BRAUER GROUP OF MODULI SPACES OF PAIRS INDRANIL BISWAS, MARINA LOGARES, AND VICENTE MUN˜OZ Abstract. We show that the Brauer group of the moduli space of stable pairs with fixed determinant over a curve is zero. 1 1 0 2 1. Introduction n a J Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A 6 holomorphic pair (also called a Bradlow pair) is an object of the form (E,φ), where E is a holomorphic vector bundle over X, and φ is a nonzero holomorphic section of E. The ] G concept of stability for pairs depends on a parameter τ ∈ R. Moduli spaces of τ-stable A pairs of fixed rank and degree were first constructed using gauge theoretic methods in . [4], and subsequently using Geometric Invariant Theory in [3]. Since then these moduli h t spaces have been extensively studied. a m Fix an integer r ≥ 2 and a holomorphic line bundle Λ over X. Let d = deg(Λ). Let [ M (r,Λ) be the moduli space of stable pairs (E,φ) such that rk(E) = r and det(E) = τ r 4 V E = Λ. This is a smooth quasi-projective variety; it is empty if d ≤ 0. Therefore, v H2(M (r,Λ),G ) is torsion, and it coincides withthe Brauer groupof M (r,Λ), defined 4 e´t τ m τ 0 by the equivalence classes of Azumaya algebras over Mτ(r,Λ). Let Br(Mτ(r,Λ)) denote 2 the Brauer group of M (r,Λ). τ 5 . We prove the following (see Theorem 3.3 and Corollary 3.5): 9 0 Theorem 1.1. Assume that (r,g,d) 6= (3,2,2). Then Br(M (r,Λ)) = 0. 0 τ 1 : Let M(r,Λ) be the moduli space of stable vector bundles over X of rank r and deter- v i minant Λ. There is a unique universal projective bundle over X ×M(r,Λ). Restricting X this projective bundle to {x} × M(r,Λ), where x is a fixed point of X, we get a pro- r a jective bundle P over M(r,Λ). We give a new proof of the following known result (see x Corollary 3.4). Corollary 1.2. Assume (r,g,d) 6= (2,2,even). The Brauer group of M(r,Λ) is gener- ated by the Brauer class of P . x This was first proved in [2]. We show that it follows as an application of Theorem 1.1. For convenience, we work over the complex numbers. However, the results are still valid for any algebraically closed field of characteristic zero. Date: 6 June 2010. Revised: 15 December 2010. 2000 Mathematics Subject Classification. 14D20, 14F22, 14E08. Key words and phrases. Brauer group, moduli of pairs, stable bundles, complex curve. 1 2 I.BISWAS, M. LOGARES,ANDV. MUN˜OZ Acknowledgements. We are grateful to Norbert Hoffmann and Peter Newstead for helpful comments; specially, Peter Newstead pointed out a mistake in a previous version of Proposition 2.4. We thank the referee for a careful reading and useful comments. The second and third authors are grateful to the Tata Institute of Fundamental Research (Mumbai), where this work was carried out, for its hospitality. Second author supported by (Spanish MICINN) research project MTM2007-67623 and i-MATH. First and third author partially supported by (Spanish MCINN) research project MTM2007-63582. 2. Moduli spaces of pairs We collect here some known results about the moduli spaces of pairs, taken mainly from [4], [5], [9], [11] and [14]. Let X be a smooth projective curve defined over the field of complex numbers, of genus g ≥ 2. A holomorphic pair (E,φ) over X consists of a holomorphic bundle on X and a nonzero holomorphic section φ ∈ H0(E). Let µ(E) := deg(E)/rk(E) be the slope of E. There is a stability concept for a pair depending on a parameter τ ∈ R. A holomorphic pair (E,φ) is τ-stable whenever the following conditions are satisfied: • for any subbundle E′ ⊂ E, we have µ(E′) < τ, • for any subbundle E′ ⊂ E such that φ ∈ H0(E′), we have µ(E/E′) > τ. The concept of τ-semistability is defined by replacing the above strict inequalities by the weaker inequalities “≤” and “≥”. A critical value of the parameter τ = τ is one for c which there are strictly τ-semistable pairs. There are only finitely many critical values. Fix an integer r ≥ 2 and a holomorphic line bundle Λ over X. Let d be the degree of Λ. We denote by M (r,Λ) (respectively, M (r,Λ)) the moduli space of τ-stable τ τ (respectively, τ-semistable) pairs (E,φ) of rank rk(E) = r and determinant det(E) = Λ. The moduli space M (r,Λ) is a normal projective variety, and M (r,Λ) is a smooth τ τ quasi-projective variety contained in the smooth locus of M (r,Λ) (cf. [11, Theorem τ 3.2]). Moreover, dimM (r,Λ) = d+(r2 −r−1)(g −1)−1. τ For non-critical values of the parameter, there are no strictly τ-semistable pairs, so M (r,Λ) = M (r,Λ) and it is a smooth projective variety. For a critical value τ , the τ τ c variety M (r,Λ) is in general singular. τc Denote τ := d and τ := d . The moduli space M (r,Λ) is empty for τ 6∈ (τ ,τ ). m r M r−1 τ m M In particular, this forces d > 0 for τ-stable pairs. Denote by τ < τ < ... < τ 1 2 L the collection of all critical values in (τ ,τ ). Then the moduli spaces M (r,Λ) are m M τ isomorphic for all values τ in any interval (τ ,τ ), i = 0,...,L; here τ = τ and i i+1 0 m τ = τ . L+1 M However, the moduli space changes when we cross a critical value. Let τ be a critical c value (note that for us, a critical value τ 6= τ ,τ ). Denote τ+ := τ +ǫ and τ− := τ −ǫ c m M c c c c for ǫ > 0 small enough such that (τ−,τ+) does not contain any critical value other than c c τc. We define the flip loci Sτc± as the subschemes: • S = {(E,φ) ∈ M (r,Λ)|(E,φ) is τ−-unstable}, τc+ τc+ c • Sτc− = {(E,φ) ∈ Mτc−(r,Λ)|(E,φ) is τc+-unstable}. BRAUER GROUP OF MODULI SPACES OF PAIRS 3 When crossing τ , the variety M (r,Λ) undergoes a birational transformation: c τ Mτc−(r,Λ)\Sτc− = Mτc(r,Λ) = Mτc+(r,Λ)\Sτc+. Proposition 2.1 ([10, Proposition 5.1]). Suppose r ≥ 2, and let τ be a critical value c with τ < τ < τ . Then m c M • codimS ≥ 3 except in the case r = 2, g = 2, d odd and τ = τ + 1 (in which τc+ c m 2 case codimS = 2), τc+ • codimSτc− ≥ 2 except in the case r = 2 and τc = τM − 1 (in which case codimSτc− = 1). Moreover we have that codimSτc− = 2 only for τc = τM −2. The codimension of the flip loci is then always positive, hence we have the following corollary: Corollary 2.2. The moduli spaces M (r,Λ), τ ∈ (τ ,τ ), are birational. τ m M The moduli spaces for the extreme values of the parameter τ+ and τ− are known m M explicitly. Let M(r,Λ) be the moduli space of stable vector bundles or rank r and fixed determinant Λ. Define (2.1) U (r,Λ) = {(E,φ) ∈ M (r,Λ)|E is a stable vector bundle}, m τm+ and denote S := M (r,Λ)\U (r,Λ) τm+ τm+ m (not to be confused with the definition above for Sτc±, which refers only to critical values τ 6= τ ,τ ). Then there is a map c m M (2.2) π : U (r,Λ) −→ M(r,Λ), (E,φ) 7→ E, 1 m whose fiber over E is the projective space P(H0(E)). When d ≥ r(2g − 2), we have that H1(E) = 0 for any stable bundle, and hence (2.2) is a projective bundle (cf. [9, Proposition 4.10]). Regarding the rightmost moduli space Mτ−(r,Λ), we have that any τM−-stable pair M (E,φ) sits in an exact sequence φ 0 −→ O −→ E −→ F −→ 0, where F is a semistable bundle of rank r −1 and det(F) = Λ. Let UM(r,Λ) = {(E,φ) ∈ Mτ−(r,Λ)|F is a stable vector bundle}, M and denote Sτ− := Mτ−(r,Λ)\UM(r,Λ). M M Then there is a map (2.3) π : U (r,Λ) −→ M(r −1,Λ), (E,φ) 7→ E/φ(O), 2 M whose fiber over F ∈ M(r−1,Λ) is the projective spaces P(H1(F∗)) (cf. [6]). Note that H0(F∗) = 0 since d > 0. So (2.3) is always a projective bundle. In the particular case of rank r = 2, the rightmost moduli space is (2.4) Mτ−(2,Λ) = P(H1(Λ−1)), M 4 I.BISWAS, M. LOGARES,ANDV. MUN˜OZ since M(1,Λ) = {Λ}. In particular, Corollary 2.2 shows that all M (2,Λ) are rational τ quasi-projective varieties. We have the following: Lemma 2.3 ([11, Lemma 5.3]). Let S be a bounded family of isomorphism classes of strictly semistable bundles of rank r and determinant Λ. Then dimM(r,Λ) −dimS ≥ (r −1)(g −1). Proposition 2.4. The following two statements hold: • Suppose d > r(2g −2). Then codimS ≥ 2 except in the case r = 2, g = 2, d τm+ even (in which case codimS = 1). τm+ • Suppose r ≥ 3. Then codimSτ− ≥ 2 except in the case r = 3, g = 2, d even (in M which case the codimSτ− = 1). M Proof. Let (E,φ) ∈ M (r,Λ), then E is a semistable bundle. As d > r(2g − 2), τm+ H1(E) = H0(E∗ ⊗ K )∗ = 0, since E∗ ⊗ K is semistable and has negative degree. X X Therefore, dimH0(E) is constant. Let F be the family of strictly semistable bundles E such that there exists some φ with (E,φ) ∈ S . Then codimS = dimM (r,Λ) − τm+ τm+ τm+ dimS ≥ dimM(r,Λ)−dimF ≥ (r −1)(g −1) (by Lemma 2.3). The first statement τm+ follows. As the dimension dimH1(F∗) is constant, the codimension of Sτ− in Mτ−(r,Λ) is M M at least the codimension of a locus of semistable bundles. Applying Lemma 2.3 to M(r −1,Λ) we have codimSτ− ≥ (r −2)(g −1). The second item follows. (cid:3) M 3. Brauer group The Brauer group of a scheme Z is defined as the equivalence classes of Azumaya algebras on Z, that is, coherent locally free sheaves with algebra structure such that, locally on the ´etale topology of Z, are isomorphic to a matrix algebra Mat(O ). If Z is Z a smooth quasi-projective variety, then the Brauer group Br(Z) coincides with H2(Z), e´t and H2(Z) is a torsion group. e´t Theorem 3.1. [8, VI.5 (Purity)] Let Z be a smooth complex variety and U ⊂ Z be a Zariskiopensubset whosecomplementhas codimensionatleast2. ThenBr(Z) = Br(U). On the moduli space of stable vector bundles M(r,Λ), there are three natural projec- tive bundles. We will describe them. We first note that there is a unique universal projective bundle over X×M(r,Λ). Fix a point x ∈ X. Restricting the universal projective bundle to {x}×M(r,Λ) we get a projective bundle (3.1) P −→ M(r,Λ). x Secondly, if d ≥ r(2g −2), then we have the projective bundle (3.2) P −→ M(r,Λ), 0 BRAUER GROUP OF MODULI SPACES OF PAIRS 5 whose fiber over any E ∈ M(r,Λ) is the projective space P(H0(E)); note that we have H1(E) = 0 because d ≥ r(2g −2). Finally, assuming d > 0, let (3.3) P −→ M(r,Λ) 1 be the projective bundle whose fiber over any E ∈ M(r,Λ) is the projective space P(H1(E∗)). Proposition 3.2. The Brauer class cl(P ) ∈ Br(M(r,Λ)) is independent of x ∈ X. x Moreover, cl(P ) = cl(P ) = −cl(P ), x 0 1 when they are defined. Proof. The moduli space M(r,Λ) is constructed as a Geometric Invariant Theoretic quotient of a Quot scheme Q by the action of a linear group GL (C) (see [13]). The N isotropy subgroup for a stable point of Q is the center C∗ ⊂ GL (C). There is a N universal vector bundle E −→ X ×Q. Let Z(GL (C)) be the center of GL (C). The action of the subgroup Z(GL (C)) on N N N Q is trivial. Therefore, Z(GL (C)) acts on each fiber of E. Identify Z(GL (C)) with N N C∗ by sending any λ ∈ C∗ to λ · Id. We note that λ ∈ Z(GL (C)) acts on E as N multiplication by λ. Let Qs ⊂ Q be the stable locus. The restriction of E to X ×Qs will be denoted by Es. Let Ex := Es|{x}×Qs −→ Qs be the restriction. Let p : X ×Qs −→ Qs be the natural projection. Define the vector 2 bundles E := p Es and E := R1p ((Es)∗). 0 2∗ 1 2∗ We noted that any λ ∈ C∗ = Z(GL (C)) acts on E as multiplication by λ. There- N x fore, λ acts on (Es)∗ as multiplication by 1/λ. Hence λ acts on E as multiplication 1 by 1/λ. Consequently, the action of Z(GL (C)) on E ⊗ E is trivial. Hence E ⊗ E N x 1 x 1 descends to a vector bundle over the quotient M(r,Λ) of Qs. Therefore, cl(P ) = −cl(P ). x 1 Any λ ∈ C∗ = Z(GL (C)) acts on E as multiplication by λ. Indeed, this follows N 0 immediately from the fact that λ acts as multiplication by λ on Es. As noted earlier, λ acts on E as multiplication by 1/λ. Hence the action of Z(GL (C)) on E ⊗ E is 1 N 0 1 trivial. Thus E ⊗E descends to M(r,Λ), implying 0 1 cl(P ) = −cl(P ). 0 1 Finally, note that it follows that cl(P ) is independent of x ∈ X for d > 0. For d ≤ 0, x P and P are not defined. In this case, we take a line bundle µ or large degree, and use 0 1 the isomorphism M(r,Λ⊗µr) ∼= M(r,Λ). For any pair x,x′ ∈ X, since cl(Px) = cl(Px′) in Br(M(r,Λ⊗µr)), the same holds for Br(M(r,Λ)). (cid:3) 6 I.BISWAS, M. LOGARES,ANDV. MUN˜OZ Theorem 3.3. Assume that d > r(2g−2). Then for the moduli space M (r,Λ) of stable τ pairs, we have that Br(M (r,Λ)) = 0. τ Proof. We will first prove it for r = 2. Recall from (2.4) that Mτ−(2,Λ) is a projective M space, hence Br(Mτ−(2,Λ)) = 0. M Moreover, all M (2,Λ) are rational varieties. Thus τ Br(M (2,Λ)) = 0 τ for non-critical values τ ∈ (τ ,τ ), since the Brauer group of a smooth rational projec- m M tive variety is zero [1, p. 77, Proposition 1]. For a critical value τ , we have c M (2,Λ) = M (2,Λ)\S . τc τc+ τc+ By Proposition 2.1, codimS ≥ 2, so the Purity Theorem implies that τc+ Br(M (2,Λ)) = 0. τc Now we assume that r ≥ 3. From Proposition 2.1 and Theorem 3.1 it follows that the Brauer group Br(M (r,Λ)) does not depend on the value of the parameter τ (for τ fixed r and Λ). As d ≥ r(2g −2), we have a projective bundle π : U (r,Λ) −→ M(r,Λ) 1 m (see (2.2)). Note that this projective bundle coincides with the projective bundle P in 0 (3.2). The projective bundle π gives an exact sequence 1 (3.4) Z·cl(P ) −→ Br(M(r,Λ)) −→ Br(U (r,Λ)) −→ 0 0 m (see [7, p. 193]). As d > r(2g −2), Proposition 2.4 and the Purity Theorem give Br(U (r,Λ)) = Br(M (r,Λ)), m τm+ so we have (3.5) Z·cl(P ) −→ Br(M(r,Λ)) −→ Br(M (r,Λ)) −→ 0. 0 τm+ We will show that the theorem follows from (3.5) if we use [2]. From Proposition 3.2 we know that cl(P ) = cl(P ), and from [2, Proposition 1.2(a)] we know that cl(P ) 0 x x generates Br(M(r,Λ)). Therefore, from (3.5) it follows that Br(M (r,Λ)) = 0. τm+ Since Br(M (r,Λ)) is independent of τ, this completes the proof using [2]. But we τ shall give a different proof without using [2], because we want to show that the above mentioned result of [2] can be deduced from our Theorem 3.3 (see Corollary 3.4). Consider the projective bundle π : U (r −1,Λ) −→ M(r −1,Λ) from (2.3). Note 2 M that this projective bundle coincides with the projective bundle P in (3.3) for rank 1 r −1. The projective bundle π gives an exact sequence 2 (3.6) Z·cl(P1) −→ Br(M(r −1,Λ)) −→ Br(UM(r,Λ)) = Br(Mτ−(r,Λ)) −→ 0, M BRAUER GROUP OF MODULI SPACES OF PAIRS 7 using Proposition 2.4, with the exception of the case (r,g,d) = (3,2,even). Let us leave this “bad” case aside for the moment. Let (3.7) Z·cl(P ) −→ Br(M(r−1,Λ)) −→ Br(U (r−1,Λ)) = Br(M (r−1,Λ)) −→ 0 0 m τm+ be the exact sequence obtained by replacing r with r−1 in (3.5); the last equality holds as (r −1,g,d) 6= (2,2,even), by Proposition 2.4. Since cl(P ) = −cl(P ) (see Proposition 3.2), comparing (3.6) and (3.7) we conclude 1 0 that the two quotients of Br(M(r −1,Λ)), namely Br(MτM−(r,Λ)) and Br(Mτm+(r−1,Λ)), coincide. In particular, Br(MτM−(r,Λ)) is isomorphic to Br(Mτm+(r −1,Λ)). Therefore, using induction, the group Br(MτM−(r,Λ)) is isomorphic to Br(Mτm+(2,Λ)). We have already shown that Br(M (2,Λ)) = 0. Hence the proof of the theorem is complete for τm+ d > r(2g −2) and (r,g,d) 6= (3,2,even). Let us now investigate the missing case of (r,g,d) = (3,2,2k). Take a line bundle ν of degree 1. Using (3.4) twice, we have Z·cl(P ) −→ Br(M(3,Λ)) −→ Br(U (3,Λ)) −→ 0 0 m ∼ ↓= || Z·cl(P ) −→ Br(M(3,Λ⊗ν3)) −→ Br(U (3,Λ⊗ν3)) −→ 0 0 m The second vertical map is induced by the isomorphism M(3,Λ) −→ M(3,Λ ⊗ ν3) defined by E 7→ E ⊗ ν, hence it is an isomorphism. This isomorphism preserves the class cl(P ), and hence the class cl(P ), by Proposition 3.2. Therefore, Br(U (3,Λ)) = x 0 m Br(U (3,Λ⊗ν3)). But deg(Λ⊗ν3) is odd, hence m 3 Br(U (3,Λ)) = Br(U (Λ⊗ν )) = 0. m m By the Purity Theorem, Br(M (3,Λ)) = 0 for any τ. (cid:3) τ Note that the proof of Theorem 3.3 works in the following cases: • r = 2, any d ; • (r,g,d) 6= (3,2,even), d > (r −1)(2g −2) ; and • r = 3, g = 2, d > 6 . Before proceeding to remove the assumption d > r(2g −2) in Theorem 3.3, we want to show that Theorem 3.3 implies Proposition 1.2(a) of [2]. Corollary 3.4. Suppose that (r,g,d) 6= (2,2,even). The Brauer group Br(M(r,Λ)) is generated by the Brauer class cl(P ) ∈ Br(M(r,Λ)) in (3.1). x Proof. Without loss of generality we can assume that d is large (since we have an iso- morphism M(r,Λ) −∼→ M(r,Λ⊗µr), E 7→ E ⊗µ, where µ is a line bundle. First, we have Br(U (r,Λ)) = Br(M (r,Λ)) by the Purity Theorem and Proposition m τm+ 2.4. Second, Br(M (r,Λ)) = 0 by Theorem 3.3, so Br(U (r,Λ)) = 0. Finally, we τm+ m 8 I.BISWAS, M. LOGARES,ANDV. MUN˜OZ use the exact sequence in (3.4) to see that cl(P ) generates Br(M(r,Λ)). Now from 0 Proposition 3.2 it follows that cl(P ) generates Br(M(r,Λ)). (cid:3) x Corollary 3.5. Suppose (r,g,d) 6= (3,2,2). Then we have that Br(M (r,Λ)) = 0. τ Proof. Forr = 2, thisresult isproved asinTheorem 3.3. Aswe knowit ford > r(2g−2), we assume that d ≤ r(2g−2). Letr ≥ 3. Supposefirstthat(r,g,d) 6= (3,2,even), thatis, (r,g,d) 6= (3,2,2),(3,2,4), (3,2,6). As d > 0, we still have a projective bundle π : U (r,Λ) −→ M(r − 1,Λ). 2 M Therefore there is an exact sequence as in (3.6). Note that Proposition 2.4 and the Purity Theorem imply that Br(Mτ−(r,Λ)) = Br(UM(r,Λ)). Now using Proposition 3.2 M and Corollary 3.4 and (3.6) it follows that Br(Mτ−(r,Λ)) = 0. The result follows. M Let us deal with the missing cases (r,g,d) = (3,2,4), (3,2,6). We start with the case (r,g,d) = (3,2,4). Let 1 Z = {E ∈ M(3,Λ)|H (E) 6= 0}. For E ∈ M(3,Λ) \ Z, we have that dimH0(E) = 4 + 3(1 − g) = 1. So the projective bundle π : U (3,Λ)\π−1(Z) −→ M(3,Λ)\Z 1 m 1 is actually an isomorphism. In this situation, the exact sequence (3.8) Z·cl(P ) −→ Br(M(3,Λ)\Z) −→ Br(U (3,Λ)\π−1(Z)) −→ 0 0 m 1 satisfies that cl(P ) = 0. The proof of Proposition 3.2 works also for M(3,Λ) \ Z, so 0 cl(P ) = 0. We shall see below that x (3.9) codimZ ≥ 2 and codimπ−1(Z) ≥ 2. 1 From this, Br(M(3,Λ)\Z) = Br(M(3,Λ)) and Br(U (3,Λ)\π−1(Z)) = Br(U (3,Λ)) = m 1 m Br(M (3,Λ)). By Corollary 3.4, cl(P ) = 0 generates Br(M(3,Λ)), so Br(M(3,Λ)) = 0 τm+ x and Br(M (3,Λ)) = 0, as required. τm+ To see the codimension estimates (3.9), we work as follows. Let E ∈ Z ⊂ M(3,Λ). So H1(E) 6= 0, i.e. H0(E∗ ⊗K ) 6= 0. Thus there is an exact sequence X (3.10) 0 −→ O −→ E′ = E∗ ⊗K −→ F −→ 0, X for some sheaf F. Note that deg(F) = deg(E′) = 2, and E′ is stable (since E stable =⇒ E∗ stable =⇒ E′ = E∗ ⊗ K stable). Here F must be a rank 2 semistable X sheaf, since any quotient F → Q with µ(Q) < µ(F) = 1, would satisfy that µ(Q) < µ(E′) = 2, violating the stability of E′. In particular, F is a (semistable) bundle, and it 3 is parametrized by an irreducible variety of dimension dimM(2,Λ) = 3(g−1) = 3 (recall that dimM(r,Λ) = (r2−1)(g−1)). Now the bundle E′ in (3.10) is given by anextension inP(H1(F∗)). AsH0(F∗) = 0(bysemistability), wehavethatdimP(H1(F∗)) = −(−2+ 2(1−g))−1 = 3. So the bundles E′ are parametrized by a 6-dimensional variety, and therefore dimZ = 6 and codimZ = 3. Now let us see that dimπ−1(Z) ≤ 7. Let E ∈ Z and F as in (3.10), and note that 1 the determinant of F is fixed. Recalling that dimH1(F∗) = 4, we see that we have to check that 0 dimF +3+dimH (E)−1 ≤ 7, BRAUER GROUP OF MODULI SPACES OF PAIRS 9 whereF isthefamilyofthebundlesF. NowdimH0(E) = dimH1(E)+1 = dimH0(E′)+ 1 ≤ dimH0(F)+2. Hence we only need to show that 0 (3.11) dimF +dimH (F) ≤ 3, i for F ∈ Fi, where F = FFi is the family (suitably stratified) of the possible bundles F. We have the following possibilities: (1) F = L ⊕L , where L ,L arelinebundles ofdegree one, L = det(F)⊗L−1. The 1 2 1 2 2 1 generic such F moves in a 2-dimensional family, and H0(F) = 0. If dimH0(F) 6= 0, then it should be either L = O(p) or L = O(q), p,q ∈ X. In this case F 1 2 moves in a 1-dimensional family, and dimH0(F) ≤ 2, so (3.11) holds. (2) F is a non-trivial extension L → F → L, where L is a line bundle of degree one. As det(F) = L2 is fixed, then there are finitely many possible L. Now 1 dimExt (L,L) = 2, so the bundles F move in a 1-dimensional family. Again dimH0(F) ≤ 2, so (3.11) is satisfied. (3) F is a non-trivial extension L → F → L , where L ,L are non-isomorphic 1 2 1 2 1 line bundles of degree one. As dimExt (L ,L ) = 1, we have that F moves in 2 1 2-dimensional family. If dimH0(F) = 1 then (3.11) holds. Otherwise, it must be L = O(p) and L = O(q), hence F moves in a 1-dimensional family and 1 2 dimH0(F) ≤ 2. So (3.11) holds again. (4) F a rank 2 stable bundle and H0(F) = 0. This is clear, since dimM(2,Λ) = 3. (5) F a rank 2 stable bundle and H0(F) = 1. Then we have an exact sequence O → F → L, where L is a (fixed) line bundle of degree two. As dimH1(L∗) = 3, we have that F moves in a 2-dimensional family and (3.11) holds. (6) F a rank 2 stable bundle, O → F → L, dimH0(L) = 1 and dimH0(F) = 2. The connecting map H0(L) = C → H1(O) is given by multiplication by the extension class in H1(L∗) defining F. To have dimH0(F) = 2, this connecting map must be zero, hence the extension class is in ker(H1(L∗) → H1(O)). This kernel is one-dimensional (since the map is surjective). So the family of such F is zero-dimensional, and (3.11) is satisfied. (7) F a rank 2 stable bundle, O → F → L, dimH0(L) = 2 and dimH0(F) ≥ 2. Now it must be L = K . The connecting map X c : H0(K ) → H1(O) = H0(K )∗ ξ X X is given by multiplication with the extension class ξ in H1(L∗) = H0(K2 )∗ X defining F. So c ∈ Hom(H0(K ),H0(K )∗) = H0(K )∗ ⊗ H0(K )∗ is the ξ X X X X image of ξ under H0(K2 )∗ → H0(K )∗ ⊗ H0(K )∗. But this map is the X X X inclusion H0(KX2 )∗ = Sym2H0(KX)∗ ⊂ N2H0(KX)∗. This means that cξ ∈ Sym2H0(K )∗. X If dimH0(F) = 2, then c is not an isomorphism. The condition det(c ) = 0 ξ ξ gives a 2-dimensional family of ξ ∈ H1(L∗). So the family of such bundles F is one-dimensional and (3.11) is satisfied. If dimH0(F) = 3, then c = 0, and so ξ ξ = 0, which is not possible (since F does not split). Finally, we tackle the case (r,g,d) = (3,2,6). Now U (3,Λ) → M(3,Λ) is a projective m fibration (with fibers P2), so Corollary 3.4 and the exact sequence (3.4) imply that 10 I.BISWAS, M. LOGARES,ANDV. MUN˜OZ Br(U (3,Λ)) = 0. To complete the proof that Br(M (3,Λ)) = 0, it only remains to m τm+ show that codimS ≥ 2. τm+ Consider the family F of strictly semistable bundles E with (E,φ) ∈ S . We stratify τm+ F = FFj, such that dimH0(E) is constant on each Fj. We have to prove that dimF +dimH0(E)−1 ≤ dimM (3,Λ)−2 = 10−2 = 8. j τm+ For E strictly semistable, we have either anexact sequence L → E → F or F → E → L, where L ∈ Jac2X, and F is a semistable bundle of rank 2 and determinant Λ′ = Λ⊗L−1 (which is of degree 4). Both cases are similar, so we assume the first one. There are three possibilities: 1 (1) Suppose that dimHom(F,L) = 0. Then dimExt (F,L) = 2. We stratify Jac2X depending on dimH0(L). For L 6= K , dimH0(L) = 1; for L = K , X X dimH0(L) = 2. So for each stratum F′ ⊂ Jac2X, we have that dimF′ + dimH0(L) ≤ 3. We also stratify the family of rank 2 semistable bundles F, according to dimH0(F). For any such stratum F′′, we have that (3.12) dimF′′ +dimH0(F)−1 ≤ 4. Assuming (3.12), and noting that dimH0(E) ≤ dimH0(L)+dimH0(F), we have that,forthecorrespondingstratumF ,dimF +dimH0(E)−1 ≤ 3+4+2−1 = 8. 0 0 Let us prove (3.12). For F stable, we have that dimF′′ + dimH0(F) − 1 ≤ dimM (2,Λ′) = 4. For F strictly semistable, there is an exact sequence L′ → τm+ F → Λ′⊗L′−1, withL′ ∈ Jac2X. IfL′ isgeneric, thendimH0(L′) = dimH0(Λ′⊗ L′−1) = 1 and dimExt1(Λ′ ⊗ L′−1,L′) = 1. So dimF′′ + dimH0(F) − 1 ≤ 2+2−1 = 3. For L′ = K , Λ′ ⊗L′−1 = K or L′2 = Λ′, we have the bounds X X dimH0(L′) ≤ 2, dimH0(Λ′ ⊗L′−1) ≤ 2 and dimExt1(Λ′ ⊗L′−1,L′) ≤ 2, giving that dimF′′ +dimH0(F)−1 ≤ 1+4−1 = 4. 1 (2) Suppose that dimHom(F,L) = 1. Then dimExt (F,L) = 3. There is an exact sequence Λ ⊗ L−2 → F → L. If dimH0(L) = 1 and dimH0(Λ ⊗ L−2) = 1 then dimH0(E) = 3. This case is covered by Lemma 2.3. Otherwise L = K or X Λ⊗L−2 = K ,sotherearefinitelymanychoicesforL. UsingthatdimH0(E) ≤ 6 X and dimExt1(L,Λ⊗L−2) ≤ 2, we get that, for the corresponding stratum F , it 1 is dimF +dimH0(E)−1 ≤ 1+2+6−1 = 8. 1 1 (3) Suppose that dimHom(F,L) = 2. Then F = L ⊕ L and dimExt (F,L) = 4. The extension is unique, because the group of automorphisms of such F is of dimension 4. Note also that there are finitely many choices for L. So for the corresponding family F , we have dimF +dimH0(E)−1 ≤ 6−1 = 5. 2 2 This completes the proof of the corollary. (cid:3) Remark 3.6. Note that Br(U (r,Λ)) = 0 for (r,g,d) 6= (3,2,even) (use Corollary 3.5 M and Proposition 2.4). Also, if d > r(2g −2), then Br(U (r,Λ)) = 0 for (r,g,d) 6= (2,2,even) (use Corollary m 3.5 and Proposition 2.4). Actually, in the range d > r(2g−2), the proof of Theorem 3.3 shows that Br(U (r,Λ)) = Br(U (r +1,Λ)), for any (r,g,d). m M