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On Picard bundles over Prym varieties

We study the birationality (onto its image) of the Abel-Prym morphism associated with a Prym-Tuyrin variety. We use such result to prove that Picard bundles over Prym varieties are simple and moreover they are stable when the Abel-Prym morphism is not birational. As a consequence we obtain that Picard bundles over moduli spaces of stable vector bundles with fixed determinant are simple. We prove that Picard bundles over moduli spaces of rank 2 vector bundles on curves of genus 2 bundles are stable.


Introduction
Let X be a non-singular irreducible projective curve of genus g X ≥ 2. We denote by M(n, d) the moduli space of stable vector bundles over X of rank n and degree d. For n = 1, M(1, d) is the Jacobian J d (X). Ein and Lazarsfeld in [EL] proved the stability of the Picard bundle W J when d > 2g X − 1 and Kempf (see [Ke]) for d = 2g X − 1. For n ≥ 2, Li proved the stability of W over M(n, d) when d > 2g X n (see [Li]). Denote by W ξ the restriction of W to the subvariety M ξ ⊂ M(n, d) determined by stable bundles with fixed determinant ξ ∈ J d (X). Balaji and Vishwanath proved in [BV] that for rank 2, W ξ is simple. Actually, they compute the deformations of W ξ . In this paper we prove (see Theorem 3.1) that for any rank W ξ is simple when g X ≥ 2 and d > n(n + 1)(g X − 1) + 6 and in the case g X = 2 and n = 2 then we prove that W ξ is stable for d > 10.
The result follows from studying the restriction of the Picard bundle over the Jacobian of a curve (which is a covering space of X) to some natural subvarieties,namely the Prym variety and the Jacobian of the base space. That is, let π : Y → X be a covering between non singular irreducible projective curves. Let (J(Y ), Θ Y ) be the principally polarized Jacobian of degree zero of Y and P Y the Poincaré bundle over Y × J(Y ).
We denote the restriction of W J to the subvarieties P and π * (J(X)) of J(Y ) by W P and W π respectively.
We deduce that W π is stable with respect to the polarization Θ Y | π * (J(X)) .
The restriction Θ P of Θ Y to P need not be a multiple of a principal polarization.
However, it is possible to construct (see [We]) a principally polarized abelian variety ( P , Ξ) and an isogeny f : P → P with f −1 (Θ P ) ≡ nΞ such that there exists a map ρ : Y → P with ρ(Y ) and Ξ m−1 numerically equivalent where m = dim P = dim P .
We prove that if n = 2 and g X > 2, the morphism ρ : Y → P is birational onto its image (Theorem 1.2) and we use some properties of the Fourier-Mukai transform to prove that in this case W P is simple.
Moreover, we give explicitly the cases where ρ may not be birational. In this case, we prove that the Prym variety is the image in J(Y ) of the Jacobian of the normalization of the curve ρ(Y ). As a consequence of stability of W π , we have that the Picard bundle W P is stable with respect to Θ Y | P when ρ is not birational. In particular, for g X ≥ 2 and d > 2g Y + 4, W P is simple (Theorem 2.9).
Acknowledgments : We thank Laura Hidalgo for previous discussions and G.P. Pirola for useful suggestions concerning Theorem 1.2. The first and the third named authors acknowledge the generous hospitality of the Departamento de Matemáticas of the University of Salamanca. The third author wishes to thank the Centro de Investigación en Matemáticas (CIMAT), Guanajuato for the warm hospitality and support.

Prym varieties
We shall denote by J(X) the Jacobian of degree zero of a non-singular projective irreducible curve X and by Θ X the natural polarization on J(X) given by the Riemann theta divisor.
Let π : Y → X be a covering of degree n between non-singular irreducible projective curves of genus g Y and g X respectively. We have the norm map Nm π : J(Y ) → J(X) and the pull back map π * : J(X) → J(Y ). It is known that the map h : J(X) → π * (J(X)) induced by π * is an isogeny such that h −1 Θ Y | π * (J(X)) ≡ nΘ X . We will denote by Θ π the restriction of the theta divisor of J(Y ) to π * (J(X)).
The Prym variety P associated with the covering is the abelian subvariety of J(Y ) defined by P = Im(n Id J(Y ) −π * •Nm π). We shall denote by µ the morphism (n Id J(Y ) −π * • Nm π) : J(Y ) −→ P .
On P there exists a natural polarization Θ P given by the restriction of the theta divisor Θ Y to P . In general Θ P is not a multiple of principal polarization (see [LB], Theorem 3.3, pp. 376). The following theorem gathers several standard results about Prym-Tyurin varieties for which we refer to [LB], [We] and [Go].
Theorem 1.1. Let π : Y → X be a covering of degree n. There exists a principally polarized abelian variety ( P , Ξ), an isogeny f : P → P and a morphism µ : J(Y ) → P such that: 2. The map f • µ : J(Y ) → P coincides with the morphism µ defined as above.
3. The restriction µ| P is surjective and the composition map f • µ| P coincides with the multiplication by n on P .
Moreover, the following relation holds where ≡ denotes algebraic equivalence between divisors.
From now on we shall consider a fixed isogeny f : P → P enjoying the properties listed in Theorem 1.1. Let y 0 be a fixed point on Y . Let ρ y 0 : Y → P and ρ y 0 : Y → P be the morphisms is the Abel-Jacobi map defined by y → O(y − y 0 ). We shall refer to ρ y 0 and ρ y 0 as the Abel-Prym maps of P and P respectively. When there is no confusion on the point y 0 we shall omit it in the notations.
By the theory of Prym-Tyurin varieties (see [We] and [LB]) the morphism ρ has the following property: where m = dim P = dim P . This last property is usually expressed by saying that the curve Y (or rather the morphism ρ) is of class n in ( P , Ξ).
In general the Abel-Prym map ρ : Y → P need not be birational. However, we have the following theorem.
Theorem 1.2. Let π : Y → X be a covering of degree n. If g X > 2 and n = 2 then Abel-Prym morphism ρ : Y → P is birational onto its image.
In order to prove Theorem 1.2 we need a lemma. Let (A, Θ A ) be a principally polarized abelian variety of dimension d and C a non-singular projective irreducible curve. Let ν : C → A be a morphism and let u : J(C) → A be the unique morphism (up to a translation) obtained by the universal property of the Jacobian such that ν = u • α. Using the polarizations, we define the transposed morphism u t : are the isomorphisms induced by the principal polarizations Θ A and Θ C respectively.
Recall that a curve C is of class n in (A, [We]). Proof. Let Nm p : J(C) → J(Z) be the norm map of p and u : J(C) → A the morphism corresponding to ν obtained by the universal property of the Jacobian. Similarly, Thus, for every x ∈ A r in the r-torsion of A, r · ( u • u t )(x) = 0 and therefore nx = 0.
Hence, A r ⊂ A n and r divides n. Since A is connected and A r is discrete, the morphism Proof of Theorem 1.2. Suppose that ρ is not birational onto its image and n = 2. Let Z be the normalization of the curve ρ(Y ) and let π : Y → Z be the morphism induced by ρ. Letñ be the degree ofπ. That is, Since the morphismμ is surjective, Z generatesP . Therefore the mapũ : J(Z) → P defined by the universal property of the Jacobian is surjective.
By Riemann-Hurwitz formula for the coverings π andπ, we have the following in- where R π and Rπ are the ramification divisors.
In all cases, we have that if ρ in not birational onto its image and n = 2, then g Y ≤ 5.
Now if we also suppose that g X ≥ 2, by the Riemann-Hurwitz formula for the covering π, we deduce that if deg π = 2 and g X ≥ 2, then ρ is not birational onto its image if the following cases occur: Therefore, if we assume that g X > 2 and n = 2, the theorem is proved.
Remark 1.4. From the proof of Theorem 1.2 we deduce that if g X ≥ 2, then the Abel-Prym map ρ may not be birational onto its image possibly in the cases given in (2) and when n = 2. For n = 2, it is well-known that if Y is not an hyperelliptic curve then ρ is birational onto its image and therefore ρ is birational onto its image.
We finish this section with a lemma that we will use later.

Stability and restrictions of Picard bundles
Let Y be a non singular irreducible projective curve of genus g Y ≥ 2. We fix a line bundle L 0 on Y of degree d and a point y 0 ∈ Y . Let P Y be the Poincaré bundle The Picard sheaves (relative to L 0 ) on J(Y ) are defined as the higher direct images is a vector bundle known as the Picard bundle. We shall consider the restriction of W J to some subvarieties when the curve is a covering space.
Let π : Y → X be a covering of degree n between non-singular projective irreducible curves of genus g Y and g X respectively and let π * : J(X) → J(Y ) be the pull-back morphism. As in §1 consider the Prym variety (P, Θ P ) and (π * (J(X)), Θ π ) in J(Y ).
From now on we fix a line bundle L 0 on Y of degree d > 2g Y − 2. Denote by W P and W π the restrictions of the Picard bundle W J (relative to L 0 ) to P and π * (J(X)) respectively.
Remark 2.1. We can take a line bundle L 0 ∈ J d (Y ) such that π * (L 0 ) is stable and deg π * (L 0 ) > 2ng X without loss of generality. Indeed: for a generic line bundle L on Y , A. Beauville in [Be] has proved that π * L is stable on X if |χ(L)| ≤ g X + then π * L is stable on X and π * (L ⊗ π * M) is stable for any line bundle M on X. Thus, for a generic line bundle L 0 ∈ J d (Y ) with d such that: we have that π * (L 0 ) is stable and deg π * (L 0 ) = deg L 0 − (deg R π )/2 > 2ng X . In the case deg π < max{g X (1 + √ 3) − 1, 2g X + 2}, it is sufficient that deg L 0 fulfills the condition (i).
Proof. For convenience of writing, we set up the following commutative diagrams: where the vertical morphisms are the natural projections.
Fix the Poincaré bundle P X on X × J(X) normalized with respect to the point π(y 0 ) ∈ X. Then, the vector bundle (id × π * ) * P Y on Y × J(X) is isomorphic to the bundle (π × id) * P X . From diagram (4) and the base change formula, we have Since E 0 is stable of degree d > 2ng X , p * X (E 0 ) ⊗ P X is a family of stable bundles parametrized by J(X). Such family corresponds to an embedding of the Jacobian in the moduli space M(n, d) of stable vector bundles of rank n and degree d. As in the proof of Theorem 2.5 in [Li], it follows that p J(X) * (p * X (E 0 ) ⊗ P X ) is Θ X -stable.
With the hypothesis of the previous theorem, we obtain Corollary 2.3. The restriction W π of W J to π * (J(X)) is Θ π -stable.
To study the restriction W P of W J to the Prym variety P we consider two cases, namely when the Abel-Prym map ρ is not birational and when it is. In both cases we can reduce our study to consider the corresponding vector bundle over ( P , Ξ) where f : P → P is a fixed isogeny as in Theorem 1.1. That is, if j P : P ֒→ J(Y ) is the natural inclusion, then we will denote by β : P → J(Y ) the composition map f • j P and by W P the vector bundle β * (W J ). Actually, f * (W P ) ∼ = W P .
Proof. The stability again follows from Lemma 2.1 in [BBN]. Suppose W P is simple.
Since O P ֒→ f * O P is injective, the map is injective. Hence, Hom(W P , W P ) ∼ = C and W P is simple.
Suppose ρ is not birational. As in section §1, let Z be the normalization of ρ(Y ) and π : Y → Z the morphism induced by ρ of degreeñ.
Proposition 2.5. If the Abel-Prym map ρ : Y → P is not birational onto its image and deg L 0 > 2g Y + 4, then the Picard bundle W P is Θ P -stable.
Proof. We consider separately each one of the cases where ρ is not birational (see Remark 1.4).
We have the following commutative diagrams: where u is the map induced by Z → P . Therefore, and Imμ t = P ⊂ Imπ * . Since dim P = 3 = dim J(Z) = dimπ * (J(Z)), it follows Remark 2.6. Observe that, in the previous proposition, the result holds for any line bundle L 0 (of degree d > 2g Y + 4) whereas in Remark 2.1 the condition "π * (L 0 ) is stable" holds for a generic line bundle L 0 . If the Picard bundle W L 0 ,P corresponding to L 0 is stable, then W L,P = p 2 * (p * 1 L ⊗ P Y | Y ×P ) is stable for any L (of degree d). In fact, since L ⊗ L −1 0 ∈ J(Y ), one can write L ∼ = L 0 ⊗ π * N ⊗ M where M ∈ P and N ∈ J(X), therefore the two Picard bundles are related as follows W L,P ∼ = τ * M (W L 0 ⊗ pi * N,P ), where τ M : P → P is the traslation by M. Since π * (L 0 ) is stable, π * (L 0 ⊗ π * N) is stable.
Hence W L,P is stable.
Before we consider the case when the map ρ is birational we describe the bundle W P in different way.
The abelian variety ( P , Ξ) is principally polarized, so we can identify P with its dual abelian variety. The Poincaré bundle on P × P is given by where m is the multiplication law on P and q i the canonical projections of P × P in the i = 1, 2 factor.
Proposition 2.7. The vector bundle W P is isomorphic to the bundle q 2 * (q * 1 ( ρ * (L 0 )) ⊗ Q ∨ ), where q i are the projection of P × P in the i = 1, 2 factor. Proof. Consider the commutative diagrams: I  I  I  I  I  I  I  I  I  I the natural inclusion. From the normalization of P Y , the restrictions of the line bundles P P and ( ρ × id) * Q ∨ to {y 0 } × P are trivial and for everyξ ∈ P , the restrictions of the both bundles to Y × {ξ} are isomorphic to f (ξ) by Lemma 1.5. Therefore, From the projection formula and the base-change formula, we have Proposition 2.8. If the Abel-Prym map ρ is birational onto its image then the restriction W P of the Picard bundle W J to the Prym variety is simple.
Proof. From Proposition 2.4 it is enough to prove that W P is simple. The bundle W P is the Fourier-Mukai transform of the sheaf ρ * (L 0 ) by Proposition 2.7. Since deg(L 0 ) > 2g Y − 2, the sheaf q * 1 ( ρ * (L 0 )) ⊗ Q ∨ has only one non-null direct image. Then the Fourier-Mukai transform is a complex concentrated in degree zero and, therefore, W P is simple if ρ * (L 0 ) is simple (see [Mu], Corollary 2.5).
Since j is a closed embedding, j * j * E ∼ = E for any sheaf E and applying the adjunction formula again, we obtain that j * ν * (L 0 ) ∼ = ρ * L 0 is simple.
From Proposition 2.5 and Proposition 2.8 we have that: Theorem 2.9. The restriction W P of the Picard bundle W J to the Prym variety is simple if g X ≥ 2 and deg L 0 > 2g Y + 4.

Picard bundles over moduli spaces
We shall recall the construction and properties of spectral coverings given by Beauville, Narasimhan and Ramanan in [BNR].
Let X be a non-singular projective irreducible curve of genus g X ≥ 2. Denote by M(n, d) the moduli space of stable vector bundles over X of rank n and degree d. determined by stable bundles with fixed determinant ξ ∈ J d (X).
Let K be the canonical bundle over X and W = n i=1 H 0 (X, K i ). For every element s = (s 1 , . . . , s n ) ∈ W , denote by Y s the associated spectral curve (see [BNR]). For a general s ∈ W , Y s is non-singular of genus g Ys = n 2 (g X − 1) + 1 and the morphism π s : Y s −→ X is of degree n.
In [BNR] it is proved that if δ = d + n(n−1)(g X −1) then there is an open subvariety Moreover, the direct image induces a dominant rational map defined on an open subvariety T ′ ⊆ P ′ , where P ′ is a translate of the Prym variety P s of π s (see [BNR], Proposition 5.7). The complement of the open subvariety T ′ ⊆ P ′ is of codimension at least 2. Actually, f : T ′ −→ M ξ is generically finite. Now we will focus on the case n = 2 and g X = 2. M(2, ξ) will denote the moduli space of stable rank 2 vector bundles with fixed determinant ξ (with deg ξ odd). We shall prove that it is possible to construct a spectral covering π : Y s → X of degree 2 in such way that the curve Y s is hyperelliptic. Let p : X → P 1 be the covering of degree 2 given by the canonical bundle K X . Let s be a section of H 0 (X, K 2 X ) such that the spectral covering π : Y s → X of degree 2 corresponding to (0, s) is smooth, integral and such that the induced map from the Prym variety associated to Y s → X to the moduli space M(2, ξ) is dominant. Observe that since g X = 2 we can write the section s as a product s = s 1 · s 2 where s i ∈ H 0 (X, K X ) for i = 1, 2. If P i + Q i is the divisor associated to the section s i , then p(P i ) = p(Q i ) = a i ∈ P 1 .
Following the construction of cyclic coverings of curves given in [Go1], the morphism X = P 1 → P 1 of degree 2, ramified at the points a 1 , a 2 is the cyclic covering defined by the construction data (D = a 1 + a 2 , L = O P 1 (2)). Let C be the desingularization of the curve X × P 1X . From Theorem 2.13 in [Go1] the map C → X is the cyclic covering associated to the data (p −1 (D), p * L), but since p −1 (D) = {P 1 , Q 1 , P 2 , Q 2 } and p * L ∼ = K 2 X , then Y s ∼ = C and the map Y s ∼ = C →X = P 1 is of degree 2. Therefore the curve Y s is hyperelliptic.
Let P be the Prym variety associated to the degree 2 covering Y s → X. Since the spectral curve Y s is hyperelliptic then the Abel-Prym morphism ρ : Y s → P is not birational onto its image. As in the case 3) of the proof of Proposition 2.5 its follows that, in this case, if δ = d + n(n − 1)(g X − 1) > 2g Ys + 2 = 12, then the Picard bundle on the Prym variety P is stable with respect to the restriction of the theta divisor.
From Remark 3.2 the stability of Picard bundles on M X (2, ξ) is ensured.
Theorem 3.3. Let X be a smooth projective irreducible curve of genus 2 and let M X (2, ξ) be the moduli space of of rank 2 stable vector bundles on X with fixed determinant ξ. If d = deg(ξ) > 10 and d odd, then the Picard bundle W ξ on M X (2, ξ) is stable with respect to the theta divisor.
Remark 3.4. After this paper was finished, I. Biswas and T. Gómez informed us that they have obtained the stability of the Picard bundle in the case of the moduli space M X (2, ξ) of rank 2 stable bundles over a smooth curve X of genus g X ≥ 3 such that d = deg ξ ≥ 4g X − 3 and odd ( [BG]).