Limits of the trivial bundle on a curve

We attempt to describe the rank 2 vector bundles on a curve C which are specializations of the trivial bundle. We get a complete classifications when C is Brill-Noether generic, or when it is hyperelliptic; in both cases all limit vector bundles are decomposable. We give examples of indecomposable limit bundles for some special curves.


Introduction
Let C be a smooth complex projective curve, and E a vector bundle on C, of rank r. We will say that E is a limit of O r C if there exists an algebraic family (E b ) b∈B of vector bundles on C, parametrized by an algebraic curve B, and a point o ∈ B, such that E o = E and E b O r C for b o. Can we classify all these vector bundles? If E is a limit of O 2 C clearly E ⊕ O r−2 C is a limit of O r C , so it seems reasonable to start in rank 2.
We get a complete classification in two extreme cases: when C is generic (in the sense of Brill-Noether theory), and when it is hyperelliptic. In both cases the limit vector bundles are of the form L ⊕ L −1 , with some precise conditions on L. However for large families of curves, for instance for plane curves, some limits of O 2 C are indecomposable, and those seem hard to classify.

Generic curves
Throughout the paper we denote by C a smooth connected projective curve of genus g over C.
Proposition 1. Let L be a line bundle on C which is a limit of globally generated line bundles (in particular, any line bundle of degree ≥ g + 1). Then L ⊕ L −1 is a limit of O 2 C .
Proof. By hypothesis there exist a curve B, a point o ∈ B and a line bundle L on C × B such that L |C×{o} L and L |C×{b} is globally generated for b o. We may assume that B is affine and that o is defined by f = 0 for a global function f on B; we put B * := B {o}. We choose two general sections s, t of L on C × B * ; reducing B * if necessary, we may assume that they generate L. Thus we have an exact sequence on which corresponds to an extension class e ∈ H 1 (C × B * , L −2 ). For n large enough, f n e comes from a class in H 1 (C × B, L −2 ) which vanishes along C × {o}; this class gives rise to an extension Remark 1. Let E be a vector bundle limit of O 2 C . We have det E = O C , and h 0 (E) ≥ 2 by semi-continuity. If E is semi-stable this implies E O 2 C ; otherwise E is unstable. Let L be the maximal destabilizing sub-line bundle of E; we have an extension 0 → L → E → L −1 → 0, with h 0 (L) ≥ 2. Note that this extension is trivial (so that E = L ⊕ L −1 ) if H 1 (L 2 ) = 0, in particular if deg(L) ≥ g.
Assume that (iii) holds. Brill-Noether theory implies that any line bundle L with h 0 (L) ≥ 2 is a limit of globally generated ones 1 . So (i) follows from Proposition 1.

Hyperelliptic curves Proposition Assume that C is hyperelliptic, and let H be the line bundle on
Proof. Let π : C → P 1 be the two-sheeted covering defined by |H|. Let us say that an effective divisor D on C is simple if it does not contain a divisor of the form π * p for p ∈ P 1 . We will need the following well-known lemma: Proof of Lemma 1. 1) Put := g − 1 − k and d := deg(D). Recall that K C H g−1 . Thus by Riemann-Roch, the first assertion is equivalent to h 0 (H (−D)) = h 0 (H ) − d. We have H 0 (C, H ) = π * H 0 (P 1 , O P 1 ( )); since D is simple of degree ≤ + 1, it imposes d independent conditions on H 0 (C, H ), hence our claim.
2) Let k be the greatest integer such that h 0 (L ⊗ H −k ) > 0; then L = H k (D) for some effective divisor D, which is simple since k is maximal. By 1) D is the fixed part of |L|, hence is uniquely determined, and so is k. In particular the only globally generated line bundles on C of degree ≤ g are the powers of H.
Proof of the Proposition : Let E be a vector bundle on C limit of O 2 C . Consider the exact sequence where we can assume deg(L) ≤ g (Remark 1). By Lemma 1 we have After tensor product with H k , the corresponding cohomology exact sequence reads By semi-continuity we have h 0 (E ⊗ H k ) ≥ 2h 0 (H k ) = 2k + 2; the only possibility is D = 0 and ∂ = 0. But ∂(1) is the class of the extension (1), which must therefore be trivial; hence E = H k ⊕ H −k .
1 ↑ Indeed, the subvariety W r d of Pic d (C) parametrizing line bundles L with h 0 (L) ≥ r + 1 is equidimensional, of dimension g −(r +1)(r +g −d); the line bundles which are not globally generated belong to the subvariety W r d−1 +C, which has codimension r.

Examples of indecomposable limits
To prove that some limits of O 2 C are indecomposable we will need the following easy lemma: Lemma 2. Let L be a line bundle of positive degree on C, and let be an exact sequence. The following conditions are equivalent: After tensor product with L, the cohomology exact sequence associated to (2) gives where ∂ maps 1 ∈ H 0 (O C ) to the extension class of (2). Thus (ii) implies that i is an isomorphism, hence (iii). (iii) ⇒ (i): If E is decomposable, it must be equal to L ⊕ L −1 by unicity of the destabilizing bundle. But this implies h 0 (E ⊗ L) = h 0 (L 2 ) + 1.

The following construction was suggested by N. Mohan Kumar:
Proposition 4. Let C ⊂ P 2 be a smooth plane curve, of degree d. For 0 < k < d 4 , there exist extensions such that E is indecomposable and is a limit of O 2 C .
Proof. Let Z be a finite subset of P 2 which is the complete intersection of two curves of degree k, and such that C ∩ Z = ∅. By [S,Remark 4.6], for a general extension the vector bundle E is a limit of O 2 P 2 ; therefore E |C is a limit of O 2 C . The extension (3) restricts to an exact sequence To prove that E |C is indecomposable, it suffices by Lemma 2 to prove that h 0 (E |C (k)) = h 0 (O C (2k)). Since 2k < d we have h 0 (O C (2k)) = h 0 (O P 2 (2k)) = h 0 (E(k)), so in view of the exact sequence The exact sequence (3) gives an injective map 2k − 3)). Now since Z is a complete intersection we have an exact sequence since 4k < d we have H 2 (O P 2 (d −4k −3)) = 0, hence H 1 (I Z (d −2k −3)) = 0, and finally H 1 (E(d −k −3)) = 0 as asserted.
We can also perform the Strømme construction directly on the curve C, as follows. Let L be a base point free line bundle on C. We choose sections s, t ∈ H 0 (L) with no common zero. This gives rise to a Koszul extension We fix a nonzero section u ∈ H 0 (L 2 ). Let L be the pull-back of L on C × A 1 . We consider the complex ("monad") where λ is the coordinate on A 1 . Let E := Ker β/ Im α, and let E := E |C×{0} .
Lemma 3. E is a rank 2 vector bundle, limit of O 2 C . There is an exact sequence 0 → L → E → L −1 → 0; the corresponding extension class in H 1 (L 2 ) is the product by u 2 ∈ H 0 (L 4 ) of the class e ∈ H 1 (L −2 ) of the Koszul extension (4).
Proof. The proof is essentially the same as in [S]; we give the details for completeness.
For λ 0, we get easily E |C×{λ} O 2 C ; we will show that E is a rank 2 vector bundle. This implies that E is a vector bundle on C × A 1 , and therefore that E is a limit of O 2 C . Let us denote by α 0 , β 0 the restrictions of α and β to C × {0}. We have Ker β 0 = L ⊕ N , where N is the kernel of (u, p) : L −1 ⊕ O 2 C → L. Applying the snake lemma to the commutative diagram which fits into a commutative diagram this means that the extension (5) is the pull-back by ×u : L −1 → L of the Koszul extension (4). Now since E is the cokernel of the map L −1 → L ⊕ N induced by α 0 , we have a commutative diagram so that the extension L → E → L −1 is the push-forward by ×u of (5). This implies the Lemma.
Unfortunately it seems difficult in general to decide whether the extension L → E → L −1 nontrivial. Here is a case where we can conclude: Proposition 5. Assume that C is non-hyperelliptic. Let L be a globally generated line bundle on C such that L 2 K C . Let 0 → L → E → L −1 → 0 be the unique nontrivial extension of L −1 by L. Then E is indecomposable, and is a limit of O 2 C .
Proof. We choose s, t in H 0 (L) without common zero, and use the previous construction. It suffices to prove that we can choose u ∈ H 0 (K C ) so that u 2 e 0: since H 1 (K C ) C, the vector bundle E will be the unique nontrivial extension of L −1 by L, and indecomposable by Lemma 2. Suppose that u 2 e = 0 for all u in H 0 (K C ); by bilinearity this implies uve = 0 for all u, v in H 0 (K C ). Since C is not hyperelliptic, the multiplication map S 2 H 0 (K C ) → H 0 (K 2 C ) is surjective, so we have we = 0 for all w ∈ H 0 (K 2 ). But the pairing is perfect by Serre duality, hence our hypothesis implies e = 0, a contradiction.
Remark 2. In the moduli space M g of curves of genus g ≥ 3, the curves C admitting a line bundle L with L 2 K C and h 0 (L) even ≥ 2 form an irreducible divisor [T2]; for a general curve C in this divisor, the line bundle L is unique, globally generated, and satisfies h 0 (L) = 2 [T1]. Thus Proposition 5 provides for g ≥ 4 a codimension 1 family of curves in M g admitting an indecomposable vector bundle limit of O 2 C .
Remark 3. Let π : C → B be a finite morphism of smooth projective curves. If E is a vector bundle limit of O 2 B , then clearly π * E is a limit of O 2 C . Now if E is indecomposable, π * E is also indecomposable. Consider indeed the nontrivial extension 0 → L → E → L −1 → 0 (Remark 1); by Lemma 2 it suffices to show that the class e ∈ H 1 (B, L 2 ) of this extension remains nonzero in H 1 (C, π * L 2 ). But the pull-back homomorphism π * : H 1 (B, L 2 ) → H 1 (C, π * L 2 ) can be identified with the homomorphism H 1 (B, L 2 ) → H 1 (B, π * π * L 2 ) deduced from the linear map L 2 → π * π * L 2 , and the latter is an isomorphism onto a direct factor; hence π * is injective and π * e 0, so E is indecomposable.
Thus any curve dominating one of the curves considered in Propositions 4 and 5 carries an indecomposable vector bundle which is a limit of O 2 C .