ON A RELATIVE MUMFORD-NEWSTEAD THEOREM SURATNOBASU ABSTRACT. In this paper, we prove a relative version of the classical 6 Mumford-Newsteadtheoremforafamilyofsmoothcurvesdegeneratingto 1 areduciblecurvewithasimplenode. WealsoproveaTorelli-typetheorem 0 byshowingthatcertainmodulispacesoftorsionfreesheavesonareducible 2 curveallowsustorecoverthecurvefromthemodulispace. y a M 6 1. INTRODUCTION 1 LetX beasmooth,projectivecurveofgenusg ≥2overC. Wefixalinebun- ] dleL ofodddegreeover X. LetM bethemodulispaceofrank2,stablevector G X bundlesE such thatdetE ≃L. Itis known that M is asmooth, projective and A X unirationalvariety. Consequently it follows, by [20, Lemma 1], that the Hodge h. numbers h0,p =hp,0=0. Therefore, we have the following Hodge decomposi- at tion: m H3(M ,C)=H1,2⊕H1,2, X [ where α is the complex conjugate of α ∈ H3(M ,C) = H3(M ,R)⊗C and X X 3 H1,2≃H2(M ,Ω1 ). Let pr :H3(M ,C)→H1,2 be the first projection. Since v X MX 1 X 7 H3(MX,R)∩H1,2={0}, we get that the image pr1(H3(MX,Z)) is a fulllattice in 4 H1,2. We associate a complex torus corresponding to the above Hodge struc- 3 ture: 7 H1,2 0 J2(M ):= (1.0.1) . X pr (H3(M ,Z)) 1 1 X 0 It is known as the second intermediate Jacobian of M . We remark that the X 5 complex torus, defined above, varies holomorphically in an analytic family of 1 smooth projective, unirational varieties and is a principally polarised abelian : v variety. It is known that the second Betti number b (M )=1 ([15]). Let ω be i 2 X X theuniqueample,integral,KählerclassonM . Thentheprincipalpolarisation X r on J2(M )isinducedbythefollowingparing: a X (α,β)7→ ωn−3∧α∧β, (1.0.2) Z MX where α,β ∈ H1,2 and n = dim M . We denote this polarisation on J2(M ) C X X by θ′. The theorem of Mumford and Newstead ([12, Theorem in page 1201]) assertsthatthereisanaturalisomorphismφ:J(X)→J2(M )suchthatφ∗(θ′)= X θ,whereJ(X)istheJacobianofthecurveandθisthecanonicalpolarisationon J(X). In[2,Section 5,page625])thereisadetailedproofofthefactφ∗(θ′)=θ. Hence, appealing to the classical Torelli theorem one can recover the curve X fromthemodulispace M . X Let X be a projective curve with exactly two smooth irreducible compo- 0 nents X and X meeting at a simple node p. Fix two rational numbers 0< 1 2 1 2 SURATNOBASU a ,a <1 such that a +a =1 and let χ be an odd integer. Under some nu- 1 2 1 2 merical conditions, Nagaraj and Seshadri construct in [13, Theorem 4.1], the modulispace M(2,(a ,a ),χ) ofrank2,(a ,a )-semistable torsion freesheaves 1 2 1 2 on X withEulercharachteristicχ. Moreover,theyshowthat M(2,(a ,a ),χ)is 0 1 2 theunionoftwosmooth,projectivevarietiesintersectingtransversallyalonga smoothdivisor. Wewillobservethatthereexistsadeterminantmorphismdet: M(2,(a ,a ),χ) → Jχ−(1−g)(X ) where Jχ−(1−g)(X ) is the Jacobian parametris- 1 2 0 0 ing the line bundles with Euler characteristic χ−(1−g) over X (see Propo- 0 sition 6.4 in Appendix). We further observe that the fibres of the morphism det isagaintheunionoftwosmoothprojectivevarietiesintersectingtransver- sally (see Proposition 6.5 in Appendix). Fix ξ∈ Jχ−(1−g). We denote the fibre det−1(ξ) by M . Since M is a singular variety, a priori H3(M ,C) has an 0,ξ 0,ξ 0,ξ intrinsic mixed Hodge structure. Let g be the arithmetic genus of X . Note 0 that g =g +g , where g is the genus of X for i =1,2. Underthe assumption 1 2 i i g >3,i =1,2,wewillshowthatH3(M ,Q)≃Q2g,andthatithasapureHodge i 0,ξ 3,0 0,3 structurewithHodgenumbersh =h =0. ThuswehaveanintermediateJa- cobian J2(M ), as defined earlier, corresponding to the Hodge structure on 0,ξ H3(M ,C)whichisapriorionlyacomplextorusofdimension g. 0,ξ Letπ:X →C beaproper,flatandsurjectivefamilyofcurves,parametrised by a smooth, irreducible curveC. Fix 0∈C. We assume that π is smooth out- side the point 0 and π−1(0)=X , where X is as above, g >3 for i =1,2. Let 0 0 i X be the fibre π−1(t) over t ∈C. Fix a line bundle L over X such that the t restrictions L to X are line bundles with Euler characteristics χ−(1−g) for t t t6=0andL isisomorphictothelinebundleξ. Inthissituation,itisobserved 0 in[13,Lemma7.2]thatthereisafamilyπ′:M →C suchthatthefibreπ′−1(t) L over a point t 6=0 is M , the moduli space of rank 2, semistable locally free t,Lt sheaves with det ≃L over the smooth projective curve X and π′−1(0)=M t t 0,ξ (see Section 6.3.1). We should mention a related work by X Sun [23]. In [23] theauthorconstructsafamilyofrankr fixeddeterminant,semistablebundles over smooth projective curves degenerating to a “fixed determinant” moduli spaceofrankr torsionfreesheavesover X . Thoughhismethodsaredifferent 0 webelieve,inrank2case,therelativemodulispacein[23]coincideswithM . L We consider an analytic disc ∆ around the point 0 and we denote the family π′:π′−1(∆)→∆by{M } . t,Lt t∈∆ Withthesenotationswestateoneofthemainresultsofthispaper: Theorem1.1. (1) There is a holomorphic family {J2(M )} of intermediate Jacobians t,Lt t∈∆ corresponding to the family {M } . In other words, there is a surjec- t,Lt t∈∆ tive,proper,holomorphicsubmersion π :J2(M )−→∆ 2 L such that π−1(t) = J2(M ) ∀ t ∈ ∆∗ := ∆\{0} and π−1(0) = J2(M ). 2 t,Lt 2 0,ξ Further, there exists a relative ample class Θ′ on J2(ML)|∆∗ such that Θ′ =θ′,whereθ′ istheprincipalpolarisationon J2(M ). |J2(M ) t t t,Lt t,Lt ONARELATIVEMUMFORD-NEWSTEADTHEOREM 3 (2) Thereisanisomorphism Φ J0(X) // J2(M ) (1.0.3) ❉❉π❉1❉❉❉❉❉❉"" ∼{{①①①①①①π①2①① L ∆ suchthatΦ∗Θ′ =θ forallt∈∆∗,whereπ :J0(X)→∆istheholo- |π−1(t) t 1 1 morphicfamily{J0(X )} ofJacobiansandθ isthecanonicalpolarisa- t t∈∆ t tionon J0(X ). Inparticular J2(M ) :=π−1(0)isanabelianvariety. t L 0 2 Bytheabovetheoremwededucethefollowing: Corollary1.2. Let X be a projectivecurve withexactlytwosmooth irreducible 0 components X and X meeting at a simple node p. We further assume that 1 2 g >3,i =1,2. Then,thereisanisomorphismJ0(X )≃J2(M ),whereξ∈Jχ(X ). i 0 0,ξ 0 Inparticular, J2(M )isanabelianvariety. 0,ξ Since J0(X )isisomorphicto J0(X )×J0(X ),weobservetheJacobian J0(X ) 0 1 2 0 is independent of the nodal point in X . Hence, the classical Torelli theorem 0 fails for such curves (see [10, Page 6 ]). On the other hand, it is known that under suitable choice of the polarisation on the Jacobian J0(X ), one can re- 0 coverthenormalization X˜ of X ,butnotthecurve X . Inotherwordsonecan 0 0 0 recoverboththecomponentsof X butnotthenodalpoint(see[8,page125]). 0 WeseethatthemodulispaceM ofrank2torsionfreesheavescarriesmore 0,ξ informationthan the Jacobian J(X ). In fact, we show thatwe can actually re- 0 cover the curve X from M byfollowing a strategyof [3]. More precisely, we 0 0,ξ′ willprovethefollowing analogueoftheTorellitheoremforreduciblecurves: Theorem 1.3. Let X ( resp. Y ) be the projective curve with two smooth irre- 0 0 duciblecomponents X (resp. Y ), i =1,2 meeting at a simple node p (resp. q). i i We assume that genus(X )=genus(Y ), for i =1,2, and X ≇X (resp. Y ≇Y ). i i 1 2 1 2 Let M (resp. M ) be the moduli space of rank 2, semistable torsion free 0,ξX0 0,ξY0 sheaves E with detE ≃ξ , ξ ∈ Jχ(X ), on X (resp. on Y ). If M ≃M X0 X0 0 0 0 0,ξX0 0,ξY0 thenwehave X ≃Y . 0 0 Acknowledgements:IamextremelythankfultoProfV.Balajiwhointroduced metothisproblemanddiscussedthisworkwithme. IthankProf. C.SSeshadri forsuggestingawaytodefineacertain”determinant”morphismandforseveral helpfuldiscussions.IthankBNarasimhaCharyforaverycarefulreadingofthe previous draft and suggesting many changes. I have greatly benefited from the discussions with Dr. Ronnie Sebastian. I also thank Prof. D.S Nagaraj, Rohith Varma and Krisanu Dan for many helpful discussions. Finally I wish to thank the referee for being extremely patient with the previous manuscript and gen- erouslysuggestingmany changes. The proofs of the Proposition 6.4 and6.6 are suggestedbythereferee. 2. PRELIMINARIES In this section, we briefly recall the main results in [13] which will be ex- tensively used in the present work. Before proceeding further we will fix the followingnotations: 4 SURATNOBASU 2.0.1. Notation. • Throughoutwe work overthefieldCofcomplex numbers. Weassume thatalltheschemesarereduced,separatedandfinitetypeoverC. • Letp :X ×···×X →X betheith projection,where X isaschemefor i 1 n i i i =1,...,n. Byabuseofnotation,wedenote p∗(E )alsobyE ,whereE i i i i isasheafofO module. Xi • Let X be a projective scheme and E be a vector bundle over X. Then we set hi(E):=dim Hi(X,E). LetS be anotherprojective scheme and C E beacoherentsheafover X×S thenwesetE :=E| , s∈S. s X×s • By cohomology of a scheme X, we mean the singular cohomology of thespace X ,theanalyticspacewithcomplexanalytictopologyasso- ann ciatedto X. E • LetE becoherentsheafover X. WedenotebyE(p):= p thefibreof m E p p Eat p∈X. • Let X be a smooth projective curve and E be a vector bundle over X. ThenwedenoteE⊗O (np)byE(np),wherep∈X isaclosedpointand X n isaninteger. • If Z is a closed subvariety of a smooth variety X, then we denote by Codim(Z,X),thecodimension of Z in X. 2.1. Triples associated to a torsion free sheaves on a reducible nodal curve. Let X be a projective curve of arithmetic genus g with exactly two smooth 0 irreduciblecomponents X andX meetingatasimplenodep. Thearithmetic 1 2 genusg ofsuchacurveis g =g +g ,where g isthegenusof X fori =1,2. 1 2 i i By a torsion free sheaf over X we always mean a coherent O -module of 0 X0 → depth 1. Let C be a category whose objects are triples (F ,F ,A) where F are 1 2 i vector bundles on X , for i =1,2 and A :F (p)→ F (p) is a linear map. Let i 1 2 → (F ,F ,A),(G ,G ,B) ∈C. We say φ : (F ,F ,A) → (G ,G ,B) is a morphism if 1 2 1 2 1 2 1 2 there are morphisms φ :F →G of O -modules for i =1,2 such that the fol- i i i Xi lowingdiagramiscommutative: F (p)φ1⊗k(p//)G (p) (2.1.1) 1 1 A B (cid:15)(cid:15) (cid:15)(cid:15) F (p)φ2⊗k(p//)G (p) 2 2 In [13, Lemma 2.8], it is shown that there is an equivalence of categories → betweenC andthecategoryoftorsionfreesheavesover X . 0 ← Remark 2.1. Similarly, we define another category C whose objects are triples (F ,F ,A) where F are vector bundles over X for i =1,2 and A:F (p)→F (p) 1 2 i i 2 1 is a linear map. The morphism between any two such triples is defined in the same way before. The category of torsion free sheaves is equivalent to the cate- ← → ← goryC (see [13, Remark 2.9]). Now if the triples (F ,F ,A)∈C and (F′,F′,B)∈C 1 2 1 2 correspondthe same torsionfree sheafF, then they are relatedby the following ONARELATIVEMUMFORD-NEWSTEADTHEOREM 5 diagram: i F (p) p // F′(p) (2.1.2) 1 1OO A B (cid:15)(cid:15) F (p) oo F′(p) 2 j 2 p where i : F → F′ (resp. j : F′ → F ) is a morphism of vector bundle which 1 1 2 2 is an isomorphism outside the point p and ker(i ) = ker(A) (resp Im(j ) = p p Im(A))(see [13, Remark 2.5]). F′ is called the Hecke-modification of F for i i i =1,2. 2.1.1. Notion of semistability. Fix an ample line bundle O (1) on X . Let X0 0 c deg(O (1)| )=c , i =1,2, and a = i . Then 0<a ,a <1 and a +a =1. X0 Xi i i c1+c2 1 2 1 2 Wesay a=(a ,a )a polarisationon X . Atorsion freesheafF on X isof rank 1 2 0 0 type(r ,r )ifthegenericrankoftherestrictionsF| arer ,i =1,2. 1 2 Xi i Definition 1. For a torsion free sheaf F of rank type (r ,r ), we define the rank 1 2 r :=a r +a r andtheslopeµ(F):= χ(F),whereχ(F):=h0(F)−h1(F). Atorsion 1 1 2 2 r freesheafF issaidtobesemistable(resp. stable)withrespecttothepolarisation a =(a ,a ) if µ(G)≤µ(F)(resp. <) for all nontrivial proper subsheavesG of F. 1 2 → WedefinetheEulercharacteristicandtheslopeofatriple(F ,F ,A)∈C tobe: 1 2 χ((F ,F ,A)) 1 2 χ((F ,F ,A))=χ(F )+χ(F )−rk(F )andµ(F ,F ,A)= . (2.1.3) 1 2 1 2 2 1 2 r A triple (F ,F ,A) is said to be semistable(resp. stable) if µ(G ,G ,B) ≤ 1 2 1 2 µ(F ,F ,A) for all nontrivial proper subtriples of (F ,F ,A) (for definition of a 1 2 1 2 subtriplesee[13,Definition2.3]). Remark 2.2. If a torsion free sheaf F is associated to a triple (F ,F ,A) then 1 2 χ(F)=χ(F ,F ,A)(see [13, Remark 2.11]). We have already remarked the cat- 1 2 egory of torsion free sheaves is equivalent to the category of triples in a fixed direction.Therefore,atorsionfreesheafF isa=(a ,a )-semistable(resp. stable) 1 2 if and only if the correspondingtriple (F ,F ,A) is a=(a ,a )-semistable (resp. 1 2 1 2 stable). 2.2. Modulispaceofrank2torsionfreesheavesoverareduciblenodalcurve. 2.2.1. EulerCharacteristicboundsforrank2semistablesheaves. Fixaninteger χandapolarisation a=(a ,a )on X suchthata χisnotaninteger. Thenwe 1 2 0 1 havethefollowingEulercharacteristicrestrictions: Lemma2.1. Letχ ,χ betheuniqueintegerssatisfying 1 2 a χ<χ <a χ+1, a χ+1<χ <a χ+2 (2.2.1) 1 1 1 2 2 2 andχ=χ +χ −2. IfF isarank2, a=(a ,a )-semistablesheafthenχ(F )=χ , 1 2 1 2 1 1 → χ(F )=χ or χ(F )=χ +1, χ(F )=χ −1 and rk(A)≥1 where (F ,F ,A)∈C is 2 2 1 1 2 2 1 2 theuniquetriplerepresentingF. MoreoverifF isnon-locallyfreethenχ(F )=χ 1 1 andχ(F )=χ . 2 2 (cid:3) Proof. See[13,Theorem3.1]. 6 SURATNOBASU For the rest of the paper we fix an odd integer χ and a polarization a := (a ,a ) (a <a ) on X such that a χ is not an integer. With these notations, 1 2 1 2 0 1 oneofthemainresultsof[13]isthefollowing: Theorem2.2. ([13, Theorem 4.1]) The moduli space M(2,a,χ) ofisomorphism classes rank 2, (a ,a ) stable torsion free sheaves exists as a reduced,connected, 1 2 projectivescheme. Moreover,ithastwosmooth,irreduciblecomponentsmeeting transversallyalongasmoothdivisorD. 2.3. Fixed determinant moduli space. Let Jχi−(1−gi)(X ) be the Jacobian of i isomorphism classes of line bundles over X with Eulercharacteristic χ −(1− i i g ), i = 1,2 and J := Jχ1−(1−g1)(X )×Jχ2−(1−g2)(X ). In the Appendix we will i 0 1 2 show that there is a well defined determinant morphism det :M(2,a,χ)→ J 0 whose fibresareagaintheunionoftwosooth,projective varietiesintersecting transversallyalongasmoothdivisor (seeProposition6.4). 2.4. Modulispaceoftriples. Fixξ∈J andletdet−1(ξ):=M . Inthissubsec- 0 0,ξ tionwe will discuss adifferentdescriptionofthemodulispaces M(2,a,χ)and M in terms of certain moduli space of triples glued along a certain divisor. 0,ξ Thisdescriptionisgiveninsection5ofthearticle[13]. Thisdescriptionwillbe usefulforthecohomology computationslater. The following facts are well known. For the completeness we shall indicate aproof. Fact 2.1. Let (X,x) be a smooth,projective curve together with a marked point x and(E,0⊂F2E(x)⊂E(x))bea parabolicvectorbundlewithweights0<β < 1 β <1. Supposetheweightssatisfy|β −β |< 1. Thenwehave- 2 1 2 2 (a)E isparabolicsemistableimpliesE isparabolicstable. (b)E isparabolicsemistableimpliesE issemistable. (c)IfE isstablethenanyquasiparabolicstructure(E,0⊂F2E(x)⊂E(x))ispar- abolicsemistablewithrespecttotheweights0<β <β <1. 1 2 Proof. Fromourassumptiononweightswegetthat|β1+β2 −β |< 1 fori =1,2. 2 i 2 SupposeE isstrictlyparabolicsemistable. LetL beaparaboliclinesubbundle ofE. Thenwehave- deg(E) β +β deg(L)= + 1 2 −β . 2 2 i Since |β1+β2 −β |< 1 and deg(L) is an integer this is not possible. This com- 2 i 2 pletes the proof of (a). Let L be a line subbundle of E. The parabolic stability ofE implies deg(E) β +β deg(L)< + 1 2 −β . 2 2 i Therefore,deg(L)< deg(E)±1. Since deg(L)is aninteger theabove inequality 2 2 deg(E) will imply deg(L)≤ . This completes the proof of (b). Let L be a sub- 2 bundleofE. IfL(x)∩F2E(x)6=0thenweassociate theweight β otherwisewe 1 associatetheweightβ . NowasE isstablewehave 2 deg(E) deg(L)< . 2 ONARELATIVEMUMFORD-NEWSTEADTHEOREM 7 Since|β1+β2 −β |< 1 anddeg(L)isanintegerweconcludethat 2 i 2 deg(E) β +β deg(L)< + 1 2 −β . 2 2 i (cid:3) Thiscompletestheproofof(c). Thefollowing resultisprovedin[13] → ← Fact 2.2. Let (F ,F ,A)∈C (resp. (F′,F′,B)∈C) be a rank 2, (a ,a )-semistable 1 2 1 2 1 2 and the Euler characteristic χ(F ), i =1,2, satisfy the inequality 2.2.1(resp. the i inequality2.4.2,thenF (resp. F′)aresemistableover X fori =1,2(see[13,The- i i i orem5.1]). Converselywehavethefollowing: Lemma2.3. Let F be rank 2 semistablebundlesover X andthe Euler charac- i i teristic χ(F ), i =1,2, satisfy the inequalities 2.2.1. Let A :F (p)→F (p) be a i 1 2 → linearmap and rk(A)=2, then (F ,F ,A)∈C is (a ,a )-semistable. Moreover,if 1 2 1 2 F andF arebothstablethen(F ,F ,A)is(a ,a )-semistableifrk(A)≥1. 1 2 1 2 1 2 Proof. Case 1: Let rk(A)= 2 The proof of the statement (1) follows from [6, Lemma 3.1.12 page 39]. Now suppose rk(A)=1. In this case we needboth F i tobestable. Since rk(A)= 1 we get a parabolic structure on F given by 0 ⊂ ker(A)⊂ 1 F (p) and a parabolic structure on F (p) given by 0⊂Im(A)⊂F (p). By Fact 1 2 2 2.1 (c)we conclude thattheabove twoquasi parabolicstructureareparabolic a a stable with respect to the weights 0< 1 < 2 <1. Thus by [13, Theorem 6.1] 2 2 (cid:3) wegetthat(F ,F ,A)issemistable. 1 2 Remark 2.3. The same resultsholdtrue for the triplesin the other directioni.e ifF aresemistableover X , i =1,2 satisfyingtheinequality2.4.2 andrk(A)=2 i i ← thenthetriple(F ,F ,A)∈C is(a ,a )-semistable. Moreover,ifF are stableand 1 2 1 2 i ← rk(A)≥1then(F ,F ,A)∈C is(a ,a )-semistable. 1 2 1 2 (I) Semistable triple of type (I): We say a rank 2, (a ,a )-semistable triple 1 2 → (F ,F ,A)∈C isoftype(I)ifχ(F ),i =1,2,satisfythefollowing inequalities: 1 2 i a χ<χ (F )<a χ+1, a χ+1<χ (F )<a χ+2 (2.4.1) 1 X1 1 1 2 X2 2 2 andrk(A)≥1. (II) Semistable triple of type (II): We say a (a ,a )-semistable triple 1 2 ← (F ,F ,B)∈C isoftype(II)ifχ(F ),i =1,2satisfythefollowing inequalities: 1 1 i a χ+1<χ (F′)<a χ+2, a χ<χ (F′)<a χ+1 (2.4.2) 1 X1 1 1 2 X2 2 2 andrk(B)≥1. Let S be a scheme. We say (F ,F ,A) a family of triples parametrised by 1 2 S if F ’s are locally free sheaves on X ×S, i =1,2 and A :F | →F | is a i i 1 p×S 2 p×S O -modulehomomorphismoflocallyfreesheaves. S Remark2.4. Givenafamilyoftriples(F ,F ,A)parametrisedbyS wecanas- 1 2 sociateafamilyoftorsionfreesheavesF parametrisedbyS i.eacoherentsheaf F on X ×S which is flat over S such that F is torsion free for all s ∈S. The 0 s associationis the following: LetG be the locallyfree subsheafof F | ⊕F | 1 p×S 2 p×S 8 SURATNOBASU F | ⊕F | generatedbythegraph ofthehomomorphismA andL := 1 p×S 2 p×S. Con- S G sidertheexactsequence- 0→F →F ⊕F →L →0. 1 2 S Since,F ⊕F andL arebothflatoverS. HenceF isflatoverS. 1 2 S In[13,Theorem5.3]itisshownthatthereisasmooth,irreducibleprojective variety which has the coarse moduli property for family of semistable triple of type I. We denote this space by M . By the same construction one can 12 constructanothersmooth,irreducible, projective varietywhichhasthecoarse modulipropertyofsemistabletriplesoftype(II). WedenotethisspacebyM . 21 Let D :={[(F ,F ,A)∈M |rk(A)=1}. 1 1 2 12 and D :={[(F′,F′,B)]∈M |rk(B)=1}. 2 1 2 21 Then, by [13, Theorem 6.1] it follows D (resp. D ) is a smooth divisor in 1 2 → M (resp. M ). Now if (F ,F ,A)∈C and rk(A)=1, then by Remark 2.1, we 12 21 1 2 ← get a unique triple (F′,F′,B) ∈C such that rk(B) = 1 and χ(F′) = χ(F )+1, 1 2 1 1 χ(F′)=χ(F )−1. Therefore,thisassociationdefinesanaturalisomorphismbe- 2 2 tween D and D . Let us denote this isomorphism by Ψ and M be the vari- 1 2 0 ety obtained by identifying the closed subschemes D and D via the isomor- 1 2 phism Ψ. Now by Remark 2.4 we get a morphism f :M →M(2,a,χ) (resp. 1 12 f :M →M(2,a,χ))byassociating atriple(F ,F ,A)tothecorrespondingtor- 2 21 1 2 sion free sheaf F. Clearly f and f are compatible with the gluing morphism 1 2 Ψ. Thuswe get a morphism M →M(2,a,χ). This morphism is bijective. Also 0 this morphism induces an isomorphism on the dense open subvariety of M 0 consistingofrank2triples. Thereforeitisabirationalmorphism. Thusby[24, Theorem 2.4] the variety M is isomorphic to the moduli space M(2,a,χ) as 0 the latter space is projective and seminormal, being the union of two smooth projectivevarietyintersectingtransversally,withoutanyonedimensionalcom- ponent. LetS beafinitetypeschemeandχ′ =χ −(1−g ). Givenafamilyoftype(I), i i i (a ,a ) semistable triples(F ,F ,A) parametrisedby S we get two families of 1 2 1 2 linebundles∧2F over X ×S,i =1,2. Thusbytheuniversalpropertyof Jχ′i(X ) i i i wegetamorphism χ′ χ′ det :M →J :=J 1(X )×J 2(X ). 1 12 0 1 2 such that det ((F ,F ,A)) =(∧2F ,∧2F ) for all closed points (F ,F ,A)∈ M . 1 1 2 1 2 1 2 12 Similarly,wegetanothermorphism: det :M →J′ :=Jχ′1+1(X )×Jχ′2−1(X ). 2 21 0 1 2 suchthatdet ((F ,F ,A))=(∧2F ,∧2F )forallclosedpoints(F ,F ,A)∈M . 2 1 2 1 2 1 2 21 Lemma 2.4. The fibres of det are smooth and the fibres of det intersect D i i i transversally,i =1,2. Proof. The group J0(X )×J0(X ) acts on M (resp. M ) by (F ,F ,A)7→(F ⊗ 1 2 21 21 1 2 1 L ,F ⊗F ⊗L ,A) and on J (resp. J′ ) by (M ,M )7→(M ⊗L ,M ×L ) where 1 2 2 2 0 0 1 2 1 1 2 2 (L ,L )∈J0(X )×J0(X ). Themorphism det (resp. det )isclearly compatible 1 2 1 2 1 2 ONARELATIVEMUMFORD-NEWSTEADTHEOREM 9 with the above actions. Thus det (resp. det ) is smooth. As M (resp. M ) 1 2 12 21 and J (resp. J′)aresmooth,thefibresofdet (resp. det )aresmooth. Clearly, 0 0 1 2 thedivisorD (resp. D )isinvariantundertheaboveaction. Therefore,det | 1 2 i Di are smooth, i =1,2. Thus the fibres of det | are also smooth. Clearly, the i Di intersectionofafibreofdet withD isthefibreofdet | . Hencewearedone. i i i Di (cid:3) Fix ξ=(ξ ,ξ )∈ Jχ′1(X )×Jχ′2(X ). Let det−1(ξ):=Mξ and det−1(ξ′):=Mξ′ 1 2 1 2 1 12 2 21 where ξ′ = (ξ(p),ξ(−p)). By Lemma 2.4 the fibre det−1(ξ) (resp. det−1(ξ′)) 1 2 intersects D (resp. D ) transversally. Hence Dξ :=det−1(ξ)∩D and Dξ′ := 1 2 1 1 1 2 det−1(ξ′)∩D . Let M be the closed subvariety of M obtainedby gluing Mξ 2 2 0,ξ 0 12 andMξ′ alongtheclosedsubschemesDξ andDξ′ viatheisomorphismΨ. 21 1 2 Let det be the morphism defined in Proposition 6.4. We can easily show that det−1(ξ), ξ∈ J is isomorphic to the variety M . In the next section we 0 0,ξ willcomputesomeofthecohomology groupsof M . 0,ξ 2.4.1. Notation. Henceforth, we will denote by M , the moduli space of rank 0,ξ 2,(a ,a )-semistablebundleswithdet≃ξanditscomponentsbyM andM . 1 2 12 21 Wealsodenotethesmoothdivisor Dξ in M byD andthesmoothdivisorDξ 1 12 1 2 in M byD . 21 2 Weconcludethissectionbyprovingageometricfactaboutthemodulispace M (resp. M ). 12 21 Lemma2.5. ThemodulispaceM (resp,M )isaunirationalvariety. 12 21 Proof. Toprove M isunirationalwe canassume, aftertensoringbyline bun- 12 dles,itconsistsofalltriples(F ,F ,A),whereF issemistableover X suchthat 1 2 i i deg(F )>2(2g −1)i =1,2. Then,anysuchF canbeobtainedasanextension: i i i 0→O →F →ξ →0, Xi i i whereξ =det(F )fori =1,2. Theexactsequencesofthistypeareclassifiedby i i V :=Ext1(O ,ξ )=H1(X ,ξ∗). LetE betheuniversalextensionover X ×V . ξi Xi i i i i i ξi We denote the restriction E by E . Clearly, Hom(E ,E ) parametrises a i|p×Vξi ip 1 2 family of triplesin the sense we havedefinedfamily of triples andif (F ,F ,A) 1 2 be a triple corresponding to the closed point A ∈ Hom(E ,E ) then F ’s are 1 2 i the extensions of the type described before. Now as F ’s are semistable if we i choose an isomorphism A : F (p) → F (p) then by Lemma 2.2, (F ,F ,A) is 1 2 1 2 semistable. Thus we conclude the set of points W where the corresponding triple is semistable is a nonempty Zariski open set of Hom(E ,E ). Therefore, 1 2 by the coarse moduli property of M , we get a morphism from W to M . 12 12 Clearly the morphism W →M is surjective. Hence, M is a unirational va- 12 12 riety. The same argument shows the moduli space M is also a unirational 21 (cid:3) variety. 3. TOPOLOGY OF M 0,ξ In this section, our main aim is to outline a strategy to compute the coho- mologygroupsofM andcomputeexplicitlythethirdcohomologygroup. We 0,ξ make the following convention: Let X be a topological space. By Hk(X) we meanthecohomology groupsof X withthecoefficientsinQ,k≥0. Whenever 10 SURATNOBASU we obtain anyresults forother coefficients, e.g Z, we will specifically mention it. Suppose X andY bevaritiesover C. Wheneverwe say X →Y a topological fibre bundle, we assume the underlying topology of X and Y to be complex analytictopology. Let Y be a smooth,projective curve of genus g ≥2 and M be the moduli Y Y space of rank 2 semistable bundles with fixed determinant. The cohomology groups of M are quite well studied in the literature. When the determinant Y is odd M is a smooth projective variety of dimension 3g −3 and the coho- Y Y mology groups with integralcoefficients are completely known. Whenthe de- terminantisevenM neednotbesmooth. Infactitisknownthatthesingular Y locusofM ispreciselythecomplementM \Ms ifg ≥3whereMs istheopen Y Y Y Y Y subset consisting of stable bundles (see [14,Theorem1]). In thiscase also the Betti numbers are determined in the work of [5]. We will summarize some of theresultsconcerningthecohomology groupsof M inboththecasesi.eodd Y determinantandevendeterminant: Lemma3.1. (1) Let M be the moduli space of rank 2 semistable bundles Y with odd determinant. Then M is a smooth, projective rational vari- Y ety ([16]) and hence it is simply connected and H3(M ,Z) =0. Fur- Y tor thermore, b (M )=0, b (M )=1, b (M )=2g , where b are the Betti 1 Y 2 Y 3 Y Y i numbers([15]). (2) Let M be the moduli space of rank 2 semistablebundleswith even de- Y terminant. ThenMs isasimplyconnectedvariety([4,Proposition 1.2]). Y Furthermore,we have b (M )=0, b (M )=1 and b (Ms)=2g , where 1 Y 2 Y 3 Y Y b aretheBettinumbers([17],[5,Section3]). i LetM (resp. M′)bethemodulispaceofrank2,semistablebundlesover X 1 1 1 withdet≃ξ (resp. withdet≃ξ (p))andM (resp. M′)bethemodulispaceof 1 1 2 2 rank2,semistable bundesover X withdet≃ξ (resp. det≃ξ (−p))whereξ ’s 2 2 2 i are line bundles of degree d =χ −2(1−g ) for i =1,2 and the integers χ , χ i i i 1 2 satisfytheinequality2.2.1. Sinceχisodd,oneoftheintegerinthepair(d ,d ) 1 2 is odd andthe other is even. We assume that d is odd andd is even. There- 1 2 fore,M andM′ aresmoothprojectivevarieties. LetMs betheopensubvariety 1 2 2 of M consisting of all the isomorphism classes of stable bundles over X and 2 2 M′s be the open subvariety of M′ consisting of all the isomorphism classes of 1 1 stable bundlesover X . Note that M \Ms isprecisely thesingular locus of M 1 2 2 2 if g ≥3andM′\M′s ispreciselythesingularlocusof M′ if g ≥3. 2 1 1 1 1 Let us denote the open subvariety M ×Ms of M ×M by B. We will show 1 2 1 2 thefollowing, Proposition 3.2. There is a surjective morphism p :M →M ×M . Moreover, 12 1 2 p:P→B isatopologicalP3-bundlewhereP:=p−1(B). Proof. Let S be a finite type scheme and (F ,F ,A) be a family of triples 1 2 parametrised by S such that (F ,F ,A ) is (a ,a )-semistable of type (I) for 1s 2s s 1 2 all s ∈ S where F := F | . We also assume ∧2F ≃ ξ , i = 1,2. Then is i X0×s is i by Fact 2.2 F , i = 1,2, are semistable for all s ∈ S. Thus we get a mor- is phism p :M →M ×M . Let ([F ],[F ])∈M ×M . Choose any isomorphism 12 1 2 1 2 1 2 A:F (p)→F (p). Then, by Fact 2.3 (F ,F ,A) is (a ,a )-semistable. Therefore, 1 2 1 2 1 2 p issurjective.