Rationality and Brauer group of a moduli space of framed bundles
aa r X i v : . [ m a t h . AG ] O c t RATIONALITY AND BRAUER GROUP OF A MODULI SPACE OFFRAMED BUNDLES
INDRANIL BISWAS, TOM ´AS L. G ´OMEZ, AND VICENTE MU ˜NOZ
Abstract.
We prove that the moduli spaces of framed bundles over a smooth projectivecurve are rational. We compute the Brauer group of these moduli spaces to be zero undersome assumption on the stability parameter. Introduction
Let X be a compact connected Riemann surface of genus g , with g ≥
2. A framedbundle on X is a pair of the form ( E , φ ), where E is a vector bundle on X , and φ : E x −→ C r is a non–zero C –linear homomorphism, where r is the rank of E . The notion of a(semi)stable vector bundle extends to that for a framed bundle. But the (semi)stabilitycondition depends on a parameter τ ∈ R > . Fix a positive integer r , and also fix aholomorphic line bundle L over X . Also, fix a positive number τ ∈ R . Let M τL ( r ) bethe moduli space of τ –stable framed bundles of rank r and determinant L .In [BGM], we investigated the geometric structure of the variety M τL ( r ). The followingtheorem was proved in [BGM]: Assume that τ ∈ (0 , r − r − ) . Then the isomorphism class of the Riemann surface X is uniquely determined by the isomorphism class of the variety M τL ( r ) . Our aim here is to investigate the rationality properties of the variety M τL ( r ). We provethe following (see Theorem 2.3 and Corollary 3.2): The variety M τL ( r ) is rational.If τ ∈ (0 , r − r − ) , then Br( M τL ( r )) = 0 , where Br( M τL ( r )) is the Brauer group of M τL ( r ) . The rationality of M τL ( r ) is proved by showing that M τL ( r ) is birational to the totalspace of a vector bundle over the moduli space of stable vector bundles E on X togetherwith a line in the fiber of E over a fixed point. The rationality of M τL ( r ) also follows from[Ho1], Example 6.9, taking D to be the point x . Mathematics Subject Classification.
Key words and phrases.
Brauer group, rationality, framed bundle, stable bundle.
The Brauer group of M τL ( r ) is computed by considering the morphism to the usualmoduli space that forgets the framing.2. Rationality of moduli space
Let X be a compact connected Riemann surface of genus g , with g ≥
2. Fix aholomorphic line bundle L over X , and take an integer r >
0. Fix a point x ∈ X . Aframed coherent sheaf over X is a pair of the form ( E , φ ), where E is a coherent sheaf on X of rank r , and φ : E x −→ C r is a non–zero C –linear homomorphism. Let τ > τ – stable (respectively, τ – semistable ) if for all proper subsheaves E ′ ⊂ E ,we have(2.1) deg E ′ − ǫ ( E ′ , φ ) τ < rk E ′ deg E − τ rk E (respectively, deg E ′ − ǫ ( E ′ , φ ) τ ≤ rk E ′ (deg E − τ ) / rk E ), where ǫ ( E ′ , φ ) = (cid:26) φ | E ′ x = 0 , φ | E ′ x = 0 . A framed bundle is a framed coherent sheaf (
E, ϕ ) such that E is locally free.We remark that the framed coherent sheaves considered here are special cases of theobjects considered in [HL], and hence from [HL] we conclude that the moduli space M τL ( r )of τ –stable framed bundles of rank r and determinant L is a smooth quasi–projectivevariety.Let ( E , φ ) be a τ –semistable framed coherent sheaf. We note that if τ <
1, then E isnecessarily torsion–free, because a torsion subsheaf of E will contradict τ –semistability,hence in this case E is locally free. But if τ is large, then E can have torsion. In particular,the natural compactification of M τL ( r ) using τ –semistable framed coherent sheaves couldhave points which are not framed bundles. Lemma 2.1.
There is a dense Zariski open subset (2.2) M τL ( r ) ⊂ M τL ( r ) corresponding to pairs ( E , φ ) such that E is a stable vector bundle of rank r , and φ is anisomorphism.The moduli space M τL ( r ) is irreducible.Proof. From the openness of the stability condition it follows immediately that the locusof framed bundles (
E , φ ) such that E is not stable is a closed subset of the moduli space M τL ( r ) (see [Ma, p. 635, Theorem 2.8(B)] for the openness of the stability condition). Itis easy to check that the locus of framed bundles ( E , φ ) such that φ is not an isomorphismis a closed subset of M τL ( r ). Therefore, M τL ( r ) is a Zariski open subset of M τL ( r ). ODULI SPACES OF FRAMED BUNDLES 3
We will now show that this open subset M τL ( r ) is dense. Let ( E, ϕ ) be a τ –stableframed bundle. The moduli stack of stable vector bundles is dense in the moduli stackof coherent sheaves, and both stacks are irreducible (see, for instance, [Ho2, Appendix]).Therefore we can construct a family { E t } t ∈ T of vector bundles parametrized by an irre-ducible smooth curve T with a base point 0 ∈ T such that the following two conditionshold:(1) E ∼ = E , and(2) the vector bundle E t is stable for all t = 0.Shrinking T if necessary (by taking a nonempty Zariski open subset of T ), we get afamily of frames { φ t } t ∈ T such that φ is the given frame φ , and φ t : E t,x −→ C r is anisomorphism for all t = 0. Since E t is stable, and φ t is an isomorphism, it is easy tocheck that ( E t , φ t ) is τ –stable. Therefore, M τL ( r ) is dense in M τL ( r ).To prove that M τL ( r ) is irreducible, first note that M τL ( r ) is irreducible because themoduli stack of stable vector bundles of fixed rank and determinant is irreducible. Since M τL ( r ) ⊂ M τL ( r ) is dense, it follows that M τL ( r ) is irreducible. (cid:3) Let N P be the moduli space of pairs of the form ( E , ℓ ), where E is a stable vectorbundle on X of rank r with determinant L , and ℓ ⊂ E x is a line. Consider M τL ( r ) defined in (2.2). Let(2.3) β : M τL ( r ) −→ N P be the morphism defined by ( E , φ ) ( E , φ − ( C · e )), where the standard basis of C r is denoted by { e , . . . , e r } . Proposition 2.2.
The variety M τL ( r ) is birational to the total space of a vector bundleover N P .Proof. We will first construct a tautological vector bundle over N P . Let N L ( r ) be themoduli space of stable vector bundles on X of rank r and determinant L . Consider theprojection(2.4) f : N P −→ N L ( r )defined by ( E , ℓ ) −→ E . Let P PGL −→ N L ( r ) be the principal PGL( r, C )–bundlecorresponding to f ; the fiber of P PGL over any E ∈ N L ( r ) is the space of all linearisomorphisms from P ( C r ) (the space of lines in C r ) to P ( E x ) (the space of lines in E x );since the automorphism group of E is the nonzero scalar multiplications (recall that E isstable), the projective space P ( E x ) is canonically defined by the point E of N L ( r ). Let Q ⊂ PGL( r, C )be the maximal parabolic subgroup that fixes the point of P ( C r ) representing the line C · e . The principal PGL( r, C )–bundle f ∗ P PGL −→ N P I. BISWAS, T. L. G ´OMEZ, AND V. MU ˜NOZ has a tautological reduction of structure group e E Q ⊂ f ∗ P PGL to the parabolic subgroup Q ; the fiber of e E Q over any point ( E , ℓ ) ∈ N P is the space ofall linear isomorphisms ρ : P ( C r ) −→ P ( E x )such that ρ ( C · e ) = ℓ . The standard action of GL( r, C ) on C r defines an action of Q on( C · e ) ∗ N C C r . Let(2.5) W := f ∗ P PGL (( C · e ) ∗ ⊗ C r ) −→ N P be the vector bundle over N P associated to the principal PGL( r, C )–bundle f ∗ P PGL forthe above PGL( r, C )–module ( C · e ) ∗ N C C r . The action of Q on ( C · e ) ∗ N C C r fixesId C · e ∈ ( C · e ) ∗ ⊗ C C r = Hom( C · e , C r ) . Therefore, the element Id C · e defines a nonzero section(2.6) σ ∈ H ( N P , W ) , where W is the vector bundle in (2.5). Note that the fiber of W over ( E, ℓ ) is ℓ ∗ ⊗ E x ,and the evaluation of σ at ( E, ℓ ) is Id ℓ .The projective bundle P ( W ) −→ N P parametrizing lines in W is identified withthe pullback f ∗ N P of the projective bundle N P to the total space of N P , where f isconstructed in (2.4). The tautological section N P −→ f ∗ N P of the projection f ∗ N P −→N P coincides with the section given by σ in (2.6).Let U ⊂ N P be some nonempty Zariski open subset such that there exists V ⊂ W | U , a direct summand of the line subbundle of W | U generated by σ . Consider the vectorbundle W := V ∗ ⊗ C C r −→ U .
The total space of W will also be denoted by W . Consider the map β defined in (2.3).Let γ : M τL ( r ) ⊃ β − ( U ) −→ W be the morphism that sends any y := ( E , φ ) ∈ β − ( U ) to the homomorphism V β ( y ) −→ C r defined by v φ ( v ) /λ , where λ ∈ C ∗ − { } satisfies the identity φ ( σ ( β ( y ))) = λ · e .The morphism γ is clearly birational. (cid:3) Theorem 2.3.
The moduli space M τL ( r ) is rational. ODULI SPACES OF FRAMED BUNDLES 5
Proof.
Since any vector bundle is Zariski locally trivial, the total space of a vector bundleof rank n over N P is birational to N P × A n . Therefore, from Proposition 2.2 we concludethat M τL ( r ) is birational to N P × A n , where n = dim M τL ( r ) − dim N P .The variety N P is known to be rational [BY, p. 472, Theorem 6.2]. Hence N P × A n isrational, implying that M τL ( r ) is rational. Now from Lemma 2.1 we infer that M τL ( r ) isrational. (cid:3) Brauer group of moduli of framed bundles
We quickly recall the definition of Brauer group of a variety Z . Using the naturalisomorphism C r ⊗ C r ′ ∼ −→ C rr ′ , we have a homomorphism PGL( r, C ) × PGL( r ′ , C ) −→ PGL( rr ′ , C ). So a principal PGL( r, C )–bundle P and a principal PGL( r ′ , C )–bundle P ′ on Z together produce a principal PGL( rr ′ , C )–bundle on Z , which we will denote by P ⊗ P ′ .The two principal bundles P and P ′ are called equivalent if there are vector bundles V and V ′ on Z such that the principal bundle P ⊗ P ( V ) is isomorphic to P ′ ⊗ P ( V ′ ). Theequivalence classes form a group which is called the Brauer group of Z . The additionoperation is defined by the tensor product, and the inverse is defined to be the dualprojective bundle. The Brauer group of Z will be denoted by Br( Z ).As before, fix r and L . Define τ ( r ) := 1( r − r − . Henceforth, we assume that τ ∈ (0 , τ ( r )) , where τ is the parameter in the definition of a (semi)stable framed bundle. As before, let M τL ( r ) be the moduli space of τ –stable framed bundles of rank r and determinant L .Let N L ( r ) be the moduli space of semistable vector bundles on X of rank r and deter-minant L . As in the previous section, the moduli space of stable vector bundles on X ofrank r and determinant L will be denoted by N L ( r ).If E is a stable vector bundle of rank r and determinant L , then for any nonzerohomomorphism φ : E x −→ C r , the framed bundle ( E , φ ) is τ –stable (see [BGM, Lemma 1.2(ii)]). Also, if ( E , φ ) is any τ –stable framed bundle, then E is semistable [BGM, Lemma 1.2(i)]. Therefore, we havea morphism(3.1) δ : M τL ( r ) −→ N L ( r )defined by ( E , φ ) −→ E . Define(3.2) M τL ( r ) ′ := δ − ( N L ( r )) ⊂ M τL ( r ) , where δ is the morphism in (3.1). From the openness of the stability condition (mentionedin the proof of Lemma 2.1) it follows that M τL ( r ) ′ is a Zariski open subset of M τL ( r ). I. BISWAS, T. L. G ´OMEZ, AND V. MU ˜NOZ
Lemma 3.1.
The Brauer group of the variety M τL ( r ) ′ vanishes.Proof. We noted above that (
E , φ ) is τ –stable if E is stable. Therefore, the morphism δ := δ | M τL ( r ) ′ : M τL ( r ) ′ −→ N L ( r )defines a projective bundle over N L ( r ), where δ is constructed in (3.1); for notationalconvenience, this projective bundle M τL ( r ) ′ will be denoted by P . The homomorphism δ ∗ : Br( N L ( r )) −→ Br( P )is surjective, and the kernel of δ ∗ is generated by the Brauer classcl( P ) ∈ Br( N L ( r ))of the projective bundle P (see [Ga, p. 193]). In other words, we have an exact sequence(3.3) Z · cl( P ) −→ Br( N L ( r )) δ ∗ −→ Br( M τL ( r ) ′ ) −→ . Let P := N L ( r ) × P ( C r ) −→ N L ( r )be the trivial projective bundle over N L ( r ). Consider the projective bundle f : N P −→ N L ( r )in (2.4). Let ( N P ) ∗ −→ N L ( r )be the dual projective bundle; so the fiber of ( N P ) ∗ over any point z ∈ N L ( r ) is the spaceof all hyperplanes in the fiber of N P over z . It is easy to see that(3.4) P = ( N P ) ∗ ⊗ P (the tensor product of two projective bundles was defined at the beginning of this section).Since P is a trivial projective bundle, from (3.4) it follows thatcl( P ) = cl(( N P ) ∗ ) = − cl( N P ) ∈ Br( N L ( r )) . But the Brauer group Br( N L ( r )) is generated by cl( N P ) [BBGN, Proposition 1.2(a)].Hence cl( P ) generates Br( N L ( r )). Now from (3.3) we conclude that Br( M τL ( r ) ′ ) = 0. (cid:3) Corollary 3.2.
The Brauer group of the moduli space M τL ( r ) vanishes.Proof. Since M τL ( r ) ′ is a nonempty Zariski open subset of M τL ( r ), the homomorphismBr( M τL ( r )) −→ Br( M τL ( r ) ′ )induced by the inclusion M τL ( r ) ′ ֒ → M τL ( r ) is injective. Therefore, from Lemma 3.1 itfollows that Br( M τL ( r )) = 0. (cid:3) ODULI SPACES OF FRAMED BUNDLES 7
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