Moduli spaces of vector bundles over ruled surfaces

M. Aprodu and V. Brı̂nzǎnescu Nagoya Math. J. Vol. 154 (1999), 111–122 MODULI SPACES OF VECTOR BUNDLES OVER RULED SURFACES MARIAN APRODU and VASILE BRÎNZǍNESCU Abstract. We study moduli spaces M (c1 , c2 , d, r) of isomorphism classes of algebraic 2-vector bundles with fixed numerical invariants c1 , c2 , d, r over a ruled surface. These moduli spaces are independent of any ample line bundle on the surface. The main result gives necessary and sufficient conditions for the nonemptiness of the space M (c1 , c2 , d, r) and we apply this result to the moduli spaces ML (c1 , c2 ) of stable bundles, where L is an ample line bundle on the ruled surface. Introduction Let π : X → C be a ruled surface over a smooth algebraic curve C, defined over the complex number field C. Let f be a fibre of π. Let c1 ∈ Num(X) and c2 ∈ H 4 (X, Z) ∼ = Z be fixed. For any polarization L, denote the moduli space of rank-2 vector bundles stable with respect to L in the sense of Mumford-Takemoto by ML (c1 , c2 ). Stable 2-vector bundles over a ruled surface have been investigated by many authors; see, for example [T1], [T2], [H-S], [Q1]. Let us mention that Takemoto [T1] showed that there is no rank-2 vector bundle (having c1 .f even) stable with respect to every polarization L. In this paper we shall study algebraic 2-vector bundles over ruled surfaces, but we adopt another point of view: we shall study moduli spaces of (algebraic) 2-vector bundles over a ruled surface X, which are defined independent of any ample divisor (line bundle) on X, by taking into account the special geometry of a ruled surface (see [B], [B-St1], [B-St2] and also [Br1], [Br2], [W]). In Section 1 (put for the convenience of the reader) we present (see [B]) two numerical invariants d and r for a 2-vector bundle with fixed Chern classes c1 and c2 and we define the set M (c1 , c2 , d, r) of isomorphism classes of bundles with fixed invariants c1 , c2 , d, r. The integer d is given by the splitting of the bundle on the general fibre and the integer r is given by some normalization of the bundle. Recall that the set M (c1 , c2 , d, r) carries Received February 8, 1996. 111 Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 112 M. APRODU AND V. BRı̂NZǍNESCU a natural structure of an algebraic variety (see [B], [B-St1], [B-St2]). In Section 2 we study uniform vector bundles and we prove the existence of algebraic vector bundles given by extensions of line bundles and which are not uniform. In Section 3 the main result gives necessary and sufficient conditions for the non-emptiness of the space M (c1 , c2 , d, r) and we apply this result to the moduli space of stable bundles ML (c1 , c2 ). §1. Moduli spaces of rank-2 vector bundles In this section we shall recall from ([B], [B-St1], [B-St2]) some basic notions and facts. The notations and the terminology are those of Hartshorne’s book [Ha]. Let C be a nonsingular curve of genus g over the complex number field and let π : X→C be a ruled surface over C. We shall write X ∼ = P(E) where E V is normalized. Let us denote by e the divisor on C corresponding to 2 E and by e = − deg(e). We fix a point p0 ∈ C and a fibre f0 = π −1 (p0 ) of X. Let C0 be a section of π such that OX (C0 ) ∼ = OP(E) (1). 2 ∼ Any element of Num(X) = H (X, Z) can be written aC0 + bf0 with a, b ∈ Z. We shall denote by OC (1) the invertible sheaf associated to the divisor p0 on C. If L is an element of Pic(C) we shall write L = OC (k) ⊗ L0 , where L0 ∈ Pic0 (C) and k = deg(L). We also denote by F (aC0 + bf0 ) = F ⊗ OX (a) ⊗ π ∗ OC (b) for any sheaf F on X and any a, b ∈ Z. Let E be an algebraic rank-2 vector bundle on X with fixed numerical Chern classes c1 = (α, β) ∈ H 2 (X, Z) ∼ = Z × Z, c2 = γ ∈ H 4 (X, Z) ∼ = Z, where α, β, γ ∈ Z. Since the fibres of π are isomorphic to P1 we can speak about the generic ′ splitting type of E and we have E|f ∼ = Of (d) ⊕ Of (d ) for a general fibre ′ ′ f , where d ≤ d, d + d = α. The integer d is the first numerical invariant of E. The second numerical invariant is obtained by the following normalization: −r = inf{l| ∃L ∈ Pic(C), deg(L) = l, s.t. H 0 (X, E(−dC0 ) ⊗ π ∗ L) 6= 0}. We shall denote by M (α, β, γ, d, r) or M (c1 , c2 , d, r) or M the set of isomorphism classes of algebraic rank-2 vector bundles on X with fixed Chern classes c1 , c2 and invariants d and r. With these notations we have the following result (see [B]): Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 113 MODULI VECTOR BUNDLES OVER RULED SURFACES Theorem 1. For every vector bundle E ∈ M (c1 , c2 , d, r) there exist L1 , L2 ∈ Pic0 (C) and Y ⊂ X a locally complete intersection of codimension 2 in X, or the empty set, such that E is given by an extension (1) ′ 0→OX (dC0 + rf0 )⊗π ∗ L2 →E→OX (d C0 + sf0 )⊗π ∗ L1 ⊗IY →0, ′ ′ where c1 = (α, β) ∈ Z × Z, c2 = γ ∈ Z, d + d = α, d ≥ d , r + s = β, l(c1 , c2 , d, r) := γ + α(de − r) − βd + 2dr − d2 e = deg(Y ) ≥ 0. Remark. By applying Theorem 1 we can obtain the canonical extensions used in [Br1], [Br2]. ′ Indeed, let us suppose first that d > d . From the exact sequence (1) it follows that OC (r) ⊗ L2 ∼ = π∗ E(−dC0 ) so OX (rf0 ) ⊗ π ∗ L2 ∼ = π ∗ π∗ E(−dC0 ) and OX (dC0 + rf0 ) ⊗ π ∗ L2 ∼ = (π ∗ π∗ E(−dC0 ))(dC0 ). ′ If d = d then, by applying π∗ to the short exact sequence 0→OX (rf0 ) ⊗ π ∗ L2 →E(−dC0 )→OX (sf0 ) ⊗ π ∗ L1 ⊗ IY →0 it follows the exact sequence 0→OC (r) ⊗ L2 →π∗ E(−dC0 )→OC (s) ⊗ L1 ⊗ OC (−Z1 )→0, where Z1 is an effective divisor on C with the support π(Y ). With the notation Z = π −1 (Z1 ), by applying π ∗ (π is a flat morphism) we obtain the following commutative diagram with exact rows 0 ✲ OX (rf0 ) ⊗ π ∗ L2 ≀ ✻id ✲ E(−dC0 ) ✻ϕ ✲ OX (sf0 ) ⊗ π ∗ L1 ⊗ IY ✲ 0 ✻ψ 0 ✲ OX (rf0 ) ⊗ π ∗ L2 ✲ π ∗ π∗ E(−dC0 ) ✲ OX (sf0 ) ⊗ π ∗ L1 ⊗ IZ ✲ 0 Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 114 M. APRODU AND V. BRı̂NZǍNESCU From the injectivity of ψ we obtain the injectivity of ϕ. Because of OX (sf0 ) ⊗ π ∗ L1 ⊗ IY ⊂Z ∼ = Coker ψ ∼ = Coker ϕ we conclude. Recall that a set M of vector bundles on a C−scheme X is called bounded if there exists an algebraic C-scheme T and a vector bundle V on T × X such that every E ∈ M is isomorphic with Vt = V |t×X for some closed point t ∈ T (see [K]). For the next result see [B]: Theorem 2. The set M (c1 , c2 , d, r) is bounded. §2. Uniform bundles In what follows, we keep the notations from Section 1. Definition 3. A 2-vector bundle E is called an uniform bundle if the splitting type is preserved on all fibres of X. Theorem 1 allows us to give a criterion for uniformness. Lemma 4. Let f be a fibre of X and let us suppose that IY ∩f ⊂f ∼ = ′ Of (−n). Then E|f ∼ = Of (d + n) ⊕ Of (d − n). ′ ′ Proof. We suppose that E|f ∼ = Of (a) ⊕ Of (a ), where a ≥ a . Then we have a surjective morphism ′ E|f →Of (d ) ⊗ IY ⊗ Of in virtue of Theorem 1. On the other hand, the restriction of the sequence 0→IY →OX →OY →0 to f gives a surjective morphism IY ⊗ Of →IY ∩f ⊂f ∼ = Of (−n). So, we obtain another surjective morphism ′ ′ Of (a) ⊕ Of (a )→Of (d − n). ′ ′ ′ ′ By using the inequalities a ≥ a , d ≥ d ≥ d − n and the equality a + a = ′ ′ ′ d + d = α it follows that a = d − n and a = d + n. Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 115 MODULI VECTOR BUNDLES OVER RULED SURFACES Corollary 5. E is an uniform bundle if and only if l(c1 , c2 , d, r) = 0. By means of Corollary 5 the uniform bundles are given by extensions of line bundles. It is naturally to ask if the converse is true. Unfortunately, this question has a negative answer, as proved by the following Proposition 6. On the rational ruled surface Fe with e ≥ 1 there exist non-uniform bundles given by extensions of line bundles. For the proof we need some preparations. Let E be a 2-vector bundle given by an extension 0→F →E→G→0, (2) ′ ′ ′ ′ ′ ′ ′ where F = OX (aC0 + r f0 ) ⊗ π ∗ L2 , G = OX (a C0 + s f0 ) ⊗ π ∗ L1 (L1 , L2 ∈ Pic0 (C)) are line bundles on X. By means of Theorem 1, E sits also in a ′ canonical extension (1). If a ≥ a then E is obviously uniform. Then, we ′ shall suppose that a < a . ′ Lemma 7. With the above notations we have d ≤ a . Proof. Indeed, by the restriction of the sequence (2) to a general fibre f we obtain a surjective morphism ′ ′ Of (d) ⊕ Of (d )→Of (a ). ′ ′ ′ If d > a , then it follows that d = a which contradicts the inequalities ′ ′ a < a , d ≥ d (a + a′ = d + d′ ). ′ Lemma 8. If d = a then E is uniform. Proof. Let f be a fibre of X such that the splitting type of E|f is different from the generic splitting type of E. According to Lemma 4 ′ E|f ∼ = Of (d + n) ⊕ Of (d − n), where n > 0. By the restriction of (2) to f we obtain a surjective morphism ′ Of (d + n) ⊕ Of (d − n)→Of (d). ′ Because of d + n > d it follows d − n = d, contradiction. Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 116 M. APRODU AND V. BRı̂NZǍNESCU ′ Lemma 9. In the above hypotheses, if d = a , then E ∼ = F ⊕ G. Proof. Let us observe that we can suppose, without loss of general′ ity, that a = 0 and r = 0 (by twisting the sequences (1) and (2) with ′ ′ ′ ′ OX (−aC0 − r f0 )). Then, it follows that d = a = α > 0, s = β and d = 0. Therefore, the sequences (1) and (2) become: 0 ✻ ′ OX (αC0 + βf0 ) ⊗ π ∗ L1 χ ✒ ✻ϕ ψ (1′ ) 0 ✲ OX (αC0 + rf0 ) ⊗ π ∗ L2 ✲ E ✲ OX (sf0 ) ⊗ π ∗ L1 ⊗ IY ✲ 0 ✻ ′ π ∗ L2 ✻ 0 The computation of c2 (E) in (1′ ) gives deg(Y ) = −αs. Moreover, by means of Lemma 8, deg(Y ) = 0, so s = 0 (we supposed α > 0). The homomorphism χ = ϕψ is non-zero, otherwise OX (αC0 + βf0 ) ⊂ ′ ′ ∗ π (L2 ) (which would contradict the condition α > 0), so L2 = L1 and χ is the multiplication by a λ ∈ C∗ , and the assertion follows. In this moment, we are able to give the counter-example announced in Proposition 6. Proof of Proposition 6. Let G be OX (2C0 ) and let F be OX . Then: dim H 1 (G−1 ) = e + 1 6= 0. For E given by an extension ξ ∈ Ext1 (G, OX ), keeping the notations ′ ′ from Section 1, we have d ≤ 2 (Lemma 7) , d ≥ d , d + d = 2 and r + s = 0. There are only two possibilities: ′ (a) d = 2 , d = 0, which implies E ∼ = OX ⊕ OX (2C0 ) (Lemma 9). ′ (b) d = d = 1 and, in this case, in the canonical extension (1) of E, we have Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 MODULI VECTOR BUNDLES OVER RULED SURFACES ′ 117 ′ deg(Y ) = dd e − ds − d r = e ≥ 1. By applying Corollary 5, all vector bundles given by non-zero extensions from Ext1 (G, OX ) are non-uniform. §3. Non-emptiness of moduli spaces For a rank-2 vector bundle E, we shall denote by dE and rE the invariants of E, when confusions may appear. Theorem 10. M (c1 , c2 , d, r) is non-empty if and only if l := l(c1 , c2 , d, r) ≥ 0 and one of the following conditions holds: (I) 2d > α or, (II) 2d = α, β − 2r ≤ g + l. Proof. We observe that if M 6= ∅ then, by means of Theorem 1, the elements of M lie among 2-vector bundles given by extensions of type (1). Therefore, we conclude that M 6= ∅ if and only if in the extensions of type (1) there are 2-vector bundles with dE = d and rE = r. It is clear that all the vector bundles given by an extension of type (1) have dE = d so we shall look for bundles with rE = r. We fix L1 , L2 ∈ Pic0 (C) and Y ⊂ X a locally complete intersection (or the empty set) and we denote ′ N1 = OX (d C0 + sf0 ) ⊗ π ∗ L1 N2 = OX (dC0 + rf0 ) ⊗ π ∗ L2 and l = deg(Y ). Consider the spectral sequence of terms E2p,q = H p (X, Extq (IY ⊗ N1 , N2 )) which converges to Extp+q (IY ⊗ N1 , N2 ). We have Ext0 (IY ⊗ N1 , N2 ) ∼ = N2 ⊗ N1−1 and Ext1 (IY ⊗ N1 , N2 ) ∼ = OY . But H 2 (X, N2 ⊗ N1−1 ) = 0 so the exact sequence of lower terms becomes 0→H 1 (X, N2 ⊗ N1−1 )→Ext1 (IY ⊗ N1 , N2 )→H 0 (Y, OY )→0. Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 118 M. APRODU AND V. BRı̂NZǍNESCU Now, by a result due to Serre (see [O-S-S], Chap.I, 5, [Se]), any element in the group Ext1 (IY ⊗ N1 , N2 ) which has an invertible image in H 0 (Y, OY ) defines an extension of the desired form with E a 2-vector bundle. We write the sequence (1) under the equivalent form (3) ′′ ′ 0→OX →E(−dC0 ) ⊗ π ∗ L →OX ((d − d)C0 + (s − r)f0 ) ⊗ π ∗ (L̃) ⊗ IY →0 ′′ ′′ −1 where L̃ = L1 ⊗ L−1 2 , L = OC (−r) ⊗ L2 and deg(L ) = −r. From the definition, it follows r ≤ rE for every bundle E given by an extension (1). We distinguish three cases: ′ (I) d > d . In this case we shall prove that M is non-empty if and only if l ≥ 0. To do this we prove that all vector bundles from extension (1) have rE = r. ′ ′ We verify that for all L ∈ Pic(C) with deg(L ) < 0 we have ′′ ′ H 0 (E(−dC0 ) ⊗ π ∗ (L ⊗ L )) = 0, ′ which is true because H 0 (L ) = 0 and H 0 (OX ((d − d)C0 + (s − r)f0 ) ⊗ π ∗ (L1 ⊗ L−1 2 ⊗ L ) ⊗ IY ) = 0. ′ ′ (II) a◦ . d = d , r ≥ s. Then M is non-empty if and only if l ≥ 0. The proof ′ runs like in the first case with the remark deg(OC (s−r)⊗L1 ⊗L−1 2 ⊗L ) < 0. ′ (II) b◦ . d = d , r < s. Then M is non-empty if and only if l ≥ 0 and β − 2r ≤ g + l. Let us see first that the natural isomorphism ′ M (2d, β, γ, d, r)−→M (0, β, l, 0, r) E−→E(−dC0 ) ′ allows us to suppose d = d = 0. In this case, the sequence (3) becomes −1 ∗ 0→OX →E ⊗ OX (−rf0 ) ⊗ π ∗ L−1 2 →OX ((s − r)f0 ) ⊗ π (L1 ⊗ L2 ) ⊗ IY →0. The definition of the second invariant implies that rE = r if and only ′ ′ if E := π∗ E ⊗ OC (−rp0 ) ⊗ L−1 2 is normalised. E belong to an extension (4) ′ 0→OC →E →L→0 where L = OC ((s − r)p0 ) ⊗ L1 ⊗ L−1 2 ⊗ OC (−Z1 ) with Z1 an effective divisor on C with support π(Y ) and card(Y ) ≤ deg(Z1 ) ≤ l = deg(Y ). ′ According to a result of Nagata ([N] or [Ha] Ex.V.2.5) , if E is normalised, then Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 MODULI VECTOR BUNDLES OVER RULED SURFACES 119 ′ − deg(E ) = r − s + deg(Z1 ) ≥ −g which proves “only if” part of (II) b◦ . For “if” part we choose Y reduced, obtained by intersection between C0 and l distinct fibres of X. In this case, we have the following short exact sequence (5) 0→IZ →IY →IY ⊂Z →0 where Z1 = π(Y ) = p1 + · · · + pl , Y ⊂ Z = π −1 (Z1 ) = f1 + · · · + fl with fi distinct fibres, OZ = Of1 ⊕ · · · ⊕ Ofl , IY ⊂Z = Of1 (−1) ⊕ · · · ⊕ Ofl (−1) . So, the sequence (5) becomes 0→IZ →IY →Of1 (−1) ⊕ · · · ⊕ Ofl (−1)→0. Tensoring by KX ⊗ N2−1 ⊗ N1 and taking the long cohomology sequence we obtain an injective map: H 1 (KX ⊗ N2−1 ⊗ N1 ⊗ IZ )−→H 1 (KX ⊗ N2−1 ⊗ N1 ⊗ IY ). By dualizing, it follows that the natural map ϕ Ext1 (IY ⊗ N1 , N2 ) −→ Ext1 (IZ ⊗ N1 , N2 ) ∼ = Ext1 (L, OC ) is surjective, which shows that all bundles in (4) are coming from (1) by applying π∗ . According to [Ha] (Ex. V.2.5), there is a non-empty open set V ⊂ Ext1 (L, OC ) (don’t forget the condition s − r ≤ g + l !) such that all ξ ∈ V define normalised vector bundles on C. Now, in Ext1 (IY ⊗ N1 , N2 ) the set of vector bundles is a non-empty open set U . It is clear that ϕ−1 (V ) ∩ U 6= ∅ (being open sets in Zariski topology), so we conclude. §4. Moduli of stable bundles There is an interesting relation between the moduli spaces M (c1 , c2 , d, r) and the Qin’s sets Eζ (c1 , c2 ) (see [Q1], [Q2] for precised definitions). As in the proof of Theorem 10, case (I) we conclude that if ζ is a normalized class reprezenting a non-empty wall of type (c1 , c2 ) such that lζ (c1 , c2 ) > 0 then, for (2d−α, 2r −β) = ζ , Eζ (c1 , c2 ) and M (c1 , c2 , d, r) are coincident modulo a factor of Pic0 (C) (Qin workes with first Chern class c1 as an element in Pic(X)). This is a consequence of the following facts: Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 120 M. APRODU AND V. BRı̂NZǍNESCU (a) lζ (c1 , c2 ) = l(c1 , c2 , d, r) (b) condition ζ 2 < 0 implies 2d > α (c) in the case 2d > α the bundles L1 , L2 and the set Y from the sequence (1) are uniquely determined by E. (d) if l(c1 , c2 , d, r) > 0 then in the sequence (1) the bundles are given only by non-trivial extensions. In fact it is not hard to see that M (c1 , c2 , d, r) 6= ∅ iff Eζ (c1 , c2 ) 6= ∅ so, by means of Theorem 10, Eζ (c1 , c2 ) 6= ∅ if lζ (c1 , c2 ) > 0. But we have even more: Corollary 11. Let X be a ruled surface different from P1 × P1 and let C be a chamber of type (c1 , c2 ) different from Cf0 . Then the moduli space MC (c1 , c2 ) 6= ∅. Proof. From Theorem 1.3.3 in [Q2] it follows that F F MC (c1 , c2 ) = (MC1 (c1 , c2 ) − E(−ζ) (c1 , c2 )) Eζ (c1 , c2 ) , ζ ζ where C1 is the chamber lying above C and sharing with C a non-empty common wall W and ζ runs over all normalised classes representing W . By the above considerations, it follows that Eζ (c1 , c2 ) 6= ∅ if l(c1 , c2 , d, r) > 0. It remains the case l(c1 , c2 , d, r) = 0 and it will be sufficient to prove that h1 (X, N2 ⊗ N1−1 ) := dim H 1 (X, N2 ⊗ N1−1 ) > 0 (see the proof of Theorem 10). We have N2 ⊗ N1−1 = OX ((d − d′ )C0 + (r − s)f0 ) ⊗ π ∗ (L2 ⊗ L−1 1 ), where d − d′ = 2d − α = u and r − s = 2r − β = v. But ζ = uC0 + vf0 is a normalized class and this implies that u > 0 and v < 0 (see [Q1]). Because H 2 (X, N2 ⊗ N1−1 ) = 0, the Riemann-Roch Theorem gives the equality: χ = h0 (X, N2 ⊗N1−1 )−h1 (X, N2 ⊗N1−1 ) = 1−g+(1/2)((u+1)(2v−ue)+u(2−2g)). But ζ 2 < 0 gives u(2v − ue) < 0; it follows 2v − ue < 0. If g ≥ 1, then obviously χ < 0. If g = 0, then e ≥ 0 and χ = 1 + v + (u/2)(2(v + 1) − e(u + 1)). Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 MODULI VECTOR BUNDLES OVER RULED SURFACES 121 If e ≥ 1, then χ < 0. For e = 0 we get X = P1 × P1 , which we excluded. Thus, in all cases χ < 0; it follows h1 (X, N2 ⊗ N1−1 ) > 0 and the proof is over. Remark. Let us suppose that X = P1 × P1 and that C is a chamber of type (c1 , c2 ) lying below a non-empty wall defined by a normalized class ζ = uC0 + vf0 with v ≤ −2. Then the same conclusion as in the above corollary holds. Indeed, in this case we have χ = (1 + v)(1 + u). Since v < −1, then again χ < 0. Acknowledgements. The second named author expresses his gratitude to the Max-Planck-Institut für Mathematik Bonn for its hospitality during the final stage of this work. References [B] V. Brı̂nzǎnescu, Algebraic 2-vector bundles on ruled surfaces, Ann. Univ. Ferrara-Sez VII, Sc. Mat., XXXVII (1991), 55–64. [B-St1] V. Brı̂nzǎnescu and M. Stoia, Topologically trivial algebraic 2-vector bundles on ruled surfaces I, Rev. Roumaine Math. Pures Appl., 29 (1984), 661–673. [B-St2] , Topologically trivial algebraic 2-vector bundles on ruled surfaces II, In : Lect. Notes Math., 1056, Springer (1984). [Br1] J. E. Brossius, Rank-2 vector bundles on a ruled surface I, Math. Ann., 265 (1983), 155–168. [Br2] , Rank-2 vector bundles on a ruled surface II, Math. Ann., 266 (1984), 199–214. [Ha] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math., 49, Springer, Berlin-Heidelberg, 1977. [H-S] H. J. Hoppe and H. Spindler, Modulräume stabiler 2-Bündel auf Regelflächen, Math. Ann., 249 (1980), 127–140. [K] S. Kleiman, Les théorèmes de finitude pour les Founcteurs de Picard, In : Théories des intersections et théorème de Riemann-Roch, SGA VI, Exp. XIII, Lect. Notes in Math., 225, Springer (1971), 616–666. [N] M. Nagata, On self-intersection Number of a section on ruled surface, Nagoya Math. J., 37 (1970), 191–196. [O-S-S] C. Okonek, M. Schneider and H. Spindler, Vector bundles on complex projective spaces, Birkhäuser, Basel Boston Stuttgart, 1980. [Q1] Z. Qin, Moduli spaces of stable rank-2 bundles on ruled surfaces, Invent. Math., 110 (1992), 615–626. , Equivalence classes of polarizations and moduli spaces of sheaves, J. Diff. [Q2] Geom., 37 (1993), 397–415. Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 122 [Se] [T1] [T2] [W] M. APRODU AND V. BRı̂NZǍNESCU J. P. Serre, Sur les modules projectifs, Sém. Dubreil-Pisot 1960/1961 Exp. 2, Fac. Sci. Paris, 1963. F. Takemoto, Stable vector bundles on algebraic surfaces I, Nagoya Math. J., 47 (1972), 29–48. , Stable vector bundles on algebraic surfaces II, Nagoya Math. J., 52 (1973), 173–195. C. H. Walter, Components of the stack of torsion-free sheaves of rank-2 on ruled surfaces, Math. Ann., 301 (1995), 699–716. Marian Aprodu Institute of Mathematics of the Romanian Academy P.O. BOX 1-764, RO-70700 Bucharest Romania [email protected] Vasile Brı̂nzǎnescu Institute of Mathematics of the Romanian Academy P.O. BOX 1-764, RO-70700 Bucharest Romania [email protected] Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332