On A Relative Mumford-Newstead Theorem

0.35 MB·English

by  Suratno Basu

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.