Stability of the Picard Bundle^{∗}

I. Biswas

School of Mathematics, Tata Institute of Fundamental Research

Homi Bhabha Road, Mumbai 400005, India indranil@math.tifr.res.in

L. Brambila-Paz

CIMAT, Apdo. Postal 402, C.P. 36240, Guanajuato, México lebp@cimat.mx

T. L. Gómez

School of Mathematics, Tata Institute of Fundamental Research

Homi Bhabha Road, Mumbai 400005, India tomas@math.tifr.res.in

P. E. Newstead

Dept. of Mathematical Science, The University of Liverpool,

Peach St., Liverpool, L69 7ZL, UK newstead@liv.ac.uk

October 20, 2000

∗Mathematics Subject Classification: 14H60. All authors are members of the research group Vector Bundles on Algebraic Curves. The second author acknowledges support from CONACYT grant no. 28492-E. The third author was supported by a postdoctoral fellowship of Ministerio de Educación y Cultura (Spain).

Abstract Let X be a non-singular algebraic curve of genus g ≥ 2, n ≥ 2 an integer, ξ a line bundle over X of degree d > 2n(g − 1) with (n, d) = 1 and M_{ξ} the moduli space of stable bundles of rank n and determinant ξ over X. We prove that the Picard bundle W_{ξ} is stable with respect to the unique polarisation of M_{ξ}.

1 Introduction

Let X be a non-singular projective algebraic curve of genus g ≥ 2 defined over an algebraically closed field k, n and d integers with d > 2n(g − 1) and (n, d) = 1. Let M_{n,d} denote the moduli space of stable bundles of rank n and degree d over X. Then M_{n,d} is a smooth projective variety and there exists a universal bundle U over X × M_{n,d}. Since H^{1}(X, E) = 0 for all E ∈ M_{n,d}, the direct image sheaf p_{2∗}U over M_{n,d} is locally free; the associated vector bundle W is called the Picard bundle over M_{n,d}.

These Picard bundles have been studied for n = 1 in [4, 5] and for general n in [6], where it was proved that, for d > 2ng, the Picard bundle is stable with respect to the generalised theta divisor on M_{n,d}. The argument depends on pulling W back to an open subset of the Jacobian of a spectral curve Y over X and then using the Abel-Jacobi map Y → Pic^{0}(Y).

For n ≥ 2 and ξ a line bundle over X of degree d, one can also consider the moduli space M_{ξ} of stable vector bundles of rank n and determinant ξ over X. Again (given the assumption (n, d) = 1), M_{ξ} is a smooth projective variety and there exists a universal bundle U_{ξ} over X × M_{ξ} and a Picard bundle W_{ξ} over M_{ξ}. However there is no simple means of generalising the previous argument to prove the stability of W_{ξ}. Some useful results on M_{ξ} have been obtained in [2], but stability is proved there only for g = n = 2.

Very recently two of the authors [1] proved the stability of W_{ξ} for g ≥ 3 and n = 2 using the Hecke correspondence of M. S. Narasimhan and S. Ramanan [7, 8]. Extending the technique of [1], we prove here

Theorem. Let X be a non-singular projective algebraic curve of genus g ≥ 2 defined over an algebraically closed field, n and d integers with n ≥ 2, d > 2n(g − 1) and (n, d) = 1 and ξ a line bundle over X of degree d. Then the Picard bundle W_{ξ} over M_{ξ} is stable with respect to the unique polarisation of M_{ξ}.

In connection with this theorem, we recall that PicM_{ξ} ≅ Z [9, 3], so that indeed M_{ξ} has a unique polarisation. We denote the ample generator of PicM_{ξ} by Θ. Note that the condition g ≥ 3 is not needed.

The strategy of the proof is as follows. Given a point x ∈ X and an appropriate vector bundle F of determinant ξ(x), and using the Hecke correspondence, we construct an embedding

ψ_{F,x} : P(F_{x}^{∗}) ⟶ M_{ξ};

we call the pair (P(F_{x}^{∗}), ψ_{F,x}) a Hecke space. We show that there is a homomorphism from the twisted restriction ψ_{F,x}^{∗}W_{ξ}(−j) (for a suitable integer j) to the stable vector bundle Ω_{P(F_{x}^{∗})}(1). Let F be a saturated subsheaf of W of rank r with 0 < r < rkW_{ξ}. We show that x and F can be chosen so that the restriction of F to the corresponding Hecke space is locally free outside some subvariety of codimension at least 2, and furthermore the composition

ψ_{F,x}^{∗}F(−j) ⟶ ψ_{F,x}^{∗}W_{ξ}(−j) ⟶ Ω_{P(F_{x}^{∗})}(1)

is non-zero. A computation of degrees, involving the stability of Ω_{P(F_{x}^{∗})}, now gives the result.

Notations. Given any ample line bundle L over a smooth projective variety Y of dimension m, we recall that the degree of a torsion-free sheaf F with respect to L is defined by

deg F = c_{1}(F).c_{1}(L)^{m−1}[Y].

When PicY ≅ Z and c_{1}(L) is the ample generator of PicY, we can write c_{1}(F) = λ.c_{1}(L) for some integer λ and then

deg F = λN, N = c_{1}(L)^{m}[Y].

When Y = M_{ξ}, we shall always take c_{1}(L) = Θ; when Y is a projective space, we shall always take L = O(1), so that N = 1.

We assume throughout that the hypotheses of the theorem hold.

Acknowledgments. The second author would like to thank Andrei Tyurin for some useful discussions on the subject and the University of Liverpool for its hospitality.

2 Hecke spaces

In this section we recall some facts about the Hecke correspondence of [7, 8]. Full details are contained in [8]; for the convenience of the reader, we include sketches of the proofs of the results of [8] that we need.

Let x ∈ X and let F be a bundle over X of rank n and determinant ξ(x). For each non-zero homomorphism ϕ : F → k_{x} from F to the torsion sheaf k_{x} of length 1 supported at {x}, Kerϕ is a torsion-free sheaf over X. Since X is a non-singular curve, Kerϕ is locally free and can be identified with a vector bundle E of rank n and determinant ξ. We have an exact sequence

0 → E → F → k_{x} → 0.  (1)

Moreover the set of all such homomorphisms ϕ forms (up to scalar multiples) the projective space P(F_{x}^{∗}) (we adopt the notation that P(F_{x}^{∗}) stands for the projective space of lines in F_{x}^{∗}). For fixed F and x, the sequences (1) then form a family

0 → E → p_{1}^{∗}F → O_{{x}×P(F_{x}^{∗})}(1) → 0  (2)

over X × P(F_{x}^{∗}).

If all the bundles E arising in this family are stable, the universal property of M_{ξ} gives us a morphism

ψ_{F,x} : P(F_{x}^{∗}) → M_{ξ}

such that

E ≅ (id × ψ_{F,x})^{∗}U_{ξ} ⊗ p_{2}^{∗}(O(−j))  (3)

for some integer j. We shall call the pair (P(F_{x}^{∗}), ψ_{F,x}) a Hecke space.

Remark. The good Hecke cycles of [8] arise from a more general form of this construction which allows the point x to vary. We shall not need this, nor shall we need the fact that two distinct Hecke spaces have distinct images (compare Theorem 5.13 in [8]).

In order to use this construction, we need to determine conditions on F which will ensure that all the bundles E in the family (2) are stable.

Definition. ([8], Definition 5.1 and Remark 5.2) Let l, m be integers. A vector bundle F over X is (l, m)-stable if, for every proper subbundle G of F,

\frac{deg G + l}{rkG} < \frac{deg F + l − m}{rkF}.

Note that (0, 0)-stability is the same as ordinary stability. Apart from this, we shall use only (0, 1)-stability.

Lemma 1 If the bundle F in the sequence (1) is (0, 1)-stable, then E is stable.

Proof. (See [8], Lemma 5.5.) Let G be a proper subbundle of E which contradicts the stability of E. Then the subbundle of F generated by G contradicts the (0, 1)-stability of F.

Remark. In fact it is easy to verify that the condition of (0, 1)-stability on F is precisely what is needed for ψ_{F,x} to be defined. Thus the Hecke spaces correspond exactly to the pairs (F, x) with F (0, 1)-stable.

Lemma 2 The (0, 1)-stable bundles F form a non-empty open subset of the moduli space of stable bundles of rank n and determinant ξ(x).

Proof. (See [8], Propositions 5.3 and 5.4.) For any stable bundle F ∈ M_{ξ(x)}, the ranks and degrees for the quotients of F which contradict (0, 1)-stability form a finite set. The openness of the subset of (0, 1)-stable bundles follows from the properness of the Quot scheme.

To prove this subset is non-empty, we estimate the dimension of its complement. F fails to be (0, 1)-stable if and only if it has a subbundle G of rank r < n and degree e such that ne ≥ r((d + 1) − 1), i.e. ne ≥ rd. Moreover

h := dim H^{1}(X, Hom(F/G, G)) = r(d + 1) − ne + r(n − r)(g − 1).

The corresponding component of the subvariety of non-(0, 1)-stable bundles in M_{ξ(x)} therefore has dimension at most

(r^{2}(g − 1) + 1) + ((n − r)^{2}(g − 1) + 1) − g + (h − 1)
= (n^{2} − nr + r^{2} − 1)(g − 1) + r(d + 1) − ne.

This is smaller than (n^{2} − 1)(g − 1) provided

r(n − r)(g − 1) > r(d + 1) − ne.

Since ne ≥ rd, this fails only if g = 2, r = n−1 and ne = rd. Since (n, d) = 1, this never happens.

Lemma 3 For any Hecke space, ψ_{F,x} is an isomorphism onto its image.

Proof. This follows from Lemma 5.9 in [8]. We prove just the injectivity (sufficient for our purpose) following the proof of [8], Lemma 5.6.

It is sufficient to prove that, for any sequence (1) with F (0, 1)-stable,

dim H^{0}(X, Hom(E, F)) = 1.

For this, note first that any non-zero homomorphism ϕ : E → F is of maximal rank; otherwise the image of E in F would contradict (0, 1)-stability. By taking determinants we see further that ϕ is an isomorphism away from x.

Now fix y ∈ X, y ≠ x. If ϕ_{1} : E → F, ϕ_{2} : E → F are linearly independent, then there is a non-zero linear combination ϕ of ϕ_{1}, ϕ_{2} which is singular at y. This is a contradiction.

3 Proof of Theorem

Note first that, from (1), we obtain the following commutative diagram of exact sequences:

\begin{aligned}& & 0 & & 0 \\ & & ↑ & & ↑ \\ 0 & ⟶ & k_{x}^{n−1} & ⟶ & F_{x} & ⟶ & k_{x} & ⟶ & 0 \\ & & ↑ & & ↑ & & ∥ \\ 0 & ⟶ & E & ⟶ & F & ⟶ & k_{x} & ⟶ & 0 \\ & & ↑ & & ↑ \\ & & F (−x) & = & F (−x) \\ & & ↑ & & ↑ \\ & & 0 & & 0 \end{aligned}  (4)

Just as the sequences (1) combine to form a family (2), the diagrams (4) combine to form

0  0
↑  ↑
0 ⟶ Ω_{{x}×P(F_{x}^{∗})}(1) ⟶ F_{x} ⊗ O_{{x}×P(F_{x}^{∗})} ⟶ O_{{x}×P(F_{x}^{∗})}(1) ⟶ 0
↑  ↑  ∥
0 ⟶  E  ⟶  p_{1}^{∗}F  ⟶ O_{{x}×P(F_{x}^{∗})}(1) ⟶ 0
↑  ↑
p_{1}^{∗}(F (−x)) =  p_{1}^{∗}(F (−x))
↑  ↑
0  0

(5)

Here the top sequence is the Euler sequence on {x} × P(F_{x}^{∗}), the kernel of which is indeed Ω_{{x}×P(F_{x}^{∗})}(1), where Ω_{{x}×P(F_{x}^{∗})} is the cotangent bundle.

Now suppose F is (0, 1)-stable. Since d > n(2g−2), we have H^{1}(F (−x)) = 0; hence, taking the direct image of (5) by p_{2} and using (3), we obtain the following diagram of exact sequences over P(F_{x}^{∗}):

0  0
↑  ↑
0 ⟶  Ω_{P(F_{x}^{∗})}(1)  ⟶  F_{x} ⊗ O  ⟶ O(1) ⟶ 0
↑  ↑  ∥
0 ⟶ ψ_{F,x}^{∗}W_{ξ}(−j) ⟶ H^{0}(F) ⊗ O ⟶ O(1) ⟶ 0
↑  ↑
H^{0}(F (−x)) ⊗ O = H^{0}(F (−x)) ⊗ O
↑  ↑
0  0

(6)

Now let F be a subsheaf of the Picard bundle W_{ξ} of rank r with 0 < r < rkW_{ξ} and such that the quotient W_{ξ}/F is torsion-free. We must prove that

\frac{deg F}{r} < \frac{deg W_{ξ}}{rkW_{ξ}} .  (7)

Consider the projective bundle

P_{x} := P(U_{ξ}|_{{x}×M_{ξ}})

over M_{ξ}. Each point of P_{x} corresponds to a pair (E, l) with E ∈ M_{ξ} and l ∈ P(E_{x}). We have

P(E_{x}) ≅ P(Hom(E, k_{x})^{∗}) ≅ P(Ext^{1}(k_{x}, E)),

and hence a point of P_{x} also corresponds to a short exact sequence (1). Indeed, after identifying scalar multiples, there is a universal family of nontrivial sequences (1) with E ∈ M_{ξ} parametrised by P_{x}. Let H_{x} be the open subset of P_{x} such that F is (0, 1)-stable and let V denote the open subset of (0, 1)-stable bundles in M_{ξ(x)}. Note that, by Lemma 2, V is non-empty.

There are morphisms

H_{x} ⟶^{q} V ⊂ M_{ξ(x)}
↓^{p}
M_{ξ}

The morphism q is surjective and, by Lemma 1, its fibre can be identified with P(F_{x}^{∗}). In particular H_{x} is an irreducible variety of dimension dim M_{ξ}+n−1. On the other hand, the fibre of H_{x} over E ∈ M_{ξ} is an open subset of P(E_{x}). Since p is the restriction of the projection morphism P_{x} → M_{ξ}, the image of H_{x} in M_{ξ} is a non-empty open subset. Note also that the restriction of p to P(F_{x}^{∗}) is precisely ψ_{F,x}, so, by Lemma 3, P(F_{x}^{∗}) maps isomorphically to its image P (F, x) := ψ_{F,x}(P(F_{x}^{∗})).

Lemma 4 Let x_{1}, …, x_{m} ∈ X. There exists a non-empty open subset U of M_{ξ} such that, if E ∈ U, then

1. F is locally free at E;

2. the homomorphism of fibres F_{E} → W_{ξ,E} is injective;

3. for all x_{i} and for the generic line ℓ in E_{x_{i}}, the corresponding bundle F is (0, 1)-stable and F is locally free at every point of P (F, x_{i}) outside some subvariety of codimension at least 2.

Proof. The first two conditions clearly define non-empty open subsets of M_{ξ}.

For the third condition, let S denote the singular set of F. It is clearly sufficient to prove that, for any fixed x and generic F, either P (F, x) ∩ S is empty or

dim(P (F, x) ∩ S) ≤ n − 3.

One can reduce immediately to the case where S is irreducible. The inverse image S′ = p^{−1}(S) of S in H_{x} is then either empty or an irreducible variety of dimension dim S + n − 1. Note that the fibre of S′ over F can be identified with P (F, x)∩S. If q(S′) is of positive codimension, then certainly P (F, x)∩S is empty for generic F. On the other hand, if q(S′) is dense in V, the generic fibre of the projection of S′ to V is of dimension

dim S + n − 1 − dim V = dim S + n − 1 − dim M_{ξ} ≤ n − 3,

which is exactly what we wanted to prove.

Proof of theorem. Fix points x_{1}, …, x_{m} ∈ X with m > \frac{d}{n} and choose E ∈ M_{ξ} satisfying the conditions of Lemma 4. Let v be a non-zero element of F_{E}. The image s of v in W_{ξ,E} = H^{0}(X, E) is non-zero. If s(x_{i}) = 0 for all i, then s ∈ H^{0}(X, E(−x_{1} − … − x_{m})). But E(−x_{1} − … − x_{m}) has negative degree, which contradicts the stability of E. Hence we can choose i such that s(x_{i}) ≠ 0. By Lemma 4, we can also choose a line ℓ in E_{x_{i}} such that s(x_{i}) ∉ ℓ and the corresponding F has the property that F is locally free on P (F, x_{i}) outside some subvariety of codimension at least 2.

Note that ℓ is just the image of F (−x_{i})_{x_{i}} in E_{x_{i}} in the diagram (4), so we conclude that

s ∉ H^{0}(F (−x_{i})).  (8)

Let F_{1} denote the image of ψ_{F,x_{i}}^{∗}F(−j) in ψ_{F,x_{i}}^{∗}W_{ξ}(−j). By (8) and (6), the induced homomorphism

F_{1} → Ω_{P(F_{x}^{∗})}(1)  (9)

has non-zero image. Now Ω_{P(F_{x}^{∗})} is the cotangent bundle of P(F_{x_{i}}^{∗}), so Ω_{P(F_{x}^{∗})}(1) is stable of degree −1. Hence the image of (9) has degree ≤ −1. Moreover from (6) the kernel of (9) is a subsheaf of a trivial sheaf and therefore has degree ≤ 0. So

deg F_{1} ≤ −1.

On the other hand, F_{1} is isomorphic to ψ_{F,x_{i}}^{∗}F(−j) away from some subvariety of codimension 2; so

deg ψ_{F,x_{i}}^{∗}F(−j) = deg F_{1}.

By (6), the bundle ψ_{F,x_{i}}^{∗}W_{ξ}(−j) has degree −1; so

\frac{deg ψ_{F,x_{i}}^{∗}F(−j)}{r} ≤ −\frac{1}{r} < −\frac{1}{rkW_{ξ}} = \frac{deg ψ_{F,x_{i}}^{∗}W_{ξ}(−j)}{rkW_{ξ}}.

Finally, by Lemma 3, ψ_{F,x_{i}}^{∗}Θ = O(δ) for some positive δ, so

\frac{deg F}{r} = \frac{N}{δ} \frac{deg ψ_{F,x_{i}}^{∗}F}{r} < \frac{N}{δ} \frac{deg ψ_{F,x_{i}}^{∗}W_{ξ}}{rkW_{ξ}} = \frac{deg W_{ξ}}{rkW_{ξ}} .

This completes the proof of (7) and hence of the theorem.

Remark. Let L be a line bundle over X. Under the same hypotheses as for our theorem, the bundle W_{ξ}(L) := p_{2∗}(U_{ξ} ⊗ p_{1}^{∗}(L)) over M_{ξ} is stable. For the proof we just replace E and F in (1) by E ⊗ L and F ⊗ L and argue as before.

References

[1] I. Biswas and T. L. Gómez, Stability of Picard bundle over moduli space of stable vector bundles of rank two over a curve, preprint, 2000.

[2] L. Brambila-Paz, E. Gómez-González and F. Pioli, On Picard bundles over Prym varieties, preprint, 2000.

[3] J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des variétés de modules be fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), 53-94.

[4] L. Ein and R. K. Lazarsfeld, Stability and restrictions of Picard bundles with an application to the normal bundles of elliptic curves, in Complex Projective Geometry (ed. G. Ellingsrud, C. Peskine, G. Sacchiero and S. A. Stromme), London Math. Soc. Lecture Notes Series 179, Cambridge Univ. Press, 1992.

[5] G. Kempf, Towards the inversion of abelian integrals I, Ann. of Math. 110 (1979), 243-273.

[6] Y. Li, Spectral curves, theta divisors and Picard bundles, Internat. J. Math. 2 (1991), 525-550.

[7] M. S. Narasimhan and S. Ramanan, Deformations of the moduli space of vector bundles over an algebraic curve, Ann. of Math. 101 (1975), 391-417.

[8] M. S. Narasimhan and S. Ramanan, Geometry of Hecke cycles I, in C. P. Ramanujan - A Tribute, Studies in Math. No. 8, T.I.F.R., Bombay, 1978.

[9] S. Ramanan, The moduli spaces of vector bundles on an algebraic curve, Math. Ann. 200 (1973), 69-84.