We prove that the moduli spaces of framed bundles over a smooth projective
curve are rational. We compute the Brauer group of these moduli spaces to be
zero under some assumption on the stability parameter.Comment: 7 pages, to appear in Tbilisi Math. J; v2. reference adde

Biswas, I.

Gómez, T.

Muñoz, V.

Tbilisi Mathematical Journal

English

arXiv

We prove that the moduli spaces of framed bundles over a smooth projective curve are rational. We compute the Brauer group of these moduli spaces to be zero under some assumption on the stability parameter.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu

Biswas, Indranil

Gómez , Tomás L.

Muñoz, Vicente

Docta Complutense

arXiv:1110.0384v2  [math.AG]  5 Oct 2011RATIONALITY AND BRAUER GROUP OF A MODULI SPACE OFFRAMED BUNDLESINDRANIL BISWAS, TOMÁS L. GÓMEZ, AND VICENTE MUÑOZAbstract. 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.1. IntroductionLet 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 , andφ : Ex0 −→ Cris 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>0. Fix a positive integer r, and also fix aholomorphic line bundle L over X . Also, fix a positive number τ ∈ R. Let MτL(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 MτL(r). The followingtheorem was proved in [BGM]:Assume that τ ∈ (0 , 1(r−1)!(r−1)). Then the isomorphism class of the Riemann surfaceX is uniquely determined by the isomorphism class of the variety MτL(r).Our aim here is to investigate the rationality properties of the variety MτL(r). We provethe following (see Theorem 2.3 and Corollary 3.2):The variety MτL(r) is rational.If τ ∈ (0 , 1(r−1)!(r−1)), thenBr(MτL(r)) = 0 ,where Br(MτL(r)) is the Brauer group of MτL(r).The rationality of MτL(r) is proved by showing that MτL(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 MτL(r) also follows from[Ho1], Example 6.9, taking D to be the point x0.2000 Mathematics Subject Classification. 14H60, 14F05.Key words and phrases. Brauer group, rationality, framed bundle, stable bundle.12 I. BISWAS, T. L. GÓMEZ, AND V. MUÑOZThe Brauer group of MτL(r) is computed by considering the morphism to the usualmoduli space that forgets the framing.2. Rationality of moduli spaceLet 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 x0 ∈ X . Aframed coherent sheaf over X is a pair of the form (E , φ), where E is a coherent sheaf onX of rank r, andφ : Ex0 −→ Cris a non–zero C–linear homomorphism. Let τ > 0 be a real number. A framed coherentsheaf is called τ–stable (respectively, τ–semistable) if for all proper subsheaves E ′ ⊂ E,we have(2.1) degE ′ − ǫ(E ′, φ)τ < rkE ′degE − τrkE(respectively, degE ′ − ǫ(E ′, φ)τ ≤ rkE ′(degE − τ)/rkE), whereǫ(E ′, φ) ={1 if φ|E′x0 6= 0,0 if φ|E′x0 = 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 spaceMτL(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 MτL(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) MτL(r)0 ⊂ MτL(r)corresponding to pairs (E , φ) such that E is a stable vector bundle of rank r, and φ is anisomorphism.The moduli space MτL(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 spaceMτL(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 MτL(r). Therefore, MτL(r)0 is a Zariski open subset of MτL(r).MODULI SPACES OF FRAMED BUNDLES 3We will now show that this open subset MτL(r)0 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 {Et}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) E0 ∼= E, and(2) the vector bundle Et is stable for all t 6= 0.Shrinking T if necessary (by taking a nonempty Zariski open subset of T ), we get afamily of frames {φt}t∈T such that φ0 is the given frame φ, and φt : Et,x0 −→ Cr is anisomorphism for all t 6= 0. Since Et is stable, and φt is an isomorphism, it is easy tocheck that (Et, φt) is τ–stable. Therefore, MτL(r)0 is dense in MτL(r).To prove that MτL(r) is irreducible, first note that MτL(r)0 is irreducible because themoduli stack of stable vector bundles of fixed rank and determinant is irreducible. SinceMτL(r)0 ⊂ MτL(r) is dense, it follows that MτL(r) is irreducible. Let NP 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 ⊂ Ex0 is a line. Consider MτL(r)0defined in (2.2). Let(2.3) β : MτL(r)0 −→ NPbe the morphism defined by (E , φ) 7−→ (E , φ−1(C · e1)), where the standard basis of Cris denoted by {e1 , . . . , er}.Proposition 2.2. The variety MτL(r)0 is birational to the total space of a vector bundleover NP .Proof. We will first construct a tautological vector bundle over NP . Let NL(r) be themoduli space of stable vector bundles on X of rank r and determinant L. Consider theprojection(2.4) f : NP −→ NL(r)defined by (E , ℓ) −→ E. Let PPGL −→ NL(r) be the principal PGL(r,C)–bundlecorresponding to f ; the fiber of PPGL over any E ∈ NL(r) is the space of all linearisomorphisms from P (Cr) (the space of lines in Cr) to P (Ex0) (the space of lines in Ex0);since the automorphism group of E is the nonzero scalar multiplications (recall that E isstable), the projective space P (Ex0) is canonically defined by the point E of NL(r). LetQ ⊂ PGL(r,C)be the maximal parabolic subgroup that fixes the point of P (Cr) representing the lineC · e1. The principal PGL(r,C)–bundlef ∗PPGL −→ NP4 I. BISWAS, T. L. GÓMEZ, AND V. MUÑOZhas a tautological reduction of structure groupẼQ ⊂ f∗PPGLto the parabolic subgroup Q; the fiber of ẼQ over any point (E , ℓ) ∈ NP is the space ofall linear isomorphismsρ : P (Cr) −→ P (Ex0)such that ρ(C · e1) = ℓ. The standard action of GL(r,C) on Cr defines an action of Q on(C · e1)∗⊗CCr. Let(2.5) W := f ∗PPGL((C · e1)∗ ⊗ Cr) −→ NPbe the vector bundle over NP associated to the principal PGL(r,C)–bundle f∗PPGL forthe above PGL(r,C)–module (C · e1)∗⊗CCr. The action of Q on (C · e1)∗⊗CCr fixesIdC·e1 ∈ (C · e1)∗ ⊗C Cr = Hom(C · e1 ,Cr) .Therefore, the element IdC·e1 defines a nonzero section(2.6) σ ∈ H0(NP , W ) ,where W is the vector bundle in (2.5). Note that the fiber of W over (E, ℓ) is ℓ∗ ⊗ Ex0,and the evaluation of σ at (E, ℓ) is Idℓ.The projective bundle P (W ) −→ NP parametrizing lines in W is identified withthe pullback f ∗NP of the projective bundle NP to the total space of NP , where f isconstructed in (2.4). The tautological section NP −→ f ∗NP of the projection f ∗NP −→NP coincides with the section given by σ in (2.6).Let U ⊂ NP be some nonempty Zariski open subset such that there existsV ⊂ W |U ,a direct summand of the line subbundle of W |U generated by σ. Consider the vectorbundleW := V ∗ ⊗C Cr −→ U .The total space of W will also be denoted by W. Consider the map β defined in (2.3).Letγ : MτL(r)0 ⊃ β−1(U) −→ Wbe the morphism that sends any y := (E , φ) ∈ β−1(U) to the homomorphismVβ(y) −→ Crdefined by v 7−→ φ(v)/λ, where λ ∈ C∗ − {0} satisfies the identity φ(σ(β(y))) = λ · e1.The morphism γ is clearly birational. Theorem 2.3. The moduli space MτL(r) is rational.MODULI SPACES OF FRAMED BUNDLES 5Proof. Since any vector bundle is Zariski locally trivial, the total space of a vector bundleof rank n over NP is birational to NP ×An. Therefore, from Proposition 2.2 we concludethat MτL(r)0 is birational to NP × An, where n = dimMτL(r)0 − dimNP .The variety NP is known to be rational [BY, p. 472, Theorem 6.2]. Hence NP × An isrational, implying that MτL(r)0 is rational. Now from Lemma 2.1 we infer that MτL(r) isrational. 3. Brauer group of moduli of framed bundlesWe quickly recall the definition of Brauer group of a variety Z. Using the naturalisomorphism Cr ⊗ Cr′ ∼−→ Crr′, 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′ onZ 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 Vand 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. Defineτ(r) :=1(r − 1)!(r − 1).Henceforth, we assume thatτ ∈ (0 , τ(r)) ,where τ is the parameter in the definition of a (semi)stable framed bundle. As before, letMτL(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 NL(r).If E is a stable vector bundle of rank r and determinant L, then for any nonzerohomomorphismφ : Ex0 −→ Cr ,the framed bundle (E , φ) is τ–stable (see [BGM, Lemma 1.2(ii)]). Also, if (E , φ) is anyτ–stable framed bundle, then E is semistable [BGM, Lemma 1.2(i)]. Therefore, we havea morphism(3.1) δ : MτL(r) −→ N L(r)defined by (E , φ) −→ E. Define(3.2) MτL(r)′ := δ−1(NL(r)) ⊂ MτL(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 MτL(r)′ is a Zariski open subset of MτL(r).6 I. BISWAS, T. L. GÓMEZ, AND V. MUÑOZLemma 3.1. The Brauer group of the variety MτL(r)′ vanishes.Proof. We noted above that (E , φ) is τ–stable if E is stable. Therefore, the morphismδ1 := δ|MτL(r)′ : MτL(r)′ −→ NL(r)defines a projective bundle over NL(r), where δ is constructed in (3.1); for notationalconvenience, this projective bundle MτL(r)′ will be denoted by P. The homomorphismδ∗1 : Br(NL(r)) −→ Br(P)is surjective, and the kernel of δ∗1 is generated by the Brauer classcl(P) ∈ Br(NL(r))of the projective bundle P (see [Ga, p. 193]). In other words, we have an exact sequence(3.3) Z · cl(P) −→ Br(NL(r))δ∗1−→ Br(MτL(r)′) −→ 0 .LetP := NL(r)× P (Cr) −→ NL(r)be the trivial projective bundle over NL(r). Consider the projective bundlef : NP −→ NL(r)in (2.4). Let(NP )∗ −→ NL(r)be the dual projective bundle; so the fiber of (NP )∗ over any point z ∈ NL(r) is the spaceof all hyperplanes in the fiber of NP over z. It is easy to see that(3.4) P = (NP )∗ ⊗ 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((NP )∗) = −cl(NP ) ∈ Br(NL(r)) .But the Brauer group Br(NL(r)) is generated by cl(NP ) [BBGN, Proposition 1.2(a)].Hence cl(P) generates Br(NL(r)). Now from (3.3) we conclude that Br(MτL(r)′) = 0. Corollary 3.2. The Brauer group of the moduli space MτL(r) vanishes.Proof. Since MτL(r)′ is a nonempty Zariski open subset of MτL(r), the homomorphismBr(MτL(r)) −→ Br(MτL(r)′)induced by the inclusion MτL(r)′ →֒ MτL(r) is injective. Therefore, from Lemma 3.1 itfollows that Br(MτL(r)) = 0. MODULI SPACES OF FRAMED BUNDLES 7References[BBGN] V. Balaji, I. Biswas, O. Gabber and D. S. Nagaraj, Brauer obstruction for a universal vectorbundle. Comp. Rend. Acad. Sci. Paris 345 (2007), 265–268.[BGM] I. Biswas, T. Gómez and V. Muñoz, Torelli theorem for the moduli space of framed bundles,Math. Proc. Camb. Phil. Soc. 148 (2010), 409–423.[BY] H. U. Boden and K. Yokogawa, Rationality of moduli spaces of parabolic bundles, Jour. LondonMath. Soc. 59 (1999), 461–478.[Ga] O. Gabber, Some theorems on Azumaya algebras, in: The Brauer Group, pp. 129–209, LectureNotes in Math., Vol. 844, Springer, Berlin–New York, 1981.[Ho1] N. Hoffmann, Rationality and Poincaré families for vector bundles with extra structure on acurve, Int. Math. Res. Not. 2007, no. 3, Art. ID rnm010, 30 pp.[Ho2] N. Hoffmann, Moduli stacks of vector bundles on curves and the King-Schofield rationalityproof, in: Cohomological and geometric approaches to rationality problems, pp. 133–148, Progr.Math., 282, Birkhäuser Boston, Inc., Boston, MA, 2010.[HL] D. Huybrechts and M. Lehn, Framed modules and their moduli, Int. Jour. Math. 6 (1995),297–324.[Ma] M. Maruyama, Openness of a family of torsion free sheaves, Jour. Math. Kyoto Univ. 16 (1976),627–637.School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,Bombay 400005, IndiaE-mail address : indranil@math.tifr.res.inInstituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera 15, Cam-pus Cantoblanco UAM, 28049 Madrid, SpainE-mail address : tomas.gomez@icmat.esFacultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza Cien-cias 3, 28040 Madrid, SpainE-mail address : vicente.munoz@mat.ucm.es

Rationality and Brauer group of a moduli space of framed bundles.

Rationality and Brauer group of a moduli space of framed bundles

Rationality and Brauer group of a moduli space of framed bundles

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