Lobachevskii Journal of Mathematics

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Vol. 19, 2005, 3–12 \begin{matrix}c \\ ◯ \end{matrix} E. Ballico

E. Ballico

QUIVERS, VECTOR BUNDLES AND COVERINGS OF SMOOTH CURVES

(submitted by B. N. Shapukov)

Abstract. Fix a finite quiver Q and consider quiver-bundles on smooth and connected projective curves. Let f : X → Y be a degree m morphism between such curves and Ẽ a quiver bundle on Y. We prove that Ẽ is semistable (resp. polystable) if and only if f^{∗}(Ẽ) is semistable. Then we construct many stable quiver-bundles on bielliptic curves.

1. Introduction

Here we consider a problem related to stable and semistable quiverbundles on a smooth and connected projective curve X, i.e. representations of a finite quiver into the the category of all coherent sheaves on X ([1], [6], [11], [12], [13], [14]). We assume the existence of a finite covering f : X → Y between smooth and connected projective curves and we want to use informations on quiver-bundles on Y to obtain informations on quiver-bundles on X (for the same quiver). For the case of plain vector bundles this approach was used several times (see e.g. [3] for hyperelliptic curves and [10] for bielliptic curves). In the case of plain vector bundles the starting point is the following result ([8], Lemmas 3.2.2

curve. (Italy).

2000 Mathematical Subject Classification. 14H60.

Key words and phrases. holomorphic triples on curves, decorated vector bundle, vector bundles on curves, stable vector bundles, quiver; bielliptic

The author was partially supported by MIUR and GNSAGA of INdAM

and 3.2.3), which was a key step in the usual proof of the (generalized) Grauert-Mülich restriction theorem ([8], [9]).

Proposition 1. Let X, Y integral projective varieties, O_{Y} (1) an ample line bundle on Y, E a vector bundle on Y and f : X → Y a finite covering. Set O_{X}(1) := f^{∗}(O_{Y} (1)).

(a) f^{∗}(E) is O_{X}(1)-semistable if and only if E is O_{Y} (1)-semistable.

(b) f^{∗}(E) is O_{X}(1)-polystable if and only if E is O_{Y} (1)-polystable.

Of course, only the “ if ” parts are not trivial. For our quiver-bundles we do not study or use the existence of moduli spaces (see e.g. [1], [5], [6], [11], [12], [13], [14]) and hence we may allow fairly general oriented finite quivers (e.g. with multiple paths). Here we give our set-up. Let Z be a smooth and connected projective curve and Q = (S, A, s, t) (or just Q = (S, A) for short) a finite quiver, i.e. two finite sets V, A, V ≠ ∅, equipped with two functions s : A → V (the source), t : A → V (the target). Contrary to the assumptions made in [1], [6], [12] and [13] we allow multiple arrows, i.e. we allow the existence of a, b ∈ A such that s(a) = s(b), t(a) = t(b) and a ≠ b: recall that our aims are more modest: we only consider curves and do not consider moduli spaces. A quiverbundle Ẽ of Z is given by a finite set {E_{v}}_{v∈V} of vector bundles on Z and a finite set f_{a} : E_{s(a)} → E_{t(a)} of homomorphisms. For every v ∈ V fix m_{v} ∈ ℝ, m_{v} > 0, and call m_{v} the weight of the node v. We will require that each E_{v} has non-zero rank, so that its slope μ_{E_{v}} := deg(E_{v})/rank(E_{v}) is well-defined and call μ(Ẽ) := ∑_{v∈V} m_{v}μ(E_{v}) the slope (or the total slope or the total weighted slope) of Ẽ. A subobject F̃ = {F_{v}, f′_{a}} of Ẽ will be said to be strict if all bundles F_{v}, v ∈ V, have non-zero rank and hence the total slope μ(F̃) is well-defined. By the very definition of slope stability (or stability in the sense of Mumford and Takemoto) to check if Ẽ is semistable or stable or polystable it is sufficient to check all its strict subobjects. This is one of the two reasons why we do not use the Hilbert polynomials of the bundles E_{v} to define total stability. The second reason is that degrees and slopes work very well when making a pull-back by a degree m coverings (they are just multiplied by m), while in general the genus is not multiplied by m. Hence with our definition of total slope the “ only if ” part in Theorem 1 below is obvious.

In section 2 we will prove the following result.

Theorem 1. Let f : X → Y a degree m covering between smooth and connected projective curves and Ẽ a quiver-bundle on Y.

(a) f^{∗}(Ẽ) is semistable if and only if Ẽ is semistable.

(b) f^{∗}(Ẽ) is polystable if and only if Ẽ is polystable.

On X we take m · m_{v}, v ∈ V, as weights to define the total weighted slope, but m_{v}, v ∈ V, defines the same notion of stability and semistability, because for any λ > 0 the weights (m_{v}, v ∈ V) and (λm_{v}, v ∈ V) give the same notion of stability. We stress that in the definition of polystability any vector bundle appearing in a direct factor must have positive rank, otherwise its slope is not defined.

If f : X → Y is a degree m covering and E a vector bundle on Y, then deg(f^{∗}(E)) ≡ 0 (mod m). This is a very strong restrictions for the semistable quiver-bundles on X obtained using Theorem 1. In section 3 we will show how to overcome this restriction (when m = 2 and p_{a}(Y) = 1, i.e. for bielliptic curves) for certain quivers and how to obtain stable (not just semistable or polystable) quiver-bundles on any bielliptic curve X with large genus (again, only for certain very specific quivers). In Theorem 2 we will consider the case of a multiple arrow, i.e. we fix an integer n ≥ 2 and take V := {0, 1}, A := {f_{1}, …, f_{n}}, s(f_{i}) = 0 and t(f_{i}) = 1 for all i. In Theorem 3 we will consider the case of a source, i.e. we fix an integer n ≥ 1 and take V := {0, 1, …, n}, A := {f_{1}, …, f_{n}} with s(f_{i}) = 0 and t(f_{i}) = i, for all i. Taking duals, from Theorem 3 one gets the case of a so-called sink. In Theorem 4 we will consider the case of an oriented chain, i.e. we take V := {1, …, n}, A := {f_{1}, …, f_{n−1}} with s(f_{i}) = i and t(f_{i}) = i + 1, 1 ≤ i ≤ n − 1. In Theorem 5 we will consider the case of a fork, i.e. we take V := {−1, 0, 1, …, n}, A := {f_{0}, f_{1}, …, f_{n}}, s(f_{0}) = −1, t(f_{0}) = 0, s(f_{i}) = 0 and t(f_{i}) = 1 for all 1 ≤ i ≤ n.

2. Proof of Theorem 1

Let Q = (V, S) be a finite quiver in the sense of section 1. We fix m_{v} ∈ ℝ, m_{v} > 0, for all v ∈ V. All quiver-bundles will be with respect to the quiver Q and semistability (resp. stability, resp. polystability) will be the total slope semistability (resp. stability, resp. polystability) with respect to the weights m_{v}, v ∈ V discussed in section 1. Let Z be any smooth and connected projective curve.

Remark 1. Let Ẽ = {E_{v}, f_{a}} and F̃ = {F_{v}, f′_{a}} be stable quiver-bundles with the same total slope and ũ : Ẽ → F̃ a non-zero morphism. Call u_{v} : E_{v} → F_{v}, v ∈ V, the associated morphism. By the definition of stability ũ is an isomorphism if u_{v} ≠ 0 for all v ∈ V. We do not know if this is true in general. It is true in certain cases, e.g. the case of triples ([7], Cor. 2.1).

Lemma 1. Let Ẽ be a quiver-bundle on Z. Then Ẽ has a Harder-Narasimhan filtration; we do not claim its uniqueness.

Proof. If Ẽ is semistable, then there is nothing to prove. Hence we may assume that Ẽ is not semistable. Since Q is finite and each bundle E_{v}, v ∈ V, associated to Ẽ has a well-defined and finite rank and degree, there is a quiver-subsheaf F̃ = {F_{v}, f_{a}}_{v∈V,a∈A} of Ẽ with μ(F̃) > μ(Ẽ) and with μ(F̃) maximal; we stress that we may get F̃ such that to no F_{v} has rank zero and hence each μ(F_{v}) is well-defined, because only strict subobjects are used to test the semistability of a quiver-bundle. Since Z is smooth, the kernel of any homomorphism m : G → G′ of vector bundles on Z is saturated in G, i.e. either m ≡ 0 or Coker(m) ≅ Im(m) is locally free. Hence, by the very definition of subobject and the maximality of μ(F̃) we get that each F_{v} is a saturated subbundle of E_{v}, v ∈ V. If F_{v} = E_{v} for at least one v, then we stop: this chain cannot be refined and we stop. If F_{v} ⊊ E_{v} for all v, then the family {f_{a}}_{a∈A} induces a quiver-structure on the set of vector bundles {E_{v}/F_{v}}, v ∈ V. Now we may use induction on the total rank ∑_{v∈V} rank(E_{v}). □

Similarly, we have the following result.

Lemma 2. Let Ẽ be a semistable quiver-bundle on Z. Then Ẽ has a Jordan-Hölder filtration.

Although we do not claim uniqueness in Lemma 1, we are able to prove the following key lemma.

Lemma 3. Let Ẽ = {E_{v}, f_{a}} be a quiver-bundle on Z and Ã := {A_{v}, f′_{a}}, B̃ := {B_{v}, f′′_{v}} strict subobjects of Ẽ with maximal total slope and maximal total rank. Then Ã = B̃

Proof. Since Ã and B̃ have maximal slope, they are semistable. It is easy to define the subobject Ã + B̃ of Ẽ, because f_{a}(A_{s(a)}) + f_{a}(B_{s(a)}) ⊆ A_{t(a)} + B_{t(a)} for every a ∈ A. Furthermore, Ã + B̃ is a quotient of Ã ⊕ B̃, which is semistable. Since Ã + B̃ is a strict subobject of Ẽ (with our definition of “ strict ” Ẽ is a strict subobject of itself!), we easily conclude, by the maximality of the total rank of both Ã and B̃ and the fact (proved in the proof of Lemma 1) that A_{v} and B_{v} are saturated in E_{v} for all v ∈ V. □

Proof of Theorem 1. Since μ(F̃) = m · μ(F̃) for every quiver bundle F̃ on Y, the “ only if ” parts of both (a) and (b) are obvious. Furthermore, to check the “ if ” parts, it is sufficient to prove the semistabily or the polystability of a pull-back of f^{∗}(Ẽ) by another finite covering. Hence we may reduce to the case in which f is a Galois covering ; here we use the assumption char(𝕂) = 0. Call G the Galois group of the covering f. Now assume Ẽ semistable and that f^{∗}(Ẽ) is not semistable. Let F̃ be a strict subobject of f^{∗}(Ẽ) with maximal slope and (among the strict subobjects with maximal slope) with maximal total rank. Hence h^{∗}(F̃), h ∈ G, has the same properties. Hence F̃ = h^{∗}(F̃) (Lemma 3). Hence G acts on each F_{v}. Fix any v ∈ V. Notice that G acts trivially on the fiber of f^{∗}(E_{v}) over any ramification point, P, of f. Since F_{v} is saturated, in f^{∗}(E_{v}), G acts trivially also the fiber F_{v}|{P } of F_{v} at P. By descent theory we get the existence of a subbundle M_{v} of E_{v} such that F_{v} = f^{∗}(M_{v}). Also the maps f′_{a} descend and hence we get a strict subobject M̃ of Ẽ such that μ(M̃) = μ(F̃)/m, contradicting the semistability of Ẽ. Now assume Ẽ polystable. We just proved that f^{∗}(Ẽ) is semistable. Let Ã be a strict subobject of f^{∗}(Ẽ) with maximal slope and minimal total rank. Hence Ã is stable. Hence μ(Ã) = μ(f^{∗}(Ẽ)). We just proved that ∑_{h∈G} h^{∗}(Ã) descends to a quiver-bundle on Y. By the polystability of Ẽ we get that C̃ := ∑_{h∈G} h^{∗}(Ã) is stable and that, calling C_{v}, v ∈ V, the vector bundles associated to C̃, either C_{v} is a proper saturated subbundle of f^{∗}(E_{v}) for all v, or C_{v} = f^{∗}(E_{v}) for all v, i.e. that either f^{∗}(Ã) = C̃ is polystable or we may define a quotient quiver-bundle f^{∗}(Ẽ)/C̃ with all associated bundles with non-zero rank. Hence we may define the slopes of all bundles associated to f^{∗}(Ẽ)/C̃ and get part (b) by induction on the total rank of Ẽ.

3. Examples

We recall the following well-known lemma.

Lemma 4. Let f : X → Y be a double covering between smooth and projective curves and E a stable vector bundle on Y. Assume that f is not étale, i.e. assume p_{a}(X) ≥ 2p_{a}(Y). Then f^{∗}(E) is stable.

Proof. Since char(𝕂) ≠ 2, we have f_{∗}(O_{X}) ≅ O_{Y} ⊕ R for some R ∈ Pic(Y) (use the trace map). By Riemann-Hurwitz and the assumption p_{a}(X) ≥ 2p_{a}(Y) we get deg(R) < 0. By Proposition 1 f^{∗}(E) is polystable and hence f^{∗}(E) is stable if and only if it is simple. By the projection formula we have h^{0}(X, End(f^{∗}(E))) = h^{0}(Y, f_{∗}(End(f^{∗}(E)))) = h^{0}(Y, End(E)) + h^{0}(Y, End(E) ⊗ R) = 1 + h^{0}(Y, End(E) ⊗ R). Since End(E) is semistable ([8], §3.2) and deg(R) < 0, we get h^{0}(Y, End(E) ⊗ R) = 0, concluding the proof. □

From now on, in this paper we fix the following notation. Let Y be an elliptic curve, X a smooth and connected projective curve and f : X → Y a double covering. We assume that f is not étale, i.e. we assume g := p_{a}(X) ≥ 2 (Riemann-Hurwitz). Let σ : X → X denote the order two automorphism associated to f.

Theorem 2. Fix an integer n ≥ 2 and take V := {0, 1}, A := {f_{1}, …, f_{n}}, s(f_{i}) = 0 and t(f_{i}) = 1 for all i; this quiver is called a multiple arrow. Fix vector bundles E_{i} on Y, i ∈ {0, 1}, and set r_{i} := rank(E_{i}) and d_{i} := deg(E_{i}). Assume g := p_{a}(X) ≥ 6, (d_{0} + 1)/r_{0} < (d_{1} − 1)/r_{1}, that each E_{i}, i = 0, 1, is polystable and that no two of the indecomposable factors of any E_{i} are isomorphic. Fix integers a_{i}, i = 0, 1, such that a_{0} ∈ {2d_{0} − 1, 2d_{0} − 2} and a_{1} ∈ {2d_{1} + 1, 2d_{1} + 2}. Then there exists a stable quiver-bundle F̃ = {F_{0}, F_{1}, ϕ_{1}, ϕ_{n}} such that rank(F_{i}) = r_{i} and deg(F_{i}) = a_{i} for all i.

Proof. By Lemma 4 all vector bundles f^{∗}(E_{i}) are polystable and E_{i} and f^{∗}(E_{i}) have the same number of indecomposable factors. Fix a general P ∈ Y. Hence ♯(f^{−1}(P)) = 2. Set f^{−1}(P) = {P′, P′′}.

(a) Here we assume a_{0} = 2d_{0} − 1 and a_{1} = 2d_{1} + 1. Let F_{0} (resp. F_{1}) be the general bundle obtained from f^{∗}(F_{0}) (resp. f^{∗}(F_{1})) making a general negative (resp. positive) elementary transformation supported by P′. Call A_{0} (resp. A_{1}) the general bundle obtained from E_{0} making a negative (resp. positive) elementary transformation supported by P. Since the set of all bundles on X (resp. Y) obtained from a fixed bundle making a positive elementary transformation supported by P′ (resp. P) is irreducible, to prove Theorem 2 using the bundles F_{0}, F_{1} just defined we may assume f^{∗}(A_{0}) ⫋ F_{0} ⫋ f^{∗}(E_{0}) and f^{∗}(E_{1}) ⫋ F_{1} ⫋ f^{∗}(A_{1}); essentially, f^{∗}(A_{i}), i = 0, 1, is obtained from f^{∗}(E_{i}) making two “ exchanged by the involution σ ” negative elementary transformations at P′ and P′′ := σ(P′), and we impose that F_{i} is obtained from one of them, say the one supported by P′. By [4], Cor. 2.4 and its dual, A_{0} and A_{1} are polystable and no two of the indecomposable factors of of one of them are isomorphic. Hence we may apply Lemma 4 to their indecomposable factors. Since A_{0} is a subsheaf of E_{0} and E_{1} is a subsheaf of A_{1}, the maps f_{i} : E_{0} → E_{i}, 1 ≤ i ≤ n, induce maps u_{i} : A_{0} → A_{1}. Set v_{i} := f^{∗}(u_{i}). Since F_{0} is a subsheaf of f^{∗}(E_{0}) and f^{∗}(E_{1}) of F_{1}, each map f^{∗}(f_{i}) induces a map ϕ_{i} : F_{0} → F_{i}. Set F̃ := {F_{0}, F_{1}, ϕ_{1}, …, ϕ_{n}} in order to obtain a contradiction we assume that F̃ is not stable with respect to the parameters 2m_{0} and 2m_{1}. Take a strict subobject G̃ = {G_{0}, G_{1}, τ_{1}, …, τ_{n}} of F̃ with maximal slope.

First Claim: We may find F_{0}, F_{1} as above with both F_{0} and F_{1} stable

Proof of the First Claim: We will only prove the stability of F_{0}, because the case of F_{1} is very similar. Assume that F_{0} is not stable and take a proper subsheaf A of F_{0} with maximal slope and (among these subsheaves) with minimal rank. Since A has maximal slope, it is semistable and saturated in F_{0}, i.e. F_{0}/A is locally free. Since A has minimal rank among the subsheaves of F_{0} with maximal slope, it is stable. We see F_{0} as a subsheaf of the σ-invariant vector bundle f^{∗}(E_{0}). Hence we may see A as a subsheaf of f^{∗}(E_{0}). With this identification the subsheaf σ^{∗}(A) of f^{∗}(E_{0}) is defined. Since F_{0} is obtained from f^{∗}(E_{0}) making a negative elementary transformation supported by a non-ramification point of f, for any ramification point O of f the fiber F_{0}|{O} ≅ 𝕂^{⊕r_{E_{0}}} of F_{0} at O maps isomorphically onto f^{∗}(E_{0})|{O}. Since A is saturated in F_{0}, the fiber A|{O} maps injectively into f^{∗}(E_{0})|{O}. Similarly, we see that the fibers of A ∩ σ^{∗}(A) and A + σ^{∗}(A) over O maps injectively into the vector space f^{∗}(E_{0})|{O}. Since f^{∗}(E_{0}) comes from Y, σ acts trivially on the vector space f^{∗}(E_{0})|{O} and hence on each of linear subspaces. Let B be any σ-invariant subsheaf B of f^{∗}(E_{0}) such that the natural map B|{O} → f^{∗}(E_{0})|{O} is injective for all ramification point O of f. Descent theory implies the existence of a subsheaf B′ of E_{0} such that B = f^{∗}(B′). In particular B has even degree. The semistability of E_{0} implies μ(B′) ≤ μ(E_{0}) with equality if and only if B′ is a direct factor of E_{0} and hence μ(B) ≤ μ(f^{∗}(E_{0})) with equality if and only if B is a direct factor of f^{∗}(E_{0}) (for the “ equality ” part of the latter statement we use Lemma 2). At this point we have all the ingredients to copy [4], pp. 543–544, using our assumption on p_{a}(X), i.e. that there are “ sufficiently many ” ramification points on f. However, we may take a shortcut. Set r := rank(E_{0}), δ := deg(A) and ρ := rank(A). By [4], Lemmas 3.2 and 3.1, the vector bundles F_{i}, i = 0, 1, are semistable. Hence μ(A) = μ(F_{0}) = (2d_{0} − 1)/r). First assume that A is not saturated in f^{∗}(E_{0}) and call Ā its saturation. Hence μ(Ā) = μ(A) + 1/ρ > μ(f^{∗}(E_{0})), contradicting the semistability of f^{∗}(E_{0}). Hence A is saturated in f^{∗}(E_{0}). Thus σ^{∗}(A) is saturated in f^{∗}(E_{0}). Notice that f^{∗}(G_{0}) = F_{0} ∩ σ^{∗}(F_{0}) (as subsheaves of f^{∗}(E_{0})). Since A⊕σ^{∗}(A) is polystable and D := A+σ^{∗}(A) is a quotient of it, μ(D) ≥ μ(A), with equality if and only if D is isomorphic to a direct factor of A ⊕ σ^{∗}(A); since A ⊕ σ^{∗}(A) has only two direct factors, we get μ(A) = μ(D) if and only if either A = σ^{∗}(A) or A ∩ σ^{∗}(A) = 0. Let K be the saturation of the sheaf D inside f^{∗}(E_{0}). Since D is σ-invariant, K is σ-invariant. Since both A + σ^{∗}(A) and K are σ-invariant, deg(K)−deg(A+σ^{∗}(A)) is an even integer. Since r > ρ, μ(A) = (2d_{0} − 1)/r = μ(f^{∗}(E_{0})) − 1/r, and f^{∗}(E_{0}) is polystable, we get K = D, i.e. D is saturated in f^{∗}(E_{0}). Since it is also σ-invariant, there is a subbundle D′ of E_{0} such that D = f^{∗}(D′). First assume D ≠ f^{∗}(E_{0}). Hence E_{0}/D′ is a non-zero vector bundle.

Second Claim: h^{0}(X, f^{∗}(E_{0}/D′)) = h^{0}(Y, E_{0}/D′).

Proof of the Second Claim: Since char(𝕂) ≠ 2, there is R ∈ Pic(Y) such that deg(R) = 1 − g and f_{∗}(O_{X}) = O_{Y} ⊕ R (Riemann-Hurwitz). Hence h^{0}(X, f^{∗}(E_{0}/D′)) = h^{0}(Y, E_{0}/D′) + h^{0}(Y, (E_{0}/D′) ⊗ R (projection formula). Hence by Atiyah’s classification of indecomposable vector bundles on any elliptic curve ([2], Part II), it is sufficient to show that every indecomposable factor of E_{0}/D′ has slope < 1 − g. Let t denote the maximal slope of an indecomposable factor of E_{0}/D′. Since g ≥ 4, it is sufficient to check the inequality t < 3. In the proof of the semistability of F_{0} given in [4] the statement corresponding to the Claim is [4], Prop. 3.5; in that set-up it was proved the inequality t < 2, but with the stronger assumption (with our notation) μ(A) > μ(F_{0}); here instead we only have μ(A) = μ(F_{0}). Following the proof of [4], Prop. 3.5, in our set-up we get t ≤ 2, concluding the proof of the Second Claim.

Consider the exact sequence

0 → A ∩ σ^{∗}(A) → A ⊕ σ^{∗}(A) → A + σ^{∗}(A) → 0  (1)

By construction the vector bundle f^{∗}(E_{0}/D′) splits into two direct factors, A/(A∩σ^{∗}(A)) and σ^{∗}(A)/(A∩σ^{∗}(A)). The projection of f^{∗}(E_{0}/D′) onto its factor A/(A∩σ^{∗}(A)) does not come from an element of H^{0}(Y, E_{0}/D′), contradicting the Claim.

By the Second Claim we have μ(G_{i}) ≤ μ(F_{i}) for i = 0, 1, with strict inequality for at least one index i. Since m_{0} > 0 and m_{1} > 0, we have μ(G̃) > μ(F̃), contradiction.

Now assume D = f^{∗}(E_{0}), i.e D′ = D. Hence 2ρ ≥ r. Since deg(A) = deg(σ^{∗}(A)) = ρ(2d_{0} −1)/r, while deg(A+σ(A)) = 2d_{0}/r, we obtain 2ρ ≠ r. Hence ρ > r/2, i.e. A ∩ σ^{∗}(A) ≠ 0. Notice that A ∩ σ^{∗}(A) ⊂ f^{∗}(A_{0}). Since f^{∗}(A_{0}) is polystable, we get deg(A ∩ σ^{∗}(A)) = rank(A ∩ σ^{∗}(A)) · μ(A ∩ σ^{∗}(A)) ≤ (2ρ − r)μ(f^{∗}(A_{0})) = (2ρ − r)(2d_{0} − 2)/r. By Lemma 4 every direct factor of f^{∗}(E_{0}) is σ-invariant. Apply the proof of the Claim directly to the vector bundles A + σ^{∗}(A) and A ∩ σ^{∗}(A) and use again the exact sequence (1).

(b) Here we assume a_{0} = 2d_{0} − 2 and a_{1} = 2d_{1} + 2. Fix two general P_{1}, P_{2} ∈ Y. Hence ♯(f^{−1}(P_{1})) = ♯(f^{−1}(P_{2})) = 2. Set {P_{i}′, P_{i}′′} := f^{−1}(P_{i}), i = 1, 2. Copy the proof of part (a) with the following modifications. Here A_{0} (resp. A_{1}) is obtained from E_{0} (resp. E_{1}) making two general negative (resp. positive) elementary transformations supported by P_{1} and P_{2}. F_{0} (resp. F_{1}) is obtained from f^{∗}(E_{0}) (resp. f^{∗}(E_{1})) making two general negative (resp. positive) elementary transformations supported by P′_{1} and P′_{2}. Here we use that g is large to obtain the Claim proved in part (a) under our numerical assumptions.

(c) The cases “ a_{0} = 2d_{0} − 1 and a_{1} = 2d_{1} + 2 ” and “ a_{0} = 2d_{0} − 2 and a_{1} = 2d_{1} + 1 ” are similar and may be done as in part (b). □

Theorem 3. Fix an integer n ≥ 1 and take V := {0, 1, …, n}, A := {f_{1}, …, f_{n}} with s(f_{i}) = 0 and t(f_{i}) = i, for all i; this quiver is called a source. Fix vector bundles E_{i} on Y, i ∈ {0, 1, …, n}, and set r_{i} := rank(E_{i}), and d_{i} := deg(E_{i}). Assume g ≥ 6, (d_{0} + 1)/r_{0} < (d_{i} − 1)/r_{i} for all i ∈ {1, … }, that each E_{i} is polystable and that no two of the indecomposable factors of any E_{i} are isomorphic. Fix integers a_{i}, 0 ≤ i ≤ n, such that a_{0} ∈ {2d_{0} − 1, } and a_{j} ∈ {2d_{j}, 2d_{j} + 1} for all 1 ≤ j ≤ n. Then there exists a stable quiver-bundle F̃ = {F_{v}, ϕ_{a}} such that rank(F_{i}) = r_{i} and deg(F_{i}) = a_{i} for all i ∈ {0, 1, …, n}. Furthermore, we may find such a quiver-bundle with all F_{v}, v ∈ V, stable.

Proof. By Lemma 4 all vector bundles f^{∗}(E_{i}) are polystable and E_{i} and f^{∗}(E_{i}) have the same number of indecomposable factors. It is sufficient to copy the proof of Theorem 2 with the following modification. If a_{j} = 2d_{j} for some j, then from E_{j} and f^{∗}(E_{j}) we make first a sufficiently general negative elementary transformation and then we apply to this bundle a new sufficiently general positive elementary transformation (based on a different point of the curve). With our assumptions on g the Second Claim made in the proof of Theorems 2 is OK in our different set-up. □

In the same way we get the following two examples.

Theorem 4. Take V := {1, …, n}, A := {f_{1}, …, f_{n−1}} with s(f_{i}) = i and t(f_{i}) = i + 1, 1 ≤ i ≤ n − 1; this quiver is called an oriented chain. Fix vector bundles E_{i} on Y, i ∈ {1, …, n}, and set r_{i} := rank(E_{i}), and d_{i} := deg(E_{i}). Assume p_{a}(Y) = 1, g := p_{a}(X) ≥ 6, (d_{i} + 1)/r_{i} < (d_{i+1} − 1)/r_{i+1} for all i ∈ {1, …, n − 1}, that each E_{i} is polystable, and that no two of the indecomposable factors of any E_{i} are isomorphic. Fix a_{j} ∈ {2d_{j} − 1, 2d_{j}} for all j ∈ {1, …, n − 1}. Then there exists a stable quiver-bundle F̃ = {F_{v}, ϕ_{a}} such that rank(F_{i}) = r_{i} and deg(F_{i}) = a_{i} for all i ∈ {1, …, n}. Furthermore, we may find such a quiver-bundle with all F_{v}, v ∈ V, stable.

Theorem 5. Take V := {−1, 0, 1, …, n}, A := {f_{0}, f_{1}, …, f_{n}}, s(f_{0}) = −1, t(f_{0}) = 0, s(f_{i}) = 0 and t(f_{i}) = 1 for all 1 ≤ i ≤ n; this quiver is often called a fork. Fix vector bundles E_{i} on Y, i ∈ {−1, 01, …, n}, 12 E. BALLICO

and set r_{i} := rankE_{i} and d_{i} := deg(E_{i}). Assume g ≥ 6, (d_{−1} + 1)/r_{−1} < (d_{0}−1)/r_{0} and (d_{0}+1)/r_{0} < (d_{i}−1)/r_{i} for all i ∈ {1, …, n}. Fix integers a_{i} ∈ {2d_{i} −1, 2d_{i}}, i ∈ {−1, 0, 1, …, n}.Then there exists a stable quiverbundle F̃ = {F_{v}, ϕ_{a}} such that rank(F_{i}) = r_{i} and deg(F_{i}) = a_{i} for all i ∈ {1, …, n}. Furthermore, we may find such a quiver-bundle with all F_{v}, v ∈ V, stable.

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Department of Mathematics, University of Trento, 38050 Povo (TN), Italy E-mail address: ballico@science.unitn.it

Received August 9,2005