A note on the Picard bundle over a moduli space of vector bundles

Math. Nachr. 279, No. 3, 235 – 241 (2006) / DOI 10.1002/mana.200310358

A note on the Picard bundle over a moduli space of vector bundles

Indranil Biswas∗1 and L. Brambila–Paz2

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India2 CIMAT, Apdo. Postal 402, C.P. 36240, Guanajuato, Gto, Mexico

Received 22 July 2003, revised 1 November 2003, accepted 21 January 2004Published online 9 January 2006

Key words Picard bundle, stability, deformationMSC (2000) 14H60, 14J60

Let M(n, d) be a coprime moduli space of stable vector bundles of rank n ≥ 2 and degree d over a complexirreducible smooth projective curve X of genus g ≥ 2 and Mξ ⊂ M(n, d) a fixed determinant moduli space.Assuming that the degree d is sufficiently large, denote by K the vector bundle over X ×M(n, d) defined bythe kernel of the evaluation map H0(X, E) → Ex, where E ∈ M(n, d) and x ∈ X. We prove that K and itsrestriction Kξ to X × Mξ are stable. The space of all infinitesimal deformations of K over X × M(n, d) isproved to be of dimension 3g and that of Kξ over X ×Mξ of dimension 2g, assuming that g ≥ 3 and if g = 3then n ≥ 4 and if g = 4 then n ≥ 3.

c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Let X be a complex irreducible smooth projective curve of genus g ≥ 2, and let M(n, d) be the moduli space ofstable vector bundles of rank n ≥ 2 and degree d. Let Mξ ⊂ M(n, d) be the moduli space of all vector bundleswith determinant isomorphic to a fixed line bundle ξ.

Let E be a stable vector bundle in M(n, d). If d > n(2g − 1) then E is generated by its global sections andwe have an exact sequence

0 −→ KE −→ OX

⊗

C

H0(X,E) −→ E −→ 0 . (1.1)

Butler in [6, Theorem 1.2] proved that if d > 2gn then KE , the kernel of the evaluation map, is stable. If(n, d) = 1 and E varies over M(n, d), the family {KE} defines a vector bundle K over X × M(n, d). Wedenote by Kξ the restriction to X ×Mξ.

Under some numerical conditions the vector bundles K and Kξ are stable. We prove:

Theorem 2.1 If d > n(n− 1)(g− 1)+ n (respectively, d > 2gn) then the vector bundle K (respectively, Kξ)is stable.

Assuming that g ≥ 3 and if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3 we compute the infinitesimaldeformations of K and Kξ . We prove the following:

Theorem 3.3 The space of all infinitesimal deformations of the vector bundle K over X ×M(n, d), namelyH1(X ×M(n, d), End(K)), is canonically isomorphic to

H1(X ×M(n, d), End(K)) ∼= H1(X, OX)⊕3 .

The infinitesimal deformations of Kξ are

H1(X ×Mξ, End(Kξ)) ∼= H1(X, OX)⊕2 .

∗ Corresponding author: e-mail: [email protected]

c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

236 Biswas and Brambila–Paz: Picard bundles over moduli space

In particular,

dimH1(X ×M(n, d), End(K)) = dimH1(X ×Mξ, End(Kξ)) + g = 3g .

The vector bundle K is not quite the Picard bundle in the usual sense, rather it can be considered as a family ofPicard bundles parametrized byX (see Proposition 2.4). The vector bundle Kξ can also be interpreted as a vectorbundle over the Cartesian product of X with a certain Brill–Noether locus (see [5], [9] for Brill–Noether loci). Itwould be interesting to investigate stability and compute infinitesimal deformations of naturally occurring vectorbundles over the general Brill–Noether loci.

2 Stability

Let X be a complex irreducible smooth projective curve over C of genus g ≥ 2, and let M(n, d) be the modulispace of all stable vector bundles overX of rank n ≥ 2 and degree d. Let Mξ ⊂ M(n, d) be the moduli space ofall vector bundles with determinant isomorphic to a fixed line bundle ξ of degree d. The polarization on M(n, d)defined by a generalized theta line bundle will be denoted by θM. Let H be the polarization on X × M(n, d)defined by the sum of a positive multiple of θM with a polarization on X .

Assume that (n, d) = 1. Under this assumption there is a universal vector bundle. Fix a universal vectorbundle U overX ×M(n, d), and denote by W the Picard bundle over M(n, d). In other words,

W := pM∗U ,where pM is the natural projection to M(n, d).

Assume that d > n(2g − 1). This implies that any vector bundle E over X in M(n, d) is generated by itsglobal sections. Therefore, we have the exact sequence

0 −→ KE −→ OX

⊗

C

H0(X,E) −→ E −→ 0 (2.1)

over X , where the right-hand side homomorphism is the evaluation map.As E runs over M(n, d), this family of vector bundles {KE} defines a vector bundle K over X ×M(n, d)

which fits in the exact sequence

ρ : 0 −→ K −→ p∗MW α−→ U −→ 0 . (2.2)

Over any point (x ,E) ∈ X ×M(n, d), the homomorphism α sends any s ∈ H0(X, E) to s(x).We will denote by θξ and Wξ the restrictions to Mξ of θM and W respectively. Similarly, we will denote by

Hξ , Kξ , Uξ and pMξthe restrictions to X ×Mξ of the corresponding objects on X ×M(n, d).

In [1] it was proved that U and its restriction Uξ are stable. The infinitesimal deformations of U and Uξ werealso computed (see [1, Proposition 3.1]). From [1, Lemma 2.2], [8, Theorem 4.11] and [3] it follows that p∗MWis H-stable.

Theorem 2.1 If d > n(n + 1)(g − 1) + n (respectively, d > 2gn), then the vector bundle K (respectively,Kξ) is H-stable (respectively, Hξ-stable).

Remark 2.2 For the proof of Theorem 2.1 we recall [1, Lemma 2.2] that states that if Y and Z are twoirreducible polarized projective manifolds and Y × Z is equipped with the direct sum polarization, then a vectorbundle V over Y ×Z is stable if for the generic points z ∈ Z and y ∈ Y the restriction VY ×z over Y is semistableand Vy×Z over Z is stable.

We will consider the restriction of K to each factor and use the above criterion.Denote by ρE (respectively, ρx) the restriction of the extension (2.2) toX×{E} (respectively, x×M(n, d)).

So we have

ρE : 0 −→ KE −→ OX

⊗

C

H0(X,E) αE−−−→ E −→ 0 (2.3)

and

ρx : 0 −→ Kx −→ W αx−−→ Ux×M(n,d) −→ 0 . (2.4)

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Math. Nachr. 279, No. 3 (2006) 237

Remark 2.3 Butler in [6, p. 5, Theorem 1.2] proved that if d > 2gn, then KE is stable.

To describe K note that for any point (x ,E) ∈ X ×M(n, d) the restriction of (2.2) gives the following exactsequence

0 −→ H0(X,E(−x)) −→ H0(X,E)α(x×E)−−−−−−→ Ex −→ 0 , (2.5)

where E(−x) := E ⊗OX(−x).Consider the exact sequence of sheaves on X ×X ×M(n, d)

0 −→ q∗23U ⊗ q∗12OX×X(−∆) −→ q∗23U −→ U∆×M(n,d) −→ 0 , (2.6)

where qij , 1 ≤ i < j ≤ 3, is the projection of X ×X ×M(n, d) along the ij-th factor and ∆ ⊂ X ×X is thediagonal divisor. If d > g(2n− 1), then the exact sequence overX×M(n, d) for the direct image of (2.6) usingq13 is

0 −→ q13∗(q∗23U ⊗ q∗12OX×X(−∆)) −→ p∗MW β−→ U −→ 0 . (2.7)

By construction, the surjective homomorphism α in (2.2) coincides with β in (2.7). Consequently, we have anatural isomorphism

K ∼= q13∗(q∗23U ⊗ q∗12OX×X(−∆)) . (2.8)

For any x ∈ X , the restriction of the isomorphism in (2.8) to {x} ×M(n, d) is

Kx∼= pM∗U(−x) ,

where U(−x) := U ⊗ p∗XOX(−x). Indeed, this follows immediately from the fact that the restriction of the linebundle OX×X(−∆) to X × {x} is identified with OX(−x).

Define the morphism

f : X ×M(n, d− n) −→ M(n, d) (2.9)

that sends any (y ,E) ∈ X ×M(n, d− n) to E ⊗OX(y) ∈ M(n, d). For any x ∈ X , let

fx := f |{x}×M(n,d−n) : M(n, d− n) −→ M(n, d)

be the restriction of f ; this map fx is an isomorphism.Consider the vector bundle

U ′ := ((IdX × f)∗U) ⊗ ψ∗12OX×X(−∆)

over X ×X ×M(n, d− n), where ψ12 is the projection of X ×X ×M(n, d− n) to X ×X and f is definedin (2.9). Note that

xU ′ := U ′|X×{x}×M(n,d−n)∼= (IdX × fx)∗U(−x) (2.10)

is a universal vector bundle over X ×M(n, d− n). The isomorphism in (2.10) is constructed using the isomor-phism of OX×X(−∆)|X×{x} with OX(−x).

Let

ψ : X ×X ×M(n, d− n) −→ X ×M(n, d− n) (2.11)

be the projection defined by (y1 , y2 , E) �→ (y2 , E).Proposition 2.4 The vector bundle f∗K over X×M(n, d−n) is canonically identified with the direct image

ψ∗U ′, where f and ψ are defined in (2.9) and (2.11) respectively.The vector bundle f∗

xK over M(n, d−n) is canonically identified with the direct image of the universal vectorbundle xU ′ (defined in (2.10)). In particular, f∗

xK is a Picard bundle.

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P r o o f. The first assertion follows from the definitions U ′ and (2.8). Restricting the isomorphism to{x} ×M(n, d− n) the second assertion is obtained.

In [8, p. 548, Theorem 4.11] Li uses the spectral curves given in [2] to prove the stability of W over M(n, d).Take a spectral covering π : Y → X ; the genus g(Y ) of Y is n2(g − 1) + 1. For δ = d+ n(n− 1)(g − 1), wehave a dominant (rational) map

π∗ : Jδ(Y ) −→ M(n, d) ,

defined by L → π∗L, which is defined over a Zariski open subset T δ ⊂ Jδ(Y ) with codim(Jδ(Y ) \ T δ

) ≥ 2.Assume that δ > 2(g(Y ) − 1); this is equivalent to the condition that d > n(n + 1)(g − 1). Under thisassumption, the pull back (π∗)∗W is identified with the restriction to T δ of the Picard bundle WJ over Jδ(Y ).Now the stability of W over M(n, d) is a consequence of the fact that the Picard bundle WJ over Jδ(Y ) isstable [1, Lemma 2.1]. Note that the stability of WJ follows from the stability of the restriction WJ |φL(Y ) ofWJ to a sufficiently general embedding φL : Y −→ Jδ(Y ) (see [7, Lemma 1.2] and [8, Lemma 2.8]). Actually,WJ |φL(Y ) is precisely the kernel of an evaluation map (see [7]). The stability of the kernel was proved in [7,Lemma 1.2] and [7, Proposition 1.5].

Proposition 2.5 If d > n(n + 1)(g − 1) + n (respectively, d > n(2g − 1)), then the vector bundle Kx

(respectively, (Kξ)x) over M(n, d) is θM-stable (respectively, θξ-stable).

P r o o f. The stability of Kx over M(n, d) follows from Proposition 2.4, and [8, Theorem 4.11]. The stabilityof (Kξ)x is the main result in [3] (see [3, p. 561, Theorem]).

Now we are in a position to complete the proof of Theorem 2.1.

P r o o f o f T h e o r e m 2.1. For any E ∈ M(n, d), the vector bundle KE is stable (see Remark 2.3). ByProposition 2.5 the vector bundles Kx and (Kξ)x are stable. Consequently, Theorem 2.1 follows from [1, Lemma2.2] (see Remark 2.2). �

3 Infinitesimal deformations

We will first compute the infinitesimal deformations of Kξ. We assume that g ≥ 3 and if g = 3 then n ≥ 4 andif g = 4 then n ≥ 3.

Let ad(Kξ) ⊂ End(Kξ) denote the subbundle defined by trace zero endomorphisms, that is, ad(Kξ) is thekernel of the trace map.

Proposition 3.1 There is a natural isomorphism

H1(X ×Mξ, ad(Kξ)) ∼= H1(X, OX) .

In particular, dimH1(X ×Mξ, ad(Kξ)) = g.

P r o o f. We saw in Proposition 2.5 that for any x ∈ X , the restriction of Kξ to {x} ×Mξ is simple. That is,

H0(Mξ, ad(Kξ)|{x}×Mξ

)= 0 .

Therefore, using the Leray spectral sequence for the projection pM we have

H1(X ×Mξ, ad(Kξ)) ∼= H0(X, R1pX∗(ad(Kξ))) , (3.1)

where pX , as before, is the projection to X .Using a result of [4] it follows that

R1pX∗(ad(Kξ)) ∼= OX

⊗

C

H1(X, OX) . (3.2)


Math. Nachr. 279, No. 3 (2006) 239

To explain the isomorphism in (3.2), consider Pic−1(X), the Picard variety of degree −1 line bundles over X .Fix a Poincare line bundle L over X × Pic−1(X) (we will normalize L later). Consider the vector bundle

U ′′ := f∗13U ⊗ f∗

12L

overX × Pic−1(X)×M(n, d), where fij , 1 ≤ i < j ≤ 3, is the projection of X × Pic−1(X)×M(n, d) to theij-th factor. Let

W ′′ := R0f23∗U ′′ (3.3)

be the direct image of U ′′ over Pic−1(X) ×M(n, d). For any ξ ∈ Picd(X), set

W ′′ξ := W ′′|Pic−1(X)×Mξ

.

Let

τ : X −→ Pic−1(X)

be the embedding defined by y → OX(−y). Normalize L such that the line bundle (IdX × τ)∗L over X ×X isisomorphic to OX×X(−∆).

The isomorphism in (2.8) gives an isomorphism

K ∼= (τ × IdM(n,d))∗W ′′ ,

where W ′′ is defined in (3.3). Restricting this isomorphism to X ×Mξ we have

Kξ∼= (τ × IdMξ

)∗W ′′ξ (3.4)

for any ξ ∈ Picd(X).Consider W ′′

ξ as a family of vector bundles over Mξ parametrized by Pic−1(X). Let

ν : T Pic−1(X) −→ R1q∗ad(W ′′

ξ

)(3.5)

be the infinitesimal deformation map for this family of vector bundles, where q is the projection of Pic−1(X) ×Mξ to Pic−1(X) and T Pic−1(X) is the tangent bundle. From [4, Theorem 6.3] we know that the abovehomomorphism ν is an isomorphism (set E in [4, Theorem 6.3] to be the trivial line bundle and note thatH1(Mξ, OMξ

) = 0 in order to conclude from [4, Theorem 6.3] that ν is an isomorphism).Note that the infinitesimal deformation map commutes with base change. Since ν in (3.5) is an isomorphism,

using the isomorphism in (3.4) together with the standard fact

T Pic−1(X) ∼= OPic−1(X)

⊗

C

H1(X, OX)

we get an isomorphism as in (3.2).In other words, R1pX∗(ad(Kξ)) is identified with the trivial vector bundle over X with fiber H1(X, OX).

Finally, (3.1) and (3.2) together complete the proof of the proposition.

Proposition 3.2 There is a natural isomorphism

H1(X ×M(n, d), ad(K)) ∼= H1(X, OX) .

In particular, dimH1(X ×M(n, d), ad(K)) = g.

P r o o f. Let

γ : X ×M(n, d) −→ Picd(X)

be the projection defined by (x ,E) → ∧nE.



Since for any L ∈ Picd(X), where d > 2gn, the restriction of K to γ−1(L) is simple (Theorem 2.1), we have

R0γ∗(ad(K)) = 0 .

The isomorphism in (3.2) gives that R1γ∗(ad(K)) is identified with the trivial vector bundle over Picd(X)with fiber H1(X, OX).

Now using the Leray spectral sequence for γ we have

H1(X ×M(n, d), ad(K)) ∼= H1(X, OX)

as H0(Picd(X), OPicd(X)

) ∼= C. This completes the proof of the proposition.

Note that

H1(Picd(X), OPicd(X)

) ∼= H1(X, OX) (3.6)

with the isomorphism induced by the embedding X ↪→ Picd(X) defined by x → ζ ⊗ OX(x), where ζ is somefixed line bundle of degree d− 1 (the pull-back homomorphism of cohomologies does not depend on the choiceof ζ). Hence

H1(X ×M(n, d), OX×M(n,d)) ∼= H1(X, OX) ⊕H1(M(n, d), OM(n,d)) ∼= H1(X, OX)⊕2 ; (3.7)

the isomorphism in (3.6) and the isomorphism

H1(Picd(X), OPicd(X)

) ∼= H1(M(n, d), OM(n,d)

),

induced by the projection defined by E → ∧nE, together give the above isomorphism.

From Proposition 3.2 and Proposition 3.1 we have

Theorem 3.3 The space of infinitesimal deformations of the vector bundle K over X × M(n, d), namelyH1(X ×M(n, d), End(K)), is canonically isomorphic to

H1(X ×M(n, d), End(K)) ∼= H1(X, OX)⊕3

and for Kξ

H1(X ×Mξ, End(Kξ)) ∼= H1(X, OX)⊕2 .

In particular,

dimH1(X ×M(n, d), End(K)) = 3g

and

dimH1(X ×Mξ, End(Kξ)) = 2g .

P r o o f. Since H1(Mξ, OMξ) = 0, we have H1(X ×Mξ, OX×Mξ

) = H1(X, OX). We have

H1(X ×Mξ, End(Kξ)) ∼= H1(X ×Mξ, ad(Kξ)) ⊕H1(X ×Mξ, OX×Mξ) .

Hence from Proposition 3.1 we have that there is a natural isomorphism

H1(X ×Mξ, End(Kξ)) ∼= H1(X, OX) ⊕H1(X, OX) .

Similarly the natural isomorphism

H1(X ×M, End(K)) ∼= H1(X, OX)⊕3

is an immediate consequence of Proposition 3.2 and (3.7).


Math. Nachr. 279, No. 3 (2006) 241

Denote by U(L) the vector bundle p∗12U ⊗ p∗13L over X × Mξ × Pic0(X) and by W(L) the direct imagep23∗U(L) over Mξ ×Pic0(X), where Pic0(X) is the Jacobian of X , L a Poincare bundle overX×Pic0(X) andpij the projection of X ×Mξ × Pic0(X) to the ij-th factor.

Over X ×Mξ × Pic0(X) we have the exact sequence

0 −→ K(L) −→ p∗23W(L)γ−→ U(L) −→ 0 ,

where K(L) is the kernel of the (surjective) evaluation map γ.

Let K(L) be the bundle

K(L) := p∗123K(L) ⊗ p∗14L

over X ×Mξ × Pic0(X) × Pic0(X).The restriction of K(L) to X ×Mξ ×{OX}× {OX}, is precisely Kξ. The family K(L) is complete, and the

isomorphism in Theorem 3.3 can be interpreted as the infinitesimal deformation map for this family.Similarly, all the deformations of K over X × M arise from deformations of the universal bundle U and

tensoring by (pull-back of) line bundles of degree zero over M(n, d) and over X .

Acknowledgements We are very grateful to the referee for very detailed comments to improve the manuscript. We thankthe International Centre for Theoretical Physics, Trieste, where the work was carried out, for hospitality. The first authorthanks I.M.U. for a travel support and the second author acknowledges the support of CONACYT grant 40815-F. The authorsare members of the research group VBAC (Vector Bundles on Algebraic Curves), which is partially supported by EAGER(EC FP5 Contract No. HPRN-CT-2000-00099) and by EDGE (EC FP5 Contract No. HPRN-CT-2000-00101).

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A note on the Picard bundle over a moduli space of vector bundles