On vector bundles over reducible curves with a node

Notation 1.1

We denote by j_i : C_i ↪ C the natural inclusion of C_i in C, by O_qi the stalk of (j_i)∗O_Ci in p and by O_p the stalk of O_C in p.

Definition 2.1

A coherent sheaf E on C is of depth one if every torsion section vanishes identically on some components of C.

A coherent sheaf E on C is of depth one if and only if the stalk at the node p is isomorphic to_O⊕O₁ ⊕O_cq2 , see [23]. In particular, any vector bundle E on C is a sheaf of depth one. If E is a sheaf of depth one on C, then its restriction E|_Ci is a torsion free sheaf on C_i \ p (possibly identically zero). Moreover, any subsheaf of E is of depth one too.

Let E be a sheaf of depth one on C. We define the relative rank of E on the component C_i as the rank of the restriction E_i = E|_Ci of E to C_i

and the multirank of E as the pair (r₁, r₂).We define the relative degree of E with respect to the component C_i as the degree of the restriction E_i

where χ(E_i) is the Euler characteristic of E_i. The multidegree of E is the pair (d₁, d₂).

Definition 2.2

A polarization w of C is given by a pair of rational weights (w₁, w₂) such that 0 < w_i < 1 and w₁ + w₂ = 1. For any sheaf E of depth one on C, of multirank (r₁, r₂) and χ(E) = χ, we define the polarized slope as

Definition 2.3

Let E be a sheaf of depth one on C. E is called w-semistable if for any subsheaf F ⊆ E we have μ_w(F) ≤ μ_w(E); E is called w-stable if μ_w(F) < μ_w(E) for all proper subsheafs F of E.

For each w-semistable sheaf E of depth one on C there exists a finite filtration of sheaves of depth one on C:

such that each quotient Eⁱ/Eⁱ⁻¹ is a w-stable sheaf of depth one on C with polarized slope μ_w(Eⁱ/Eⁱ⁻¹) = μ_w(E). This is called a Jordan–Holder filtration of E. The sheaf

is called the graduate sheaf associated to E and it depends only on the isomorphism class of E. Let E and F be w-semistable sheaves of depth one on C.We say that E and F are 𝒮_w-equivalent if and only if Gr_w(E) ≃ Gr_w(F). If E and F are w-stable sheaves then 𝒮_w-equivalence is just isomorphism, as in the smooth case.

There exists a moduli space 𝒰^s_C(w, (r₁, r₂), χ) parametrizing isomorphism classes of w-stable sheaves of depth one on C of multirank (r₁, r₂) and given Euler characteristic χ, see [23]. It has a natural compactification 𝒰_C(w, (r₁, r₂), χ), whose points correspond to 𝒮_w-equivalence classes of w-semistable sheaves of depth one on C of multirank (r₁, r₂) and given Euler characteristic χ. In particular, when r₁ = r₂ = r, we denote by 𝒰_C(w, r, χ) the corresponding moduli space. In this case we have the following result (see [24] and [25]):

Theorem 2.1

Let C be a nodal curve with a single node p and two smooth irreducible components C_i of genus g_i ≥ 1, i = 1, 2. For a generic polarization w we have the following properties:

1. any w-stable vector bundle E ∈ 𝒰_C(w, r, χ) satisfies the following condition:

   (2.4) wiχ(E)≤χ(Ei)≤wiχ(E)+r,

   where E_i is the restriction of E to C_i;

2. if a vector bundle E on C satisfies the above condition for i = 1, 2 and the restrictions E₁ and E₂ are semistable vector bundles, then E is w-semistable. Moreover, if at least one of the restrictions is stable, then E is w-stable;

3. the moduli space 𝒰_C(w, r, χ) is connected, each irreducible component has dimension r²(p_a(C) − 1) + 1 and it corresponds to the choice of a multidegree (d₁, d₂) satisfying Conditions 2.4.

Definition 2.4

We denote by 𝒰_C(w, r, χ)_d1,d2 the irreducible component of 𝒰_C(w, r, χ) corresponding to the multidegree (d₁, d₂).

Lemma 3.1

Let C₁ and C₂ be smooth complex projective curves of genus g_i ≥ 1, i = 1, 2, and q_i ∈ C_i. Fix r ≥ 2 and d₁, d₂ ∈ ℤ such that r is coprime with both d₁ and d₂. Then there exists a projective bundle

such that the fiber over ([E₁], [E₂]) is ℙ(Hom(E_1,q1 , E_2,q2 )), where E_i,qi is the fiber of E_i at the point q_i.

Proof. As r and d_i are coprime, there exists a Poincaré bundle P_i for the moduli space of semistable vector bundles on C_i of rank r and degree d_i, i.e. a vector bundle P_i on 𝒰_Ci (r, d_i)× C_i such that P_i|_{[Ei ]×Ci} ≃ E_i, under the identification [E_i]× C_i ≃ C_i. This follows from a result of [20] if g_i ≥ 2 and from the isomorphism 𝒰_Ci (r, d_i)≃ C_i when g_i = 1. For i = 1, 2, consider the natural inclusion

and the pull back ι_i^∗(P_i) of the Poincaré bundle. Since 𝒰_Ci (r, d_i)× q_i is isomorphic to 𝒰_Ci (r, d_i), ι_i^∗(P_i) can be seen as a vector bundle on 𝒰_Ci (r, d_i) of rank r whose fiber at [E_i] is actually E_i,qi.

Note that the product 𝒰_C1 (r, d₁)× 𝒰_C2 (r, d₂) is a smooth irreducible variety. Let p₁ and p₂ denote the projections of the product onto factors. We define on 𝒰_C1 (r, d₁)× 𝒰_C2 (r, d₂) the following sheaf:

By construction, F is a vector bundle of rank r² whose fiber at the point ([E₁], [E₂]) is Hom(E_1,q1 , E_2,q2 ). By taking the associated projective bundle we conclude the proof.

Let C₁ and C₂ be smooth irreducible curves. We consider a nodal curve C with two smooth components and a single node p which is obtained by identifying the points q₁ ∈ C₁ and q₂ ∈ C₂. Let E_i be a stable vector bundle of rank r and degree d_i on C_i and consider a non-zero homomorphism σ : E_1,q1 → E_2,q2 between the fibres. Assume that the rank of σ is k, with 1 ≤ k ≤ r. We can associate to these data a depth one sheaf on the nodal curve C, roughly speaking, by gluing the vector bundles E₁ and E₂ along the fibers (at q₁ and q₂ respectively) with the homomorphism σ, as follows:

Let j_p be the inclusion of p in C and let j_i : C_i → C be the inclusion of C_i in C for i = 1, 2. The sheaf j_i∗E_i is a depth one sheaf on C whose stalk at p is the stalk of E_i at q_i. Hence, there is a natural surjective map given by restriction onto the fiber of E_i at q_i, i.e. the map

The sheaf j_1∗(E₁)⊕ j_2∗(E₂) is of depth one on C and we have a surjective map

The sheaf j_p∗j_p^∗j_2∗(E₂) has depth one too, and it is a skyscraper sheaf over p whose stalk is E_2,q2 . So we have again a surjective map

Let σ : E_1,q1 → E_2,q2 be a non-zero homomorphism and consider the induced surjective map

We have, moreover, the map

which sends (u, ν) to u − ν. We denote by Δ ⊂ Im(σ) ⊕ Im(σ) the diagonal. By construction we have Δ≃ℂpk.

Finally we define the map of sheaves

by requiring that the following diagram commutes.

It follows immediately by construction that ker σ̃ is a sheaf of depth one on C, which coincides with E_i on C_i \ p. One can easily see that the isomorphism class of ker σ̃ does not depend on the isomorphism classes of the E_i. Moreover, the same happens if one uses σ^' = λσ with λ ∈ ℂ^∗, instead of σ.

From now on, we assume that the hypothesis of Lemma 3.1 holds. Let ℙ(F) be the projective bundle on 𝒰_C1 (r, d₁)× 𝒰_C2 (r, d₂). We can conclude that the construction of ker σ̃ depends on the data contained in u = (([E₁], [E₂]), [σ]) ∈ ℙ(F) and not on the particular choices of E₁, E₂ and σ.

Definition 3.1

We denote by E_u the kernel of σ̃ defined by u ∈ ℙ(F).

The above construction gives the following:

Proposition 3.2

Let E_u be the sheaf defined by u = (([E₁], [E₂]), [σ]) ∈ ℙ(F). Then E_u is a depth one sheaf on C with χ(E_u) = χ(E₁) + χ(E₂)− r and multirank (r, r). It is a vector bundle if and only if σ is an isomorphism. In this case, E_u|Ci = E_i.

Proof. Let Rk(σ) = k. Since E_u is a depth one sheaf, the stalk of E_u at the node p is isomorphic to Opa⊕Oq1b⊕Oq1c where a + b = Rk(E_u|C₁) = r and a + c = Rk(E_u|C₂) = r (see Section 2). From the diagram 3.2, it follows the rank of the free part of the stalk of E_u in p is k, so a = k. Hence we have Eu|p≃Opk⊕Oq1r−k⊕Oq2r−k. In particular, E_u is a vector bundle if and only if k = r, i.e. exactly when σ is an isomorphism.

In order to obtain a w-semistable sheaf, for some polarization w, the following condition is necessary:

Lemma 3.3

Let E = E_u be the sheaf defined by u = (([E₁], [E₂]), [σ]) ∈ ℙ(F) and let k be the rank of σ. If E is w-semistable for some w, then the following conditions are satisfied:

Proof. Assume that E is w-semistable for a polarization w. Let K₁ be the kernel of the map

and let K₂ be the kernel of the map ρ₂ : j_2∗E₂ → E_2,q2 as in diagram 3.2. Since K_i is a subsheaf of E, by w-semistability of E we have μ_w(K_i)≤ μ_w(E). We also have μw(K1)=χ(K1)w1r=χ(E1)−kw1r≤χ(E)r, which implies

By replacing χ(E₁) = χ(E) − χ(E₂) + r in the above inequality, we obtain

Finally, we have μw(K2)=χ(K2)w2r=χ(E2)−rw2r≤χ(E)r, which implies

Again, by replacing χ(E₂) = χ(E) − χ(E₁) + r we obtain χ(E₁)≥ χ(E)w₁.

Given u = (([E₁], [E₂]), [σ]) and E_u defined by u, we wonder if there exists a polarization w such that the above Conditions 3.3 hold. The answer depends only on numerical assumptions on (χ(E₁), χ(E₂)) and Rk σ, as the following lemma shows.

Lemma 3.4

Let r ≥ 2 and 1 ≤ k ≤ r be integers. There exists a non-empty subsetW_r,k ⊂ ℤ² such that for any pair (χ₁, χ₂)∈ W_r,k we can find a polarization w satisfying the conditions

Proof. Note that if χ = 0, i.e. χ₁ + χ₂ = r and we assume that 0 ≤ χ₁ ≤ r, then any polarization w satisfies Conditions 3.4. We distinguish two cases according to the sign of χ. Assume that χ > 0. Then there exists a polarization w satisfying Conditions 3.4, if and only if the following system has solutions:

This occurs if and only if χ₁ > 0 and χ₂ > r − k. Likewise, if χ < 0, then we have the system

which has solutions if and only if χ₁ < k and χ₂ < r.

Remark 3.1

Let Wr=∩k=1rWr,k. Note that it is a non-empty subset and it is actually W_r,1. Moreover, if (χ₁, χ₂) ∈ W_r, then by the proof of Lemma 3.4 it follows that we can find a polarization w which satisfies the Conditions 3.4 for all k = 1, . . . , r.

Assume that Rk σ = r, i.e. E is a vector bundle. Then the necessary conditions of Lemma 3.3 are the same in Theorem 2.1. Hence, by the above theorem, they are also sufficient to give w-semistability of E. So we obtain the following:

Corollary 3.5

Let E = E_u be the sheaf defined by u = (([E₁], [E₂]), [σ]) ∈ ℙ(F). Assume that Rk σ = r and (χ(E₁), χ(E₂))∈ W_r,r. Then there exists a polarization w such that E is w-semistable. In particular, since the E_i are stable, then E is w-stable too.

Unfortunately,when E_u fails to be a vector bundle, the necessary conditions of Lemma 3.3 are not enough to ensure w-semistability, see [25] for an example. Nevertheless, we are able to produce an open subset of 𝒰_C1 (r, d₁)× 𝒰_C1 (r, d₁) such that for every u over this open subset, the sheaf E_u is w-semistable.

We recall the following definition, see [16].

Definition 3.2

Let G be a vector bundle on a smooth curve. For every integer k we set

A vector bundle G is called (m, k)-semistable (respectively stable) if for any subsheaf F we have

Proposition 3.6

Let E = E_u be the sheaf defined by u = (([E₁], [E₂]), [σ]) ∈ ℙ(F). Assume that Rk σ = k ≤ r−1. If (χ(E₁), χ(E₂)) ∈ W_r,k, E₁ is (0, k)-semistable and E₂ is (0, r)-semistable, then there exists a polarization w such that E is w-semistable. Moreover, if E₁ is (0, k)-stable or E₂ is (0, r)- stable, then E is w-stable too.

Proof. Since (χ(E₁), χ(E₂))∈ W_r,k, by Lemma 3.4 there exists a polarization w such that the necessary Conditions 3.3 hold. We claim that if E₁ is (0, k)-semistable and E₂ is (0, r)-semistable, then E is w-semistable.

Let F ⊂ E be a subsheaf; it is a sheaf of depth one too. Assume that F has multirank (s₁, s₂) and that at the node p the stalk of F OpS⊕Oq1a⊕Oq2b with s ≥ 0, s₁ = s + a ≤ r and s₂ = s + b ≤ r. Since Rk σ = k, by construction the free part of the stalk of E at p is Opk. This implies that 0 ≤ s ≤ k.

By construction, there exist two vector bundles F₁ ⊆ E₁ and F₂ ⊆ E₂ such that F is the kernel of the restriction of σ̃ to the subsheaf j_1∗(F₁)⊕ j_2∗(F₂):

Proceding as in the diagram 3.2, we deduce that F fits into an exact sequence as follows:

where G₁ is the kernel of (σ ∘ ρ₁)|_F1 and G₂ is the kernel of ρ₂|_F2 . Hence G_i ⊆ K_i. Note that if s = 0, then actually F ≃ G₁ ⊕ G₂.

For any s, we compute the w-slope of F:

Since E₁ is (0, k)-semistable, we have

Since E₂ is (0, r)-semistable, E₂(−q₂) is (0, r)-semistable too, so we have

By replacing we obtain:

By Lemma 3.3 we have μ_w(K_i)≤ μ_w(E), so we obtain:

Since s − s₂ ≤ 0, we have μ_w(F) ≤ μ_w(E).

Finally, if E₁ is (0, k)-stable or E₂ is (0, r)-stable, then the above inequality is strict.

Note that, by definition, if E_i is (0, r)-stable, then it is also (0, k)-stable for all k ≤ r.

Lemma 3.7

Let 𝒰_Ci (r, d_i) be the moduli space of semistable vector bundles of rank r and degree d_i on a smooth curve C_i of genus g_i. If d_i and r are coprime and g_i > r + 1, then the locus of vector bundles of 𝒰_Ci (r, d_i) which are (0, r)-stable is a non-empty open subset of 𝒰_Ci (r, d_i).

Proof. We consider the locus

and the subset Y_a,s of Y given by all stable vector bundles E which can be written as 0 → F → E → Q → 0, where F is a subbundle of E with deg(F) = a and Rk(F) = s ≤ r − 1 and

A deformation argument (see the proof of Proposition 1.4 of [21]) shows that if Y_a,s ≠ 0, then for a general E in Y_a,s both F and Q are stable. Moreover, since E is stable, we have Hom(Q, F) = 0. Hence we can write

Hence

Since E ∈ Y, we have μ₀(F) ≥ μ_−r(E), i.e.

which implies d_is − ar ≤ rs. Finally, if g_i > 1 + r, then for all s ≤ r − 1 we have

which concludes the proof.

Definition 4.1

We denote by U the open subset given by the complement of B_r−1 in ℙ(F).

Remark 4.1

Note that dimU = dimℙ(F) = r²(g₁ + g₂ − 1) + 1. Denote by πU the restriction of π to U . By construction,

is a fiber bundle whose fibers are isomorphic to PGL(r). More precisely,

For χ = d₁ + d₂ + r(1 − g₁ − g₂), let 𝒰_C(w, r, χ)_d1,d2 be the irreducible component of the moduli space of depth one sheaves on C of rank r and characteristic χ corresponding to the multidegree (d₁, d₂); see Section 2. Let V_C(w, r, χ)_d1,d2 ⊂ 𝒰_C(w, r, χ)_d1,d2 be the subset parametrizing classes of vector bundles.

Theorem 4.1

Let C be a nodal curve with a single node p and two smooth irreducible components C_i of genus g_i ≥ 1. Fix r ≥ 2. For any d_i ∈ ℤ we set χ_i = d_i + r(1 − g_i) and χ = d₁ + d₂ + r(1 − g₁ − g₂). Assume that r is coprime with both d₁ and d₂ and that (χ₁, χ₂)∈ W_r,r. Then there exists a polarization w such that the map

sending u to [E_u] is birational. In particular, the restriction φ|U is a an injective morphism and the image φ(U ) is contained in V_C(w, r, χ)_d1,d2.

Proof. Let u = (([E₁], [E₂]), [σ]) ∈ ℙ(F) and consider the sheaf E = E_u defined by u, as in Section 3. Since (χ₁, χ₂)∈ W_r,r, as a consequence of Lemma 3.4 and Corollary 3.5 there exists a polarization w such that E_u is w-semistable for every u ∈ U . This gives a point in the moduli space 𝒰_C(w, r, χ)_d1,d2 and it shows that φ is well defined at least on U.

We prove that φ_|U is injective. Let u=(([ E1 ],[ E2 ]),[σ]) and u′=(([ E1′ ],[ E2′ ]),[ σ′ ]) in U with φ(u) = [E] and φ(u^') = [E^']. Assume that φ(u) = φ(u^'). Since E and E^' are both w-stable and are in the same 𝒮_w- equivalence class, they have to be isomorphic (see Section 2). Let τ : E → E^' be an isomorphism. This induces an isomorphism τ_i : E_i → E_i. So we can assume that Ei′=Ei; thus σ, σ^' : E_1,q1 → E_2,q2 and τ_i : E_i → E_i are isomorphisms. As E_p (respectively Ep′ ) is obtained by glueing E_1,q1 with E_2,q2 along the isomorphism σ (respectively along σ^'), the τ_i have to satisfy a compatibility condition which is summarized in the following commutative diagram:

Since E_i is stable we have Hom(E_i , E_i) ≃ ℂ ⋅ id_Ei . Hence (τ_i)_qi is the multiplication by some λ_i ∈ ℂ^∗. In particular, σ^' is a non-zero multiple of σ and thus [σ] = [σ^'].

Now we prove that φ_|U is a morphism. It is enough to prove that φ is regular at u₀, for any u₀ ∈ U . For this, we claim that there exists a non-empty open subset W ⊆ U with u₀ ∈ W and a vector bundle E on W × C such that

Step 1: There exist two sheaves Q and R on U × C such that for each u = (([E₁], [E₂], [σ]) ∈ U we have

where j_p : p ↪ C and j_i : C_i ↪ C are the natural inclusions.

Consider the diagram

where the morphisms which appear have been defined as

As before, we denote with P_i the Poincaré bundle on 𝒰_Ci (r, d_i)× C_i and we set

Q i = Π U ∗ P i ∗ J i ∗ P i , Q = Q 1 ⊕ Q 2  and  R = J p ∗ J p ∗ Q 2 .

Note that Supp(R) = U × p. Moreover, one can verify that if we identify U × p with U we have

where ι_i : 𝒰_Ci (r, d_i)× q_i ↪ 𝒰_Ci (r, d_i)× C_i.

Step 2: There is an open subset W ⊂ U containing u₀ and a surjective map of sheaves

Q1⊕Q2|W×C→ΣWR|W×C

whose kernel is the desired vector bundle E on W × C.

Let π : ℙ(F) → 𝒰_C1 (r, d₁)×𝒰_C2 (r, d₂) be the projective bundle defined in Lemma 3.1. Consider on ℙ(F) the tautological line bundle O_ℙ(F₎(−1) which is, by definition, the subsheaf of π^∗(F) whose fiber at u ∈ ℙ(F) is

where u = (([E₁], [E₂]), [σ]).We can choose W to be an open subset of U containing the point u₀ and admitting a section s ∈ O_ℙ(F₎(−1)(W) with s(u) ≠ 0 for any u ∈ W.

In particular, s induces a map of sheaves

such that s_u : E_1,q1 → E_2,q2 is an isomorphism and [s_u] = [σ] in ℙ(Hom(E_1,q1 , E_2,q2 )). We can also define a morphism of sheaves

where id₂ is the identity of πU*p2*(l2*(P2)) )|W.

This allows us to define the map Σ_W we are looking for. Indeed, since Supp(R|_W×C) = W × p, it is enough to give the map on W × p, which can be identified with W. Using the isomorphism 4.3, we have a diagram which defines Σ_W:

By taking the kernel E of this map we conclude the second step of the proof of the claim. In particular, φ_|U is a morphism.

By construction, φ(U ) is contained in V_C(w, r, χ)_d1,d2 and it coincide with the open subset of w-semistable vector bundles whose restrictions are semistable. Moreover, V_C(w, r, χ)_d1,d2 is a dense open subset of 𝒰_C(w, r, χ)_d1,d2 , see [23]. By Remark 4.1 we have

which is the dimension of 𝒰_C(w, r, χ)_d1,d2 , see Theorem 2.1. This implies that φ is a dominant map. Hence, by a generic smoothness argument, we can conclude that φ_|U is a birational morphism.

Corollary 4.2

Let C be a nodal curve with a single node p and two smooth irreducible components C_i of genus g_i ≥ 1. Assume that the moduli space 𝒰_C(w, r, χ) has an irreducible component corresponding to the bidegree (d₁, d₂) with d₁ and d₂ coprime with r. Then this component is birational to a projective bundle over the smooth variety 𝒰_C1 (r, d₁)× 𝒰_C2 (r, d₂).

Note that φ provides a desingularization of the component 𝒰_C(w, r, χ)_d1,d2 . If the genus of the curve C_i is big enough, we can be more precise about the domain of the rational map φ. If g_i > r + 1, then by Lemma 3.7 the locus of vector bundles of 𝒰_Ci (r, d_i) which are (0, r)-stable is a non-empty open subset of 𝒰_Ci (r, d_i); let us denote it by V_i.

Definition 4.2

We denote by V the open subset π⁻¹(V₁ × V₂) in ℙ(F).

By construction, V is a projective bundle over V₁ × V₂.

Theorem 4.3

Assume that the hypothesis of Theorem 4.1 holds. Moreover, let g_i > r + 1 and (χ₁, χ₂) ∈ W_r. Then there exists a polarization w such that the map φ sending u to [E_u] is a birational map such that φ|U _∪V is a morphism.

Proof. Since (χ₁, χ₂)∈ W_r, by Remark 3.1 there exists a polarization w such that the Conditions 3.4 hold for any k = 1, . . . , r. In particular, as W_r ⊂ W_r,r, Theorem 4.1 holds: φ is a birational map which is defined on the open subset U.

Assume that u ∈ V and u ∉ 𝒰. Then u = (([E₁], [E₂]), [σ]), with ([E₁], [E₂])∈ V₁ × V₂ and Rk σ ≤ r − 1.

Since [E_i]∈ V_i, Lemma 3.6 implies that E_u is w-semistable, hence φ is defined all over the open subset V too. To prove that φ|V is a morphism, we can proceed as in the proof of Theorem 4.1, just by replacing U with V and 𝒰_Ci (r, d_i) with V_i.

Definition 5.1

Let L be the line bundle on C that is induced by the pair (L₁, L₂).We define 𝒮𝒰_C(w, r, L) as the closure of

in 𝒰_C(w, r, χ)_d1,d2.

If we assume that r and d_i are coprime, then 𝒮𝒰_Ci (r, L_i) is a smooth irreducible projective variety of dimension (r₂ − 1)(g_i − 1). As in Lemma 3.1, we can define a vector bundle F_L on 𝒮𝒰_C1 (r, L₁)× 𝒮𝒰_C2 (r, L₂) just by restricting F. Then we can consider the associated projective bundle ℙ(F_L) and

a PGL(r)-bundle on 𝒮𝒰_C1 (r, L₁)× 𝒮𝒰_C2 (r, L₂). We denote by φ_L the restriction of the morphism φ defined in Theorem 4.1 to 𝒰_L. As a consequence of Theorem 4.1, we have the following:

Corollary 5.1

Under the hypothesis of Theorem 4.1, the map

is a birational map, whose restriction φ_L|𝒰_L is an injective morphism.

Proof. φ_L|𝒰_L is a morphism and its image is the set Im φ_L = {E ∈ V_C(w, r, χ)_d1,d2 | [E_|Ci ] ∈ 𝒮𝒰_Ci (r, L_i)}. In particular, Im φ_L ⊆ 𝒮𝒰_C(w, r, L). Consider the map

sending E to (det(E|_C1 ), det(E|_C2 )), which fits into the following commutative diagramm:

It follows immediately that ψ is a surjective morphism and that Im φ_L ⊂ ψ⁻¹(L₁, L₂).

We claim that ψ has irreducible fibers of dimension (r² − 1)(g₁ + g₂ − 1).

First we prove that any two fibers of ψ are isomorphic. If (L₁, L₂) and (L^'₁, L^'₂) are in Pic^d1 (C₁)×Pic^d2 (C₂), then there exist ξ_i ∈ Pic⁰(C_i) such that Li⊗ξir≃Li′. Let ξ be the unique line bundle on C such that ξ_|Ci ≃ ξ_i. The natural map

sending E to E ⊗ ξ preserves w-semistability and gives an isomorphism of the fibers. In particular, with the fiber dimension theorem (see [13], p.95) this implies that any fiber has pure dimension (r² − 1)(g₁ + g₂ − 1).

Finally we prove that any fiber is irreducible. Let Y = V_C(w, r, χ)_d1,d2 \ φ(U ); it is a proper subvariety of V_C(w, r, χ)_d1,d2 . Assume that the fiber of ψ over (L₁, L₂) is reducible, and let F₁ be the irreducible component containing φ(𝒰_L). Then there exists an irreducible component F₂ ⊂ Y. So the restriction of ψ to Y is a surjective morphism whose fibers have dimension (r²−1)(g₁+g₂−1). This implies that dim Y = dim V_C(w, r, χ)_d1,d2 , which is impossible.

This allows us to conclude that 𝒮𝒰_C(w, r, L) is irreducible too and φ_L is a birational morphism.

Theorem 5.2

Under the hypothesis of Theorem 4.1, 𝒮𝒰_C(w, r, L) is a rational variety.

Proof. By hypothesis d_i and r are coprime, hence the moduli space 𝒮𝒰_Ci (r, L_i) is rational for any line bundle L_i ∈ Pic^di (C_i), see [14], [17] and [19]. Since 𝒰_L is a ℙ^r2−1-bundle over the product 𝒮𝒰_C1 (r, L₁)× 𝒮𝒰_C2 (r, L₂), it is a rational variety too. The assertion follows from Corollary 5.1.