Some examples of rank-2 Brill–Noether loci

In this paper, we construct some examples of rank-2 Brill–Noether loci with “unexpected” properties on general curves. The key examples are in genus 6, but we also have interesting examples in genus 5 and in higher genus. We relate some of our results to the recent proof of Mercat’s conjecture in rank 2 by Bakker and Farkas.


Introduction
Let C be a general curve of genus g defined over the complex numbers. The main focus of this paper is to study certain rank-2 Brill-Noether loci in the case g = 6 and, in particular, to show that B (2,10,4) is reducible (see below for the definitions); this is contrary to naïve expectations. We consider also similar situations in genus 5 and in higher genus and finish with some results on bundles computing the rank-2 Clifford index for low values of g. These examples are presented as a contribution to higher rank Brill-Noether theory, which is still far from fully understood even in rank 2.
We denote by M(n, d) (respectively, M(n, d)) the moduli space of stable bundles (respectively, S-equivalence classes of semistable bundles) of rank n and degree d on C and define B P. E. Newstead newstead@liv.ac.uk 1 Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK We show further that B (3,10,5) consists of a single point (Proposition 4.4). Also in Sect. 4, we relate our results for genus 6 to others in the literature and interpret them in terms of coherent systems.
In Sect. 5, we consider a somewhat analogous problem for g = 5. Finally, in Sect. 6, we obtain some results on bundles computing rank-2 Clifford indices for low values of g which extend those of [9] and relate them to the recent result of Bakker and Farkas [2] confirming Mercat's conjecture in rank 2 for general curves.
My thanks are due to the referee(s) for some useful suggestions.
minimal degree of a line bundle with h 0 ≥ 2) is gon(C) = g−1 2 + 2. It is clear that Cliff n (C) ≤ Cliff(C) for all n, and Mercat [12] conjectured that Cliff n (C) = Cliff(C) (actually Mercat's conjecture is a little stronger than this (see [8,Proposition 3.3]), but equivalent to it in rank 2). There are many counter-examples to this conjecture, but recently Bakker and Farkas [2] have proved that, for C a general curve of genus g, Cliff 2 (C) = Cliff(C). (2.1) The Brill-Noether locus B(n, d, k) has an "expected" dimension In fact, B(n, d, k) is smooth of dimension β(n, d, k) at a point E if and only if the Petri map is injective. For n = 1, one can define a Petri curve to be a curve for which is injective for all line bundles L. The general curve of any genus is a Petri curve and, if C is Petri and (For these and other results in classical Brill-Noether theory, see [1].) There is no analogue of these results for higher rank.
There is also a different Petri map (obtained by symmetrizing the usual Petri map with respect to the natural isomorphism E E * ⊗ K C ) One can then prove that, on a general curve, this Petri map is always injective for stable E and hence B(2, K C , k) is smooth at any point E for which h 0 (E) = k (see [22]). There are also partial results on non-emptiness for B(2, K C , k) for all g [21] (see also [10,24]) and complete results for small values of g [3]. Some detailed results for k ≤ 3 can be found in [7, section 7] and for k = 4 in [6]. By a subpencil of a bundle E, we mean a rank-1 subsheaf L of E such that h 0 (L) = 2. The following lemmas will be useful. Proof This is the rank-2 case of [19,Lemma 3.9]. Corollary 2.2 Let C be a general curve of genus g ≥ 6 and E a semistable bundle with d E = 2g − 2 which computes Cliff 2 (C). If g = 9, suppose in addition that E is stable. If either g is even or det E K C , then E is expressible in the form Moreover, all sections of Proof Suppose first that g = 2s, so that Cliff 2 (C) = Cliff(C) = s − 1 and h 0 (E) = s+2. If E does not admit a subpencil, then, by Lemma 2.1, h 0 (det E) ≥ 2s+1 = g+1, a contradiction. Now suppose that g = 2s + 1, so that Cliff 2 (C) = s and again h 0 (E) = s + 2. Now, by Lemma 2.1, h 0 (det E) ≥ 2s + 1 = g. Since det E K C , this is again a contradiction. So E admits a subpencil.
For g = 6, the only possibility is given by (2.2). For g ≥ 7, the existence of (2.2) follows from [9, Proposition 7.2 and Theorem 7.4]. Using Riemann-Roch, it is easy to check that so all sections of L * ⊗ K C must lift to E.

Lemma 2.4 There exist non-split exact sequences
fails to be surjective. Such extensions are classified up to isomorphism by P((Coker m) * ).
Proof The extensions for which all sections lift are classified by the kernel of the natural map The map m is the dual of this map.
The following lemma is undoubtedly well known, but I have been unable to locate a reference.

Lemma 2.5 Let F be a vector bundle on C with h 1 (F) ≥ r for some positive integer r . Then, for τ a general torsion sheaf of length t ≤ r and
Proof By induction, it s clearly sufficient to prove this when t = 1. In this case τ = C p for a general point p ∈ C. Dualising (2.3) and tensoring by K C , we obtain an exact sequence Now note that h 0 (F * ⊗ K C ) = 0 and for general p and the general homomorphism giving the required result.
Finally, we recall that a coherent system on C of type (n, d, k) is a pair (E, V ) consisting of a vector bundle E of rank n and degree d and a subspace V of H 0 (E) of dimension k. There is a concept of α-stability for coherent systems for α ∈ R and moduli spaces G(α; n, d, k) and G(α; n, d, k) exist. (For basic information on this construction, see [4].) The definition of α-stability depends on the α-slope of (E, V ) defined by μ α (E, V ) := d+αk n .

A reducible Brill-Noether locus
In this section, we prove our first main theorem. While the key case is for curves of genus 6, the theorem in fact holds for infinitely many values of the genus.

Lemma 3.1 Let C be a general curve of genus g
Moreover, if L ∈ B(1, g − λ, λ), then L is generated and h 0 (L) = λ.
For C as in Lemma 3.1, a simple calculation gives Proof The fact that B(2, K C , 2λ) and B(2, K C , 2λ + 1) are both non-empty is proved in [3] for λ = 2 and in [18,21] for λ ≥ 3. The rest of the first assertion is proved in [22]. To obtain bundles in B(2, 2g − 2, 2λ) which do not have determinant K C , we consider exact sequences where L 1 , L 2 are distinct elements of B(1, g − λ, λ) (these exist by Lemma 3.1) and the p j are distinct points of C. The general such extension gives rise to a stable bundle E by [11,; moreover the homomorphism E * → L * i obtained by dualising (3.2) is surjective. By Riemann-Roch, h 1 (L i ) = 2λ − 1, so Lemma 2.5 implies that, for a general choice of p j and (3.2), Moreover, for general p j , we have We now show that, for a generic choice of the p j , the Petri map of E is injective. This will prove that E belongs to a unique irreducible component , which is evidently not contained in B(2, K C , 4). In fact, the Petri map and It is sufficient to prove that both these maps are injective. Now we have a commutative diagram where the vertical arrows are induced by the homomorphism E * → L * 1 . The lower horizontal map is injective since C is Petri, so Since E * → L * 1 is surjective, is injective. Hence Ker μ 1 = 0 and μ 1 is injective. The same argument applies to μ 2 , completing the proof that B 0 has dimension 4λ(λ − 1) − 3.

Remark 3.3
The fact that B(2, 2g−2, 2λ) has a component of dimension 4λ(λ−1)−3 is proved in [20]. The argument in the proof above, using [11], is more precise and shows that there is a component not contained in B(2, K C , 2λ). On the other hand, it is not proved in [11] that the component B 0 has dimension 4λ(λ − 1) − 3, so we need to prove this directly.
Since d L ⊗L = 10, this implies that det E = L ⊗ L K C . It follows that every E ∈ B(2, 10, 5) can be expressed in the form (4.1) with L = L * ⊗ K C . By classical Brill-Noether theory, there are five choices for L [1, p.211 formula (1.2)]. However, since C can be embedded in a K3 surface S for which Pic S is generated by the class of C, these five line bundles all determine the same bundle E 2,10,5 (see the paragraph following the statement of [23, Théorème 0.1]).
The following proposition gives an example of a non-empty rank-3 Brill-Noether locus on C with negative Brill-Noether number.
Dualising, we obtain Since h 0 (E * ) = 0, it follows that h 0 (F) = 5. It remains to prove that F is stable. If L is a quotient line bundle of F, then L is generated and, since L * ⊂ F * and h 0 (F * ) = 0 by (4.3), h 0 (L * ) = 0. Hence h 0 (L) ≥ 2 and d L ≥ 4 > μ(F). Now suppose that G is a stable rank-2 quotient bundle of F. Then G is generated by the image V of H 0 (E) * in H 0 (G) and h 0 (G * ) = 0, so dim V ≥ 3. Let K be the kernel of the canonical surjection V ⊗ O C → G. If dim V = 3, then K is a line bundle with h 0 (K * ) ≥ 3, so d G = −d K ≥ 6. On the other hand, the homomorphism E * → K is non-zero, since otherwise E * would map into a proper direct factor of H 0 (E) * ⊗O C , a contradiction. Hence K * ⊂ E and −d K ≤ 4 by stability of E. This is a contradiction. It follows that dim V ≥ 4 and hence d G ≥ 8 since Cliff(G) ≥ Cliff 2 (C) = 2. Thus F is stable. We define E 3,10,5 := F, so that (4.3) becomes (4.2).
Conversely, let F ∈ B (3,10,5). We have already observed that h 0 (F) = 5. If F is not generated, then, applying an elementary transformation, there exists a semistable bundle of rank 3 and degree 9 with h 0 = 5; this contradicts the fact that Cliff 3 (C) = 2. We can therefore define a bundle G of rank 2 and degree 10 by the exact sequence Now suppose L is a quotient line bundle of G and let V be the image of H 0 (F) * in H 0 (L). Arguing as above, we have dim V ≥ 2. If dim V ≥ 3, then d L ≥ 6 > μ(G). If dim V = 2, then d L ≥ 4. Moreover, by stability of F, the kernel of the surjection This gives a contradiction, so G is stable and hence G E 2,10,5 . So (4.4) becomes (4.2) and F E 3,10,5 .
to fail to be surjective. Calculating dimensions, the LHS of (4.6) has dimension 6, while the RHS has dimension 7, so surjectivity does indeed fail. Moreover, by the base-point free pencil trick, the kernel of (4.6) is H 0 (L * ⊗ L * ⊗ K C ), which is zero since L L * 1 ⊗ K C . It follows that the cokernel of (4.6) has dimension 1, so, for any given L , the extension is unique up to isomorphism. By classical Brill-Noether theory, the bundles L form an irreducible variety of dimension β(1, 6, 2) = 4. Hence all such extensions belong to a single irreducible component   (2,10,4) Proof The proof is similar to that of Corollary 4.3, the key point being that any line bundle L with d L ≤ 5 has h 0 ≤ 2.

Bundles computing the Clifford index
By [2,Proposition 11], the bundles computing Cliff 2 (C) on a general curve of genus g have either h 0 = 4 or degree 2g − 2 and h 0 = 2 + g−1

2
. This substantially improves [9,Theorem 7.4]. Bakker and Farkas prove further that the second possibility does not arise when g is even and g ≥ 10 [2, Theorem 4] and conjecture that the same is true for g odd, g ≥ 15. In fact, for g odd, g ≥ 15, B(2, K C , g+3 2 ) = ∅ and all E ∈ B(2, 2g − 2, g+3 2 ) can be expressed in the form (2.2) with L L [2, Remark 13], but it is not known whether any such exist.
It is stated but not formally proved in [3] that B(2, K C , 5) is non-empty. This is proved in [21] and [9,Proposition 7.7]. The more precise statement that B(2, K C , 5) is a Fano 3-fold of Picard number 1 and genus 7 is [15, Theorem 8.1] (see also [13,Theorem 4.13]); this holds for all non-tetragonal curves of genus 7. By classical Brill-Noether theory, B(1, 6, 3) = ∅, so B(2, 12, 5) = B (2,12,5). Moreover, by Corollary 2.2, any bundle E ∈ B(2, 12, 5) \ B(2, K C , 5) can be expressed in the form (2.2) with L , L ∈ B (1,5,2), L L . It is easy to see that any bundle given by a non-trivial extension (2.2), for which all sections of L * ⊗ K C lift, is stable, but it is not known whether such extensions exist. The only way to achieve equality is with d L = 6 and h 0 (L) = 1 and then all sections of L * ⊗ K C must lift to E. On the other hand, if E ∈ B(2, 24, 8) \ B (2, K C , 8), then E is expressible in the form (2.2) with L , L ∈ B (1,8,2), L L . It is not clear whether there are any L, L for which there exist non-trivial extensions of this form for which all sections of L * ⊗ K C lift, but, if these do exist, E is necessarily stable. Note that, in this case, [2,Proposition 11] does not apply since the general curve of genus 13 cannot be embedded in a K3 surface.