Abstract. Fix a ﬁnite quiver

Q

and consider quiver-bundles on smooth and connected projective curves. Let

f : X \to Y

be a degree

m

morphism between such curves and

\tilde{E}

a quiver bundle on

Y

. We prove that

\tilde{E}

is semistable (resp. polystable) if and only if

f^{*} (\tilde{E})

is semistable. Then we construct many stable quiver-bundles on bielliptic curves.

1. Introduction

Here we consider a problem related to stable and semistable quiver-bundles 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 \to 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 and 3.2.3), which was a key step in the usual proof of the (generalized) Grauert-M

\ddot{u}

lich restriction theorem ([8], [9]).

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 \neq \emptyset

, equipped with two functions

s : A \to V

(the source),

t : A \to 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 \in A

such that

s (a) = s (b)

t (a) = t (b)

and

a \neq b

: recall that our aims are more modest: we only consider curves and do not consider moduli spaces. A quiver-bundle

\tilde{E}

Z

is given by a ﬁnite set

{E_{v}}_{v \in V}

of vector bundles on

Z

and a ﬁnite set

f_{a} : E_{s (a)} \to E_{t (a)}

of homomorphisms. For every

v \in V

ﬁx

m_{v} \in 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 (\tilde{E}) : = \sum_{v \in V} m_{v} μ (E_{v})

the slope (or the total slope or the total weighted slope) of

\tilde{E}

. A subobject

\tilde{F} = {F_{v}, {f^{'}}_{a}}

\tilde{E}

will be said to be strict if all bundles

F_{v}

v \in V

, have non-zero rank and hence the total slope

μ (\tilde{F})

is well-deﬁned. By the very deﬁnition of slope stability (or stability in the sense of Mumford and Takemoto) to check if

\tilde{E}

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.

X

we take

m \cdot m_{v}

v \in V

, as weights to deﬁne the total weighted slope, but

m_{v}

v \in V

, deﬁnes the same notion of stability and semistability, because for any

λ > 0

the weights

(m_{v}, v \in V)

and

(λ m_{v}, v \in 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.

f : X \to Y

is a degree

m

covering and

E

a vector bundle on

Y

, then

deg (f^{*} (E)) \equiv 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 \geq 2

and take

V : = {0, 1}

A : = {f_{1}, \dots, 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 \geq 1

and take

V : = {0, 1, \dots, n}

A : = {f_{1}, \dots, 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, \dots, n}

A : = {f_{1}, \dots, f_{n - 1}}

with

s (f_{i}) = i

and

t (f_{i}) = i + 1

1 \leq i \leq n - 1

. In Theorem 5 we will consider the case of a fork, i.e. we take

V : = {- 1, 0, 1, \dots, n}

A : = {f_{0}, f_{1}, \dots, f_{n}}

s (f_{0}) = - 1

t (f_{0}) = 0

s (f_{i}) = 0

and

t (f_{i}) = 1

for all

1 \leq i \leq n

2. Proof of Theorem 1

Let

Q = (V, S)

be a ﬁnite quiver in the sense of section 1. We ﬁx

m_{v} \in ℝ

m_{v} > 0

, for all

v \in 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 \in V

discussed in section 1. Let

Z

be any smooth and connected projective curve.

Proof. If $\tilde{E}$ is semistable, then there is nothing to prove. Hence we may assume that $\tilde{E}$ is not semistable. Since $Q$ is ﬁnite and each bundle $E_{v}$ , $v \in V$ , associated to $\tilde{E}$ has a well-deﬁned and ﬁnite rank and degree, there is a quiver-subsheaf $\tilde{F} = {F_{v}, f_{a}}_{v \in V, a \in A}$ of $\tilde{E}$ with $μ (\tilde{F}) > μ (\tilde{E})$ and with $μ (\tilde{F})$ maximal; we stress that we may get $\tilde{F}$ 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 $m : G \to G^{'}$ of vector bundles on $Z$ is saturated in $G$ , i.e. either $m \equiv 0$ or $Coker (m) ≅ Im (m)$ is locally free. Hence, by the very deﬁnition of subobject and the maximality of $μ (\tilde{F})$ we get that each $F_{v}$ is a saturated subbundle of $E_{v}$ , $v \in V$ . 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 ${f_{a}}_{a \in A}$ induces a quiver-structure on the set of vector bundles ${E_{v} ∕ F_{v}}$ , $v \in V$ . Now we may use induction on the total rank $\sum_{v \in V} rank (E_{v})$ . □

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

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

Proof of Theorem 1. Since

μ (\tilde{F}) = m \cdot μ (\tilde{F})

for every quiver bundle

\tilde{F}

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^{*} (\tilde{E})

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

\tilde{E}

semistable and that

f^{*} (\tilde{E})

is not semistable. Let

\tilde{F}

be a strict subobject of

f^{*} (\tilde{E})

with maximal slope and (among the strict subobjects with maximal slope) with maximal total rank. Hence

h^{*} (\tilde{F})

h \in G

, has the same properties. Hence

\tilde{F} = h^{*} (\tilde{F})

(Lemma 3). Hence

G

acts on each

F_{v}

. Fix any

v \in 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}

F_{v}

P

. By descent theory we get the existence of a subbundle

M_{v}

E_{v}

such that

F_{v} = f^{*} (M_{v})

. Also the maps

f_{a}^{'}

descend and hence we get a strict subobject

\tilde{M}

\tilde{E}

such that

μ (\tilde{M}) = μ (\tilde{F}) ∕ m

, contradicting the semistability of

\tilde{E}

. Now assume

\tilde{E}

polystable. We just proved that

f^{*} (\tilde{E})

is semistable. Let

Ã

be a strict subobject of

f^{*} (\tilde{E})

with maximal slope and minimal total rank. Hence

Ã

is stable. Hence

μ (Ã) = μ (f^{*} (\tilde{E}))

. We just proved that

\sum_{h \in G} h^{*} (Ã)

descends to a quiver-bundle on

Y

. By the polystability of

\tilde{E}

we get that

\tilde{C} : = \sum_{h \in G} h^{*} (Ã)

is stable and that, calling

C_{v}

v \in V

, the vector bundles associated to

\tilde{C}

, 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) = \tilde{C}

is polystable or we may deﬁne a quotient quiver-bundle

f^{*} (\tilde{E}) ∕ \tilde{C}

with all associated bundles with non-zero rank. Hence we may deﬁne the slopes of all bundles associated to

f^{*} (\tilde{E}) ∕ \tilde{C}

and get part (b) by induction on the total rank of

\tilde{E}

3. Examples

Proof. Since $char (𝕂) \neq 2$ , we have $f_{*} (𝒪_{X}) ≅ 𝒪_{Y} \oplus R$ for some $R \in Pic (Y)$ (use the trace map). By Riemann-Hurwitz and the assumption $p_{a} (X) \geq 2 p_{a} (Y)$ 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 $h^{0} (X, E n d (f^{*} (E))) = h^{0} (Y, f_{*} (E n d (f^{*} (E)))) = h^{0} (Y, E n d (E)) + h^{0} (Y, E n d (E) \otimes R) = 1 + h^{0} (Y, E n d (E) \otimes R)$ . Since $E n d (E)$ is semistable ([8], $§$ 3.2) and $deg (R) < 0$ , we get $h^{0} (Y, E n d (E) \otimes R) = 0$ , 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 \to Y

a double covering. We assume that

f

is not étale, i.e. we assume

g : = p_{a} (X) \geq 2

(Riemann-Hurwitz). Let

σ : X \to X

denote the order two automorphism associated to

f

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 $P \in Y$ . Hence $♯ (f^{- 1} (P)) = 2$ . Set $f^{- 1} (P) = {P^{'}, P^{''}}$ .

(a) Here we assume $a_{0} = 2 d_{0} - 1$ and $a_{1} = 2 d_{1} + 1$ . 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 $f_{i} : E_{0} \to E_{i}$ , $1 \leq i \leq n$ , induce maps $u_{i} : A_{0} \to A_{1}$ . 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 $φ_{i} : F_{0} \to F_{i}$ . Set $\tilde{F} : = {F_{0}, F_{1}, φ_{1}, \dots, φ_{n}}$ in order to obtain a contradiction we assume that $\tilde{F}$ is not stable with respect to the parameters $2 m_{0}$ and $2 m_{1}$ . Take a strict subobject $\tilde{G} = {G_{0}, G_{1}, τ_{1}, \dots, τ_{n}}$ of $\tilde{F}$ 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 $F_{0} ∣ {O} ≅ 𝕂^{\oplus r_{E_{0}}}$ 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 $A \cap σ^{*} (A)$ 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 $B ∣ {O} \to f^{*} (E_{0}) ∣ {O}$ 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 $μ (B^{'}) \leq μ (E_{0})$ with equality if and only if $B^{'}$ is a direct factor of $E_{0}$ and hence $μ (B) \leq μ (f^{*} (E_{0}))$ 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 short-cut. 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 $μ (A) = μ (F_{0}) = (2 d_{0} - 1) ∕ r$ . First assume that $A$ is not saturated in $f^{*} (E_{0})$ and call $\bar{A}$ its saturation. Hence $μ (\bar{A}) = μ (A) + 1 ∕ ρ > μ (f^{*} (E_{0}))$ , 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 $f^{*} (G_{0}) = F_{0} \cap σ^{*} (F_{0})$ (as subsheaves of $f^{*} (E_{0})$ ). Since $A \oplus σ^{*} (A)$ is polystable and $D : = A + σ^{*} (A)$ is a quotient of it, $μ (D) \geq μ (A)$ , with equality if and only if $D$ is isomorphic to a direct factor of $A \oplus σ^{*} (A)$ ; since $A \oplus σ^{*} (A)$ has only two direct factors, we get $μ (A) = μ (D)$ if and only if either $A = σ^{*} (A)$ or $A \cap σ^{*} (A) = 0$ . 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 $r > ρ$ , $μ (A) = (2 d_{0} - 1) ∕ r = μ (f^{*} (E_{0})) - 1 ∕ r$ , 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 $D \neq f^{*} (E_{0})$ . 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 $char (𝕂) \neq 2$ , there is $R \in Pic (Y)$ such that $deg (R) = 1 - g$ and $f_{*} (𝒪_{X}) = 𝒪_{Y} \oplus R$ (Riemann-Hurwitz). Hence $h^{0} (X, f^{*} (E_{0} ∕ D^{'})) = h^{0} (Y, E_{0} ∕ D^{'}) + h^{0} (Y, E_{0} ∕ D^{'}) \otimes R$ (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 $g \geq 4$ , 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 $t \leq 2$ , concluding the proof of the Second Claim.

Consider the exact sequence

0 \to A \cap σ^{*} (A) \to A \oplus σ^{*} (A) \to A + σ^{*} (A) \to 0

(1)

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

By the Second Claim we have $μ (G_{i}) \leq μ (F_{i})$ for $i = 0, 1$ , with strict inequality for at least one index $i$ . Since $m_{0} > 0$ and $m_{1} > 0$ , we have $μ (\tilde{G}) > μ (\tilde{F})$ , contradiction.

Now assume $D = f^{*} (E_{0})$ , i.e $D^{'} = D$ . Hence $2 ρ \geq r$ . Since $deg (A) = deg (σ^{*} (A)) = ρ (2 d_{0} - 1) ∕ r$ , while $deg (A + σ (A)) = 2 d_{0} ∕ r$ , we obtain $2 ρ \neq r$ . Hence $ρ > r ∕ 2$ , i.e. $A \cap σ^{*} (A) \neq 0$ . Notice that $A \cap σ^{*} (A) \subset f^{*} (A_{0})$ . Since $f^{*} (A_{0})$ is polystable, we get $deg (A \cap σ^{*} (A)) = rank (A \cap σ^{*} (A)) \cdot μ (A \cap σ^{*} (A)) \leq (2 ρ - r) μ (f^{*} (A_{0})) = (2 ρ - r) (2 d_{0} - 2) ∕ r$ . 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 $A \cap σ^{*} (A)$ and use again the exact sequence (1).

(b) Here we assume $a_{0} = 2 d_{0} - 2$ and $a_{1} = 2 d_{1} + 2$ . Fix two general $P_{1}, P_{2} \in Y$ . 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 “ $a_{0} = 2 d_{0} - 1$ and $a_{1} = 2 d_{1} + 2$ ” and “ $a_{0} = 2 d_{0} - 2$ and $a_{1} = 2 d_{1} + 1$ ” are similar and may be done as in part (b). □

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 $a_{j} = 2 d_{j}$ 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. □

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