = [0 : 1 : 0] and p 3 = [0 : 0 : 1].Without loss of generality we can also assume that d 1 ≤ d 2 , d 3 .Now it is straightforward that x d2−1 y d3−1 z a+2−d2−d3 vanish at p 1 , p 2 , p 3 with sufficient order hence A ≥ 1. Finally from this case, we obtain the following set of inequalities which completes the analysis of this case: −(a + 1) ≤ min{b i + b j }.
Notice that we considered all the possible cases up to the permutation of indices of b i .Therefore combining all possible cases we ultimately get

for all i by Corollary 4 . 9 . 2 with
Due to the same steps as in Theorem 4.8, the graded vector spaces at the left and the middle vertices are concentrated in degree 1.Therefore, we have the following quiver 1 0 the representation of dimension vector d = (d 0 , d 1 , d 2 ) where

Table 2 :
The values of h q P 2 , E(ph) for P 2 By [12, Proposition 1.6 (a)], d is an imaginary root.Then, by [12, Lemma 2.1 (e) and Lemma 2.7], d is a Schur root; that is, a general representation of a dimension vector d is indecomposable.By [16, Theorem 3.7], md is a Schur root for every m ≥ 1. Hence the general element in Rep(md) is indecomposable.