Weak point property and sections of Picard bundles on a compactified Jacobian over a nodal curve

Transcription

1 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 3, August 2016, pp DOI /s Weak point property and sections of Picard bundles on a compactified Jacobian over a nodal curve USHA N BHOSLE 1 and SANJAY SINGH 2, 1 Indian Institute of Science, Bangalore , India 2 Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland * Corresponding author. usnabh07@gmail.com; sanjayks@iiserb.ac.in MS received 13 December 2014; revised 28 May 2015 Abstract. We show that the compactified Jacobian (and its desingularization) of an integral nodal curve Y satisfies the weak point property and the Jacobian of Y satisfies the diagonal property. We compute some cohomologies of Picard bundles on the compactified Jacobian and its desingularization. Keywords. Weak point property; Picard bundle; compactified Jacobian; nodal curve Mathematics Subject Classification. Primary: 14H60; Secondary: 14D Introduction Debarre had proved that the Jacobian J 0 (X 0 ) of a smooth curve X 0 satisfies the diagonal property (Corollary 2.2 of [6]) i.e., there is a vector bundle of rank equal to the dimension of J 0 (X 0 ) over J 0 (X 0 ) J 0 (X 0 ), having a section whose zero scheme is the diagonal of J 0 (X 0 ) J 0 (X 0 ). In this note, we generalize Debarre s result to a singular curve X k with k ordinary double points (nodes). Following [12] (where the variety is assumed to be smooth), we say that a variety Z of dimension g has a weak point property if there is a vector bundle E of rank g on Z such that E has a section s whose zero scheme is a reduced point. If Z is a group variety, then this implies that Z satisfies the diagonal property. Let J 0 (X k ) be the generalized Jacobian of line bundles (locally free sheaves of rank 1) of degree 0 on X k. The variety J 0 (X k ) has a natural compactification J 0 (X k ),asthe variety of torsion-free sheaves of rank 1 and degree 0 on X k. We denote by J 0 (X k ) the natural desingularization of J 0 (X k ) (see subsection 2.1). Theorem 1.1. Let X k be an integral curve with k nodes (ordinary double points). (1) The variety J 0 (X k ) satisfies the weak point property. (2) The compactified Jacobian J 0 (X k ) satisfies the weak point property. (3) The generalized Jacobian J 0 (X k ) satisfies the diagonal property. (4) If X k is a rational curve, then J 0 (X k ) satisfies the diagonal property. As a corollary, we also prove that generalized Jacobians of some curves with many components satisfy the diagonal property. c Indian Academy of Sciences 329

2 330 Usha N Bhosle and Sanjay Singh Debarre proved the weak point property by computing the cohomology of a Picard bundle (see section 3 for definitions of Picard bundles) tensored with certain line bundles. For this, he used the identification of the Picard bundle on J 0 (X 0 ) with a symmetric product of X 0 in [9]. Unlike Debarre, we prove the weak point property by explicitly constructing suitable sections of Picard bundles on J 0 (X k ) and J 0 (X k ) and computing their zero schemes. Even in the non-singular case, our proof is simpler and more transparent than Debarre s proof. Moreover, Debarre s proof does not extend to nodal curves. On the nodal curve, the Picard bundle is not isomorphic to the symmetric product of the curve. One also uses the g-th Chern class of the Picard bundle to compute the zero scheme. The Chern classes (in the Chow group) of the Picard bundle on the Jacobian of a smooth curve were computed by Mattuck. The Chern classes of the Picard bundle on J 0 (X k ) are not known, those (modulo numerical equivalence) of the Picard bundle on J 0 (X k ) were computed by one of the authors [3]. Using that, we first prove the weak point property for J 0 (X k ) and deduce that for J As a corollary, we determine the first and top Chern classes of the Picard bundles in the Chow group (Corollary 4.4). We also compute all the cohomologies of the Picard bundle on J 0 (X k ) and the 0-th and top cohomologies of the Picard bundles on J 0 (X k ) (Proposition 4.5). 2. Preliminaries 2.1 The compactified Jacobian and its desingularization For a positive integer k, letx k denote an irreducible projective nodal curve with k nodes over an algebraically closed field C. Lety 1,...,y k be the nodes of X k.letg(x k ) be the arithmetic genus of X k. We denote by ω Xk the locally free dualizing sheaf on X k.let p k : X 0 X k be the normalization map. Let p k 1 (y i ) := {x i,z i } X 0 be the inverse image of the nodal point y i in X k. By abuse of notation, we denote by the same x i and z i, the images of these points in X j, for all i>jand y i the nodes in X j for 1 i j. Fix, once and for all, a base point t X 0 different from all x j,y j,j = 1,...,k and sufficiently general. We denote its image in X j again by t.let p k : X k 1 X k be the natural morphism obtained by identifying x k and z k to the node y k. For d Z,letJ d (X k ) be the Jacobian of line bundles (locally free sheaves of rank 1) of degree d on X k. The variety J d (X k ) has a natural compactification J d (X k ), as the variety of torsion-free sheaves of rank 1 and degree d on X k. A natural desingularization J d (X k ) of J d (X k ) can be defined inductively as follows. Let L d k 1 be the Poincaré sheaf on J d (X k 1 ) X k 1 normalised by the condition that its restriction to J d (X k 1 ) t is trivial. Let ( L d k 1 ) x k (respectively ( L d k 1 ) z k ) be the line bundle on J d (X k 1 ) obtained by restricting L d k 1 to J d (X k 1 ) x k (respectively J d (X k 1 ) z k ). Then,by Proposition 12.1, p. 64 of [11], J d (X k ) = P(( L d k 1 ) x k ( L d k 1 ) z k ).

3 Picard bundles 331 Let π k : J d (X k ) J d (X k 1 ) be the canonical projection and define π k : J d (X k ) J d (X 0 ), π k := π 1 π k. There is a birational surjective map h : J d (X k ) J d (X k ). We denote by the same letter h such a map for all d (if no confusion arises). The subset J d (X k ) is a smooth open subset of J d (X k ), it is also an open subset of J d (X k ) and h is an isomorphism over J d (X k ). 2.2 Picard bundles on J 0 (X k ) and J(X k ) Let L k be the (normalised) Poincaré sheaf on J 0 (X k ) X k. We denote the two projections of J 0 (X k ) X k by ν k and q k respectively. Let L k(d) denote the Poincaré sheaf twisted by the pullback of the line bundle O Xk (dt), where t X k is the fixed smooth point. We define the direct image sheaves Ē d,k := ν k (L k(d)). These sheaves are vector bundles on J 0 (X k ) if d 2g(X k ) 1 and are called Picard bundles on J Let L k be the pullback of L k to J 0 (X k ) X k by h id. We denote the two projections of J 0 (X k ) X k by ν k and q k respectively. The direct image sheaves E d,k := ν k ( L k (d)) are vector bundles on J 0 (X k ) if d 2g(X k ) 1 and are called Picard bundles on J One has h Ē d,k = Ed,k. We may identify J 0 (X k ) with J d (X k ) by the isomorphism L L(dt) for L J 0 (X k ) and L k (d) with L d k. Then we call the pullback of E d,k to J d (X k ) as the Picard bundle on J d (X k ) (for d 2g(X k ) 1), it is the direct image of L d k on J d (X k ). Recall that J 0 (X k ) = P(( L k 1 ) xk ( L k 1 ) zk ).Letπ k : J 0 (X k ) J 0 (X k 1 ) be the projection map, which is a P 1 bundle. There is an exact sequence i k 0 E d,k πk E d,k 1 OJ 0 (X k ) (1) 0, (Proposition 5.1 of [5]). Dualizing this sequence gives the exact sequence 0 OJ 0 (X k ) ( 1) π k i E k d,k 1 Ed,k 0. (2.1)

4 332 Usha N Bhosle and Sanjay Singh Remark 2.1. Let ψ : J 0 (X k ) J 0 (X k ) be the isomorphism defined by L ω Xk O Xk ( (2g(X k ) 2)t) L, where t X k is a smooth point. Using Serre duality on X k, it is easy to see that Ē d,k = ψ F 2g 2 d,k, where F d,k := R 1 p ν (L k (d)). 3. Sections of Picard bundles Write P k := L k (d) on J The fiber of Ē d,k over L H 0 (X k, L(d)). Hence there is an evaluation map EV k : ν (Ē d,k ) P k, J 0 (X k ) is isomorphic to defined by (L,y,φ) φ(y), for L J 0 (X k ), y X k,φ H 0 (X k, L(d)). Restricting EV k to J 0 (X k ) {t}, we get ev k : Ē d,k P k J 0 (X k ) {t}. By our assumption, we have P k J 0 (X k ) {t} = OJ Hence we have ev k : Ē d,k O J 0 (X k ). Dualizing, we get ev k : O J 0 (X k ) Ē d,k or equivalently, Let σ k H 0 ( J 0 (X k ), Ē d,k ). σ k H 0 ( J 0 (X k ), E d,k ) denote the pull back of σ k. Since h is an isomorphism over J 0 (X k ), we identify J 0 (X k ) with h 1 J Lemma 3.1. Let N be a torsion free sheaf of rank 1 and degree 2g(X k ) 2 on X k. Then dim H 0 (X k,n)= g if and only if N = ω Xk. Proof. For a torsion free sheaf N of rank 1 and degree 2g(X k ) 2, by Riemann Roch theorem, h 0 (X k,n)= g if and only if h 1 (X k,n)= 1. By Serre duality, this is equivalent to h 0 (X k,ω Xk N ) = 1. Thus ω Xk N is a torsion free sheaf of rank 1 and degree 0 with a nonzero section which must be nowhere vanishing and hence generates the trivial line bundle. Thus ω Xk N is the trivial line bundle i.e., ω Xk = N. PROPOSITION 3.2 (1) If d 2g(X k ), then σ k and σ k are nowhere vanishing sections. (2) If d = 2g(X k ) 1, then each of σ k and σ k has its zero set consisting of one point, the point p J 0 (X k ) corresponding to the line bundle ω Xk ( (2g(X k ) 2)t) on X k. Proof. Let L J Then H 1 (X k, L(d)) = 0ford 2g(X k ) 1. The section σ k vanishes at L J 0 (X k ) if and only if the restriction of ev k to the fiber over L is identically

5 Picard bundles 333 zero, or equivalently, the restriction of ev k to the fiber of Ē d,k over L is identically zero, i.e. φ(t) = 0 for all φ H 0 (X k, L(d)). (1) If d 2g(X k ), then H 0 (X k, L(d)) generates the fiber L(d) t at t for all L. Hence φ(t) = 0forsomeφ H 0 (X k, L(d)) and hence σ k does not vanish at L for any L J (2) Note that φ(t) = 0 for all φ H 0 (X k, L(d)) if and only if H 0 (X k, L(d)) = H 0 (X k, L(d) O Xk ( t)) = H 0 (X k,l(d 1)). Ford = 2g(X k ) 1, h 0 (X k, L(d)) = g and H 0 (X k, L(d)) = H 0 (X k,l(d 1)) if and only if h 0 (X k,l(d 1)) = g. By Lemma 3.1, H 0 (X k,l(d 1)) = g if and only if L(d 1) = ω Xk i.e. L = ω Xk (( 2g(X k ) + 2)t). This proves part (2). Recall that for any k, there is an exact sequence (2.1) 0 OJ 0 (X k ) ( 1) π k i E k d,k 1 Ed,k 0. This gives a map on sections H 0 (i k ) : H 0 ( J 0 (X k ), π k E d,k 1 ) H 0 ( J 0 (X k ), E d,k ). Define i k by i k = i 0 i k. Then one has i k : π k E d,0 E d,k. The following lemma shows that σ k is the image of a section coming from a section of E d,0. Lemma 3.3 (1) The map H 0 (i k ) maps π k (σ k 1) to σ k. (2) The map H 0 (i k ) maps π k (σ 0 ) to σ k. Proof. Consider (L 0,Q 1,...,Q k ) J 0 (X k ), L 0 J 0 (X 0 ). Let h(l 0,Q 1,..., Q k 1 ) = L 1,h(L 0,Q 1,...,Q k ) = L. Then there is a commutative diagram D 1 : H 0 (X k,π k L 1 (d)) = H 0 (X k 1,L 1 (d)) ev k 1 C H 0 ev k (X k, L(d)) C The diagram D 1 implies that there is a commutative diagram D 2 : πk (E d,k 1) π k ev k 1 E d,k ev k O J 0 (X k ) O J 0 (X k ). The diagram D 2 implies that π k 1 (σ k 1) is mapped to σ k by H 0 (i k ). The second assertion follows inductively from the first.

6 334 Usha N Bhosle and Sanjay Singh 4. Weak point property and diagonal property Following [12] (where the variety is assumed to be smooth), we make the following definitions. DEFINITION 4.1 A variety Z of dimension n satisfies the weak point property if there is a vector bundle E of rank n on Z such that E has a section s whose zero scheme is a reduced point. DEFINITION 4.2 A variety Z of dimension n satisfies the diagonal property if there is a vector bundle E of rank n on Z Z such that E has a section s whose zero scheme is the diagonal of Z Z. If Z is a group variety, then Z satisfies the diagonal property if it satisfies the weak point property. Let X k be an integral curve with k nodes (ordinary double points). Let σ k be the section of Ed,k defined in Section 3. Theorem 4.3. Let X k be an integral curve with k nodes and d = 2g(X k ) 1. (1) The zero scheme of σ k is the reduced point {p}. Thus J 0 (X k ) satisfies the weak point property. (2) The compactified Jacobian J 0 (X k ) satisfies the weak point property. (3) The generalized Jacobian J 0 (X k ) satisfies the diagonal property. (4) If X k is a rational curve, then J 0 (X k ) satisfies the diagonal property. Proof. (1) Let Z(σ k ) denote the zero scheme of σ k and A 0 (Z(σ k )) the Chow group of zero cycles on Z(σ k ) modulo rational equivalence. We associate to Z(σ k ) the local top Chern class Z(σ k ) A 0 (Z(σ k )) of Ed,k with respect to σ k (Chapter 14 of [8]). Since J 0 (X k ) is Cohen Macaulay and the dimension of the zero scheme of σ k equals dimension J 0 (X k ) r(ed,k ), by Example of [8], the section σ k is regular, and Z(σ k ) =[Z(σ k )], the cycle defined by the natural scheme structure on Z(σ k ). By Proposition 14.1 of [8], the image of Z(σ k ) in A 0 ( J 0 (X k )) is i Z(σ k ) [X], where i : Z(σ k ) J 0 (X k ) is the inclusion and [X] denotes the fundamental class of X. This implies that the degree of the top Chern class (i.e., the top Chern class evaluated at the fundamental cycle of J 0 (X k ))ofed,k equals the degree of the cycle associated with the zero scheme of σ k. Since c g (Ed,k ) equals the class of a (reduced) point in cycles modulo numerical equivalence (Theorem 1.1 of [3]), the degree of the cycle associated to the zero scheme of σ k is 1. Therefore it must be a reduced point. Thus σ k has a simple zero at p. (2) Since h is an isomorphism over the open set J 0 (X k ), it follows that σ k has a simple zero at p. Thus both J 0 (X k ) and J 0 (X k ) satisfy the weak point property. (3) The proofs of (1) and (2) show that, in particular, J 0 (X k ) satisfies the weak point property. Since J 0 (X k ) is a group variety and the diagonal property is equivalent to the weak point property for group varieties (Proposition 13 of [12]), it follows that

7 Picard bundles 335 J 0 (X k ) satisfies the diagonal property. More precisely, if m and i respectively denote the multiplication and inverse on J 0 (X k ) and f : J 0 (X k ) J 0 (X k ) J 0 (X k ) is the map defined by (a, b) m(m(a, i(b)), p) then the section f (σ k ) of f (Ed,k ) has zero scheme precisely the diagonal. (4) If X k is a rational curve, then J 0 (X k ) = P 1 P 1, a k-fold product (Section 9 of [3]). Since P 1 satisfies the diagonal property, its k-fold product also satisfies the diagonal property proving part (4). COROLLARY 4.4 Let d = 2g(X k ) 1. (1) The g-th Chern class c g (Ed,k ) = W 0 in the Chow group of J (2) The g-th Chern class c g (Ēd,k ) = W 0 in the Chow group of J (3) The first Chern class c 1 (E d,k ) = c 1(det E d,k ) = θ k,c 1 (Ē d,k ) = c 1(det Ē d,k ) in the Chow group. Here θ k is the pull back of the theta divisor on J 0 (X k ) to J det E d,k = O J 0 (X k ) (θ k), the line bundle associated to the divisor θ k. Proof. Since J 0 (X k ) satisfies the weak point property (Theorem 4.3), one has c g (E d,k ) =[p] CHg ( J 0 (X k )), c 1 (E d,k ) = c 1(det E d,k ) CH1 ( J 0 (X k )) (subsection 4.2 of [12]). Under the isomorphism J 0 (X k ) J 0 (X k ) defined by L ω Xk O Xk ( (2g(X k ) 2)t) L, the point W 0 J 0 (X k ) corresponding to O Xk maps to p, hence part (1) follows. Part (2) follows similarly from the fact that σ k has a simple zero at p. We shall prove part (3) by induction on k, the number of nodes. The result is true for k = 0 [9, Corollary to Theorem 3]. Assume the result for X k 1. For 0 m g, letw m,k J 0 (X k ) denote the cycle determined by the Brill-Noether locus of L J 0 (X k ) such that h 0 (X k, L(m)) > 0. We denote by W m,k its pull back to J From the proof of Lemma 3.7 of [4], for 1 m g(x k ) one has W m,k = πk 1 ( W m,k 1 ) S xk + πk 1 ( W m 1,k 1 ), where S xk is a divisor defined by a section of OJ 0 (X k ) (1). Form = g(x k) 1, using the facts that g(x k 1 ) = g(x k ) 1 and W g(xk 1 ),k 1 = J 0 (X k 1 ), one has W g(xk ) 1,k = S xk + π 1 k W g(xk 1 ) 1,k 1.

8 336 Usha N Bhosle and Sanjay Singh By induction, c 1 (Ed,k 1 ) = W g(xk 1 ) 1,k 1. Hence W g(xk ) 1,k = S xk + πk c 1(Ed,k 1 ). From the exact sequence (2.1), we have c 1 (Ed,k ) = S x k +πk c 1(Ed,k 1 ). Thus c 1(Ed,k ) = W g(xk ) 1,k = θ k. Since rational equivalence coincides with linear equivalence for divisors on normal varieties, c 1 (det Ed,k ) = θ k CH 1 ( J 0 (X k )) implies the last statement in part (3). The Chern classes of the Picard bundle E d,k,d 2g(X k) 1, modulo numerical equivalence were computed in Theorem 1.1 of [3]. It was believed that they were the same in the Chow group also. This is taken up by the first named author in a subsequent paper. The following proposition shows, in particular, that if the curve X k is not rational then the section σ k is unique up to a scalar. PROPOSITION 4.5 Let d 2g(X k ) 1. (1) π k Ed,k = Ed,k 1,πk E d,k = Ed,0,(Ri π k ) Ed,k = 0 for i>0. (2) For g(x 0 ) 1, h 0 ( J 0 (X k ), Ed,k ) = 1. (3) Let g(x 0 ) 1. Then h i ( J 0 (X k ), Ed,k ) = (g(x 0 ) 1)C i for 1 i<g(x 0 ), h i ( J 0 (X k ), Ed,k ) = 0 for i g(x 0). (4) For g(x 0 ) 1, h 0 ( J 0 (X k ), Ēd,k ) = 1. (5) For g(x 0 ) = 0, h 0 ( J 0 (X k ), Ed,k ) = d + 1, h0 ( J 0 (X k ), E d,k ) = d 2k + 1. (6) h g ( J 0 (X k ), Ēd,k ) = 0, h0 ( J 0 (X k ), Ē d,k ) = 0. Proof. (1) By taking the direct image of the exact sequence (2.1) and using the facts that π k πk E d,k = Ed,k, Ri π k O Pk ( 1) = 0 for all i 0, we have π k Ed,k = Ed,k 1, (Ri π k ) Ed,k = 0fori > 0. Since π k = π 0 π k, inductively we get π ke d,k = Ed,0,(Ri π k ) Ed,k = 0 for all i>0. (2) and (3) There is a Leray spectral sequence with Ep,q 2 = H p ( J 0 (X k ), (R q π k ) Ed,k ) which abuts to H p+q ( J 0 (X k ), Ed,k ). By Part (1), π k Ed,k = Ed,k 1, (Ri π k ) Ed,k = 0fori > 0. Hence the Leray spectral sequence gives H i ( J 0 (X k ), Ed,k ) = H i ( J 0 (X k 1 ), Ed,k 1 ) for all non-negative i and positive k. By induction on k, one has H i ( J 0 (X k ), Ed,k ) = H i ( J 0 (X 0 ), Ed,0 ), i 0. If g(x 0 ) 2, by Proposition 4.4 of [10], h 0 ( J 0 (X 0 ), Ed,0 ) = 1 and hi ( J 0 (X 0 ), Ed,0 ) = (g(x 0 ) 1)C i for 1 i < g(x 0 ),h g(x0) ( J 0 (X 0 ), Ed,0 ) = 0. Certainly, h i ( J 0 (X 0 ), Ed,0 ) = 0fori>g(X 0). If g(x 0 ) = 1, one has J 0 (X 0 ) = X 0. Since k 1,g(X k ) 2 and hence d 3. By the main theorem of [7], for d 2 the Picard bundle Ed,0 is a stable vector bundle of rank d, slope 1/d > 2g(X 0 ) 2. Hence h i ( J 0 (X 0 ), Ed,0 ) = 0, i > 0. Then by Riemann Roch theorem, h 0 ( J 0 (X 0 ), E d,0 ) = 1. It follows that h0 ( J 0 (X k ), E d,k ) = 1 and h i ( J 0 (X k ), E d,k ) = 0fori>0.

9 Picard bundles 337 This completes the proof of Parts (2) and (3). (4) Recall that Ed,k = h Ēd,k. Since J 0 (X k ) is irreducible and h is a surjective birational map, it follows that the map H 0 ( J 0 (X k ), Ēd,k ) H 0 ( J 0 (X k ), Ed,k ), defined by pull back of sections, is injective. Since 0 = σ k H 0 ( J 0 (X k ), Ēd,k ),itfollowsfrompart(2) that H 0 ( J 0 (X k ), Ẽd,k ) = 1. (5) By Proposition 9.1 of [5], E d,k = I d 2k+1 i O Pi (1), where I n denotes the trivial vector bundle of rank n for any positive integer n. Since h 0 (I n ) = n, h 0 ( J 0 (X k ), O Pi (1)) = 2, Part (5) follows. Note that d 2g(X k ) 1 = 2k 1sothath 0 ( J 0 (X k ), E d,k ) = d + 1 2k 0. (6) Since ωj 0 (X k ) = OJ 0 (X k ) [1], to prove Part (6) it suffices to show that h0 ( J 0 (X k ), Ē d,k ) = 0. For convenience, we work on J d (X k ) rather than J For every L J d 1 (X k ), we have an embedding of X k in J d (X k ) defined by x L Ix, where I x is the ideal sheaf of x. LetX k,l denote the image of X k under this embedding. If g(x 0 ) 1, then by the proof of Theorem 8.10 of [5] (or equivalently, Proposition 6.1, Lemmas 8.7, 8.8, 8.9 of [5]), the vector bundle Ē d,k Xk,L is stable if d 2g(X k ) and it is semistable if d = 2g(X k ) 1. If X k is a rational nodal curve, then by the proof of Theorem 9.3 of [5], Ē d,k Xk,L is stable. By Proposition 8.8(1) of [5], Ē d,k (t) Xk,L,t being a fixed smooth point, has degree d + 1 2g(X k ) and rank d + 1 g(x k ). Hence Ē d,k Xk,L has degree g(x k )<0. It follows that h 0 (X k,l, Ē d,k Xk,L ) = 0. Thus any section of Ē d,k vanishes on X k,l for all L J d 1 (X k ).IfN J d (X k ), then N X k,l where L = N O( t). Hence any section s of Ē d,k vanishes over J d (X k ). Since J d (X k ) is an open dense subset of J d (X k ), it follows that s = 0. Lemma 4.6. Let X be a Gorenstein projective variety of dimension g. Let E be a vector bundle of rank g on X with a section s such that the zero scheme of s is a reduced smooth point. Then all (non-zero) sections of det(e) ω X vanish at x. Proof. This follows exactly as Proposition 3.1 of [6] using the fact that ω X is locally free. COROLLARY 4.7 All (non-zero) sections of det(ē d,k ), d = 2g(X k) 1, vanish at p. Proof. Since the singularities of X k are of embedding dimension 2, by the main result of [2], the variety J 0 (X k ) is a local complete intersection and hence Gorenstein. Therefore the result follows from Theorem 4.3 and Lemma 4.6 using the fact that ω J 0 (X k ) = OJ 0 (X k ) (Lemma 4.2 of [1]). Lemma 4.8. The line bundle (det Ed,k ) ω J 0 (X k ) has no non-zero sections.

10 338 Usha N Bhosle and Sanjay Singh Proof. From the exact sequence of Proposition 5.2 of [5], taking determinants, we get det E d,k = π k (det E d,0 ) i O Pi (1).Since ω J 0 (X k ) = i O Pi ( 2), wehave (det E d,k ) ω J 0 (X k ) = π k (det E d,0 ) i O Pi ( 1). Taking direct image on J 0 (X k ), we get π k (det E d,k ω J 0 (X k ) ) = (det E d,0 ) π k ( io Pi ( 1)) = 0. Hence H 0 ( J 0 (X k ), (det Ed,k ) ω J 0 (X k ) ) = H 0 ( J 0 (X 0 ), π k(det E d,k ω J 0 (X k ) )) = Curves with many components Consider a connected reduced curve Y with many irreducible components Y i,i I such that Y has only ordinary double points as singularities. The dual graph of such a curve is a graph whose vertices v i I correspond to the Y i s and the edges e j correspond to the singular points y j. Two vertices v i and v i are connected by an edge e j if and only if Y i and Y i have y j as a node. Then a node y j lying on a single component Y i corresponds to a loop at v i. Suppose that the graph obtained by forgetting such loops in the graph of Y is a tree, or equivalently, the graph of the curve obtained by blowing up all nodes of Y, which lie on a single irreducible component (and no other component), is a tree. Simple examples of such Y are chains of smooth curves, or more generally a tree-like curve i.e., a curve whose dual graph is a tree. COROLLARY 4.9 Suppose that Y is a curve as in subsection 4.1. Then the diagonal property holds for the generalized Jacobian of Y. Proof. This follows from Theorem 4.3 as the generalized Jacobian of Y is isomorphic to the product of generalized Jacobians of Y i,i I. Acknowledgements The authors would like to thank the referee for useful comments, one of which led to improvement of results in Proposition 4.5(3). This work was done during the tenure of the first author in Indian Institute of Science, Bangalore, as a Raja Ramanna Fellow. The second author (SS) thanks Prof. Piotr Pragacz for motivating to study the diagonal and the point property, and is supported by the Post-Doctoral Research Fellowship of Institute of Mathematics, Polish Academy of Sciences. References [1] Alexeev V and Nakamura I, Mumford s construction of degenerating Abelian varieties, Tohoku Math. J. 51 (1999) [2] Altman A, Iarrobino A and Kleiman S, Irreducibility of the compactified Jacobian, Real and complex singularities, in: Proc. 9th Nordic Summer School (1977) (Oslo: Sijthoff and Noordhoff) pp

11 Picard bundles 339 [3] Bhosle Usha N, Maps into projective spaces, Proc. Indian Acad. Sci. (Math. Sci.) 123(3) (2013) [4] Bhosle Usha N and Parameswaran A J, On the Poincaré formula and Riemann singularity theorem over nodal curves, Math. Ann. 342(4) (2008) [5] Bhosle Usha N and Parameswaran A J, Picard bundles and Brill-Noether loci on the compactified Jacobian of a nodal curve, IMRN 2014(15) (2014) doi: /imrn/rnt069 [6] Debarre O, The diagonal property for abelian varieties, Curves and abelian varieties, in: Contemp. Math., 465 (2008) (Providence, RI: Amer. Math. Soc.) pp [7] Ein L and Lazarsfeld R, Stability and restrictions of Picard bundles with an application to the normal bundles of elliptic curves, in: Complex projective geometry (ed) Ellingsrud, Peskine et al LMS 179 (1992) (Cambridge University Press) [8] Fulton, Intersection theory, 2nd ed. (1998) (Berlin Heidelberg: Springer-Verlag) [9] Mattuck A, Symmetric products and Jacobians, Amer. J. Math. 83 (1961) [10] Mukai S, Duality between D(X) and D( ˆX) with its applications to Picard sheaves, Nagoya Math. J. 81 (1981) [11] Oda T and Seshadri C S, Compactifications of the generalised Jacobian variety, Transactions of the Amer. Math. Soc. 253 (1979) 1 90 [12] Pragacz P, Srinivas V and Pati V, Diagonal subschemes and vector bundles, Pure Appl. Math. Quart. 4 (2008) COMMUNICATING EDITOR: Nitin Nitsure