Nonemptiness and smoothness of twisted Brill-Noether loci

Let $V$ be a vector bundle over a smooth curve $C$. In this paper, we study twisted Brill-Noether loci parametrising stable bundles $E$ of rank $n$ and degree $e$ with the property that $h^0 (C, V \otimes E) \ge k$. We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the twisted Brill-Noether loci are nonempty, and in many cases have a component which is generically smooth and of the expected dimension. Along the way, we prove the irreducibility of certain components of both twisted and"nontwisted"Brill-Noether loci. We end with an example of a general bundle over a general curve having positive-dimensional twisted Brill-Noether loci with negative expected dimension.


Introduction
Let C be a smooth projective curve over an algebraically closed field K of characteristic zero. A fundamental feature of the geometry of C and Pic d (C) is the Brill-Noether locus (1.1) W r d (C) = {L ∈ Pic d (C) : h 0 (C, L) ≥ r + 1}. These objects have been much studied. The expected dimension of W r d (C) is the Brill-Noether number ρ(g, d, r) = g −(r +1)(g −d+r) where g is the genus of C; every irreducible component has dimension ≥ ρ(g, r, d) and a great deal is known about these loci (for details, see Section 2).
A natural generalisation of (1.1) to vector bundles of higher rank is given as follows. We denote by U (n, e) the moduli space of stable bundles of rank n and degree e over C. This is an irreducible quasiprojective variety of dimension n 2 (g − 1) + 1. The generalised Brill-Noether locus B k n,e is defined set-theoretically by B k n,e = {E ∈ U (n, e) : h 0 (C, E) ≥ k}. (In this notation, W r d (C) is written B r+1 1,d .) These loci have also been studied in much detail, although the results for the case n = 1 do not necessarily generalise. Brill-Noether loci are also closely related to moduli of coherent systems, that is, pairs (V, Λ) where V is a vector bundle and Λ a subspace of H 0 (C, V ) of a fixed dimension.
In the present work we study another generalisation of B k n,e , which to our knowledge was first defined in [TiBT92,§2]. Fix a vector bundle V over C of rank r and degree d (not necessarily semistable). Then the twisted Brill-Noether locus B k n,e (V ) is defined set-theoretically by As outlined in [TiB14,§1], the construction of B k n,e in [GTiB09,§2] is easily generalised to B k n,e (V ), substituting a vector bundle V for O C in the appropriate places. (In §2 we will give a slightly more general version of this construction.) In particular, B k n,e (V ) is a determinantal locus. Thus if h 0 (C, V ⊗ E) = k, then the expected dimension of B k n,e (V ) at E is given by the twisted Brill-Noether number ρ k n,e (V ) : = dim U (n, e) − k (k − χ(C, V ⊗ E)) = n 2 (g − 1) + 1 − k (k − dn − re + nr(g − 1)) .
Provided that this number is less than dim U (n, e), every irreducible component of B k n,e (V ) has dimension at least equal to ρ k n,e (V ). If ρ k n,e (V ) ≥ dim U (n, e) then B k n,e (V ) = U (n, e). In the case k = 1 with V of integral slope h, ρ 1 1,g−1−h = g−1 and B 1 1,g−1−h (V ) is expected to be a divisor Θ V in the Picard variety Pic g−1−h (C). When Θ V is a divisor, it is called a generalised theta divisor. These have been much studied; see [Bea06] for an overview. See also [Bri15] for results on the singular loci of Θ V . It can also happen for special V that B 1 1,h (V ) fails to be a divisor; see [Ray82], [Pop99] and [Pau08] for some examples. If V does not have integral slope, then the theta divisor of V , if it exists, belongs to U (n, e) for some n ≥ 2. See [Pop13] for a survey of results on this type of generalised theta divisor.
Note also the connection with varieties of subbundles of a vector bundle V . If we denote by M n,e (V ) the variety of stable subbundles of V of rank n and degree e, there is a natural morphism M n,e (V ) → B 1 n,e (V ). In particular, when n = 1 and e is maximal, this is a question of maximal line subbundles. In the case r = 2, these have been studied for a long time, dating back to [LN83]; for more recent work and all r, see [Oxb00], [LN03] and [Hol04]. For n > 1, see [RTiB99,Theorem 0.3] and [BPL98].
When n = 1, it turns out that the basic results of classical Brill-Noether theory generalise, at least when V is a general stable bundle; for details, see Theorem 2.1. This study was initiated in [Hir88]. Our purpose in this article is to study the case n > 1.
In §2, we give more details on some of the background material mentioned in the introduction. In §3, we construct the twisted Brill-Noether locus B k (V, E) associated to a pair of families of bundles over C, with B k n,e (V ) as a special case. After listing some elementary properties, we develop some more tools. We construct parameter spaces for certain "twisted coherent systems", generalising the loci G r d (C) in [ACGH85] and the moduli spaces of αstable coherent systems, although we do not discuss stability or moduli. Furthermore, in §3.6, we discuss the twisted Brill-Noether lociB k n,e (V ) where strictly semistable bundles are admitted.
In §4 we give two applications of the machinery set up in §3. In Theorem 4.1, we generalise Theorem 2.1(5) to families of vector bundles which are general in the sense of [TiB14]. We also find that, for a certain range of values of k, the Brill-Noether locus B k r,d possesses a uniquely determined irreducible component B k r,d PTI (Theorem 4.3); this is interesting because very little is known in general about irreducibility of B k r,d for k ≥ 2 and r > 1.
In §5, we turn to twisted Brill-Noether loci B k n,e (V ) for n > 1 and k ≥ 2, which to our knowledge remain relatively little studied. We will answer some of the basic questions on nonemptiness and smoothness in this case. Our first result is: Theorem 1.1. Let C be a smooth curve of genus g and V any vector bundle of rank r and degree d over C. Let e 0 and k 0 be integers satisfying ρ k 0 1,e 0 (V ) ≥ 1. Then for all n ≥ 2, for all e ≥ ne 0 + 1 (resp., e ≥ ne 0 ) and for 1 ≤ k ≤ nk 0 , the twisted Brill-Noether locus B nk 0 n,e (V ) (resp.,B nk 0 n,e ) is nonempty.
This directly generalises both the main result and the construction of [Mer99].
We are also interested in generically smooth components of the loci B k n,e (V ). Our approach turns out to require the existence of certain bundles with well-behaved rank-1 twisted Brill-Noether loci and which are generically generated. In §6 we construct such bundles for some values of r, g and d. We then prove in §7 our main result: Theorem 1.2. Let C be a general curve and r, l, m integers with l := g r and 0 ≤ m ≤ l − 1. If m = 0, suppose that g ≡ 0 mod r. Write k 0 = l − m and let d, e 0 be integers with d + re 0 = r(g − 2) + k 0 . Suppose that e and k are integers satisfying Then, for general V ∈ U (r, d), the twisted Brill-Noether locus B k n,e (V ) has a component B k n,e (V ) 0 which is generically smooth and of the expected dimension.
The conditions here may look rather restrictive. Note however that (1.2) is more or less equivalent to those of [Tei91] and [Mer99]. Moreover, if k < re + nd − rn(g − 1), then B k n,e (V ) = U (n, e).
To prove that B k n,e (V ) is generically smooth and of the expected dimension at a point E, we have to show that the generalised trace map is injective (details in §3). For this type of question, Teixidor i Bigas's generalisation of limit linear series to vector bundles of higher rank has been applied in many situations; for example [Tei91], [TiB08], [CMTiB18] and [TiB14]. Although we do not use limit linear series directly, several of our proofs rely on the main result of [TiB14].
Finally, in Section 8, we describe some twisted Brill-Noether loci which are non-empty but have negative expected dimension. These are closely connected with varieties of maximal subbundles and we exploit results of [Oxb00] in discussing them.
Acknowledgements: We thank André Hirschowitz for answering several questions on rank stratifications.

Notation:
We work over an algebraically closed field K of characteristic zero. We denote a locally free sheaf and the corresponding vector bundle by the same letter. If F is an O C -module, we abbreviate H i (C, F ), h i (C, F ) and χ(C, F ) respectively to H i (F ), h i (F ) and χ(F ). If D is a divisor on C, we denote F ⊗ O C (D) by F (D). The fibre of a bundle V at p ∈ C will be denoted V | p . If V → S × C is a family of bundles parametrised by S, we denote the restriction V| {s}×C by V s . We suppose throughout that k ≥ 1.

Background
In this section, we expand on the background to our paper already referred to in the introduction.The fundamental results on W r d (C) are as follows (see [ . Many of the basic questions on nonemptiness of B k n,e were answered in [Tei91] and [Mer99], and more detailed results have been obtained in several cases. However, analogues of statements (i)-(v) above may be false in higher rank. See [GTiB09] for an overview of the theory and a survey of results and techniques. For the links between Brill-Noether theory and the moduli of coherent systems, see [Bra09] and [BGMN03]. See also [New11] for a survey of results and open problems on coherent systems; note however that there are many more recent results in this area.
When B k n,e (V ) has the expected dimension, one has B k+1 n,e (V ) ⊆ Sing B k n,e (V ) . This containment may, however, be strict. See [CMTiB11] for a detailed discussion of As already remarked, for n = 1, analogues of several of the fundamental results for W r d (C) are also valid for sufficiently general bundles of higher rank: Theorem 2.1. Let C be any curve of genus g ≥ 2. Let V be a vector bundle of rank r and degree d. Let k ≥ 1 and e be integers.
(1) If ρ k 1,e (V ) ≥ 0, then B k 1,e (V ) is non-empty. The infinitesimal study of the Brill-Noether loci is the key to the proofs of (3) and (4). We do not include this here because we will describe it in detail for B k n,e (V ) (and indeed for families of bundles) in the next section.

Preliminaries on twisted Brill-Noether loci
Although generalised Brill-Noether loci are by now very familiar objects, there are fewer sources focusing primarily on twisted Brill-Noether loci. We will therefore give a detailed introduction to the subject with emphasis on functorial aspects.
3.1. The twisted Brill-Noether locus of a pair of families. Let V → S × C be a family of bundles of rank r and degree d, and E → T × C a family of bundles of rank n and degree e. Set-theoretically, we define Scheme-theoretically, this is a determinantal locus, as we will now show using a standard construction. Let D be an effective divisor of large degree on C satisfying h 1 (V s ⊗E t (D)) = 0 for all (s, t) ∈ S × T . We have a diagram of projections Then over S × T × C, we have the short exact sequence Pushing down to S × T , we obtain a complex γ : K 0 → K 1 of locally free sheaves satisfying In particular (see [ACGH85, Chapter 2]), the locus B k (V, E) has a natural scheme structure and every component of it has dimension at least From the determinantal description it also follows that B k+1 (V, E) ⊆ Sing B k (V, E) , and moreover that the loci B k (V, E) define a rank stratification on S × T . We will return to this aspect in §3.5.
Remark 3.1. This construction is symmetric in V and E. It is functorial in the sense that if φ : S ′ → S and ψ : T ′ → T are morphisms, then Definition 3.2. Let V be a vector bundle of rank r and degree d, considered as a family over Spec K × C. By [NR75, Proposition 2.4] there exists anétale cover U (n, e) → U (n, e) (which can be taken to be the identity if gcd(n, e) = 1) which carries a Poincaré family E → U (n, e) × C. Then the twisted Brill-Noether locus B k n,e (V ) is defined as the image of the moduli map B k (V, E) → U (n, e). Writing χ = re + nd − nr(g − 1), the expected dimension of B k n,e (V ) is the Brill-Noether number ρ k n,e (V ) = ρ k n,e,r,d := dim U (n, e) − k(k − χ). The following is straightforward to check: Proposition 3.3. Let V be any bundle of rank r and degree d over C.
(1) B k n,e (V ) is a proper sublocus of U (n, e) only if k > χ.
(2) If V is stable, then B k n,e (V ) is non-empty only if re + nd > 0 or (n, e) = (r, −d). In the latter case, is non-empty only if re + nd > 0 or re + nd = 0 and r ≥ n.
(4) For any line bundle L of degree ℓ, there is a canonical isomorphism 3.2. The tangent spaces of B k (V, E). We now recall some standard facts on deformations of bundles and sections. Suppose W is a vector bundle with Thus the space of deformations preserving all sections of W is exactly We are interested in the case where W is of the form V ⊗E and v is the class of a product of deformations b ∈ H 1 (End V ) and h ∈ H 1 (End E) of V and E respectively. By for example inspectingČech cocycles, we see that More generally, let us consider again the families V → S × C and where κ is the Kodaira-Spencer map.
(1) The Zariski tangent space to B k (V, E) at (s, t) is given by (2) In particular, suppose that S × T is smooth at (s, t).
we obtain (1). By (3.1) and (1), we see that B k (V, E) is smooth of the expected dimension at (s, t) if and only if By (3.3), this is equivalent to the surjectivity of ∪ • c • κ.
3.3. The Petri trace map. For any vector bundle W , it is well known that via Serre duality, ∪ : Let us use this map to reformulate (3.4). Firstly, some notation: For bundles V and E, there is a vector bundle map inducing the cohomology map (3.2) considered above. We write c V and c E for the restrictions to the first and second factors respectively. Recall also that for any bundle W , the transpose gives a canonical identification of End W and End W * , which we will use freely. Fix a vector bundle V . If we identify End V with (End V ) * by the trace pairing, a diagram chase shows that the trace map tr : which is dual to c E , and an induced map By Serre duality and the above discussion, tr E is dual to Then by linear algebra, c : We can now formulate a dual version of Proposition 3.4.
The most important corollary of this proposition is: Corollary 3.6. Let V be a bundle of rank r and degree d.
is smooth and of the expected dimension at E if and only if Proof. This follows from Proposition 3.5 applied to the family V consisting of the single bundle V and a local universal family E for E, together with the fact that the Kodaira-Spencer map for the family E at E is an isomorphism.
It will be convenient to make the following definition.
Definition 3.7. For fixed V , write µ E for the composed map tr E • µ in (3.5). We say that is injective, we say that V is Petri trace injective.
Next, as it will be central to several proofs, let us state [TiB14, Theorem 1.1] precisely.
Theorem 3.8. Let C be a general curve and V a general vector bundle over C. Then for any degree e and L ∈ Pic e (C), the Petri trace map This motivates another definition. For families V and E, let us fix the effective divisor D in §3.1 and recall the complex γ : K 0 → K 1 . As K 0 is locally free, we have a Grassmannian bundle π : Gr(k, K 0 ) → S ×T . We define G k (V, E) := {Λ ∈ Gr(k, K 0 ) : γ| Λ = 0}. This is a parameter space for triples (V, E, Λ) where Λ is a k-dimensional subspace of H 0 (V ⊗ E). It seems natural to call such a triple a "twisted coherent system", but we do not pursue questions of moduli or stability here. When the family V consists of a single vector bundle V , we write also G k (V, E).
(1) Suppose h 0 (V s ⊗ E t ) ≥ k. There is an exact sequence where the last map is defined by ∪ • c • κ as in (3.3), followed by restriction to Λ. (2) The locus G k (V, E) is smooth and of dimension (3.1) at (s, t, Λ) if and only if the restricted map in a neighbourhood of (s, t).
Proof. Statements (1) and (2)  3.5. A sufficient condition for the existence of good components. Note that is a stratification of B k (V, E) by closed subsets. The following proposition makes use of this stratification.
Proposition 3.12. Suppose S × T is smooth at (s, t) and for some k ′ ≥ χ there exists contains a component which is generically smooth and of the expected dimension.
Proof. We prove this by descending induction on k. For k = k ′ , the result follows immediately from Proposition 3.4. Now suppose χ ≤ k < k ′ and that the proposition holds Then, by Proposition 3.11, G k (V, E) is smooth of the expected dimension at (V, E, Λ).
Since k ≥ χ, it follows from (3.1) that every component of G k (V, E) has dimension greater than the dimension of π −1 (B k+1 (V, E)) at (V, E). Hence, there exists a point ( This proposition illustrates a general principle that, from the existence of just one pair of bundles with good properties, one can obtain a detailed picture of the geometry of several of the strata. This will be used on a number of occasions later. Furthermore; it was noted in [GTiB09, §4] that, for r ≥ 2, the locus B k r,0 is empty, but B k r,0 = ∅ for k ≤ r. We can generalise this example to twisted Brill-Noether loci. Suppose V ∈ U (r, d) is a stable bundle, n > r and re + nd = 0. Then, according to Proposition 3.3 (2), B k n,e (V ) = ∅ for all k. However, taking E = V * ⊕ F , where F is semistable and µ(F ) = µ(E), we see that [E] ∈B 1 n,e (V ).

Two irreducibility results
Here we will give some applications of the machinery assembled in the previous section.
Theorem 4.1. Let V → S × C be a family of Petri general vector bundles of rank r and degree d parametrised by a smooth irreducible base S. Assume that ρ k 1,e,r,d = g − k(k − (d + re) + r(g − 1)) ≥ 1. Then the locus B k (V, P) ⊆ S ×Pic e (C) is an irreducible variety of dimension dim S +ρ k 1,e,r,d which is singular precisely along B k+1 (V, P).
Proof. The fibre of B k (V, P) over each s ∈ S is exactly B k 1,e (V s ). Since ρ k 1,e,r,d ≥ 1, by Theorem 2.1 (1) and (2) this fibre is nonempty and connected. As S is irreducible, it follows that B k (V, P) is connected. As the fibres of G k (V, P) → B k (V, P) are Grassmannians, G k (V, P) is also connected.
Furthermore, as V s is Petri general for all s, by Proposition 3.11(2) in fact G k (V, P) is smooth. Therefore G k (V, P) is irreducible. As B k (V, P) is the image of G k (V, P) by a morphism, B k (V, P) is also irreducible.
The last statement follows from Petri generality and Proposition 3.5.
Remark 4.2. Suppose that C is a general curve. Since, by Theorem 3.8, a general bundle V in U (r, d) is Petri general, in particular B k 1,e (V ) is irreducible for general V . Thus we recover [Hir88, Théorème 1.2].

Irreducibility of Petri trace injective loci.
Here we give an application to "nontwisted" Brill-Noether loci. {V ∈ B k r,d :μ : Λ ⊗ H 0 (K C ⊗ V * ) → H 0 (K C ) is injective for some Λ ∈ Gr(k, H 0 (V ))} In particular, the locus of Petri trace injective bundles in B k r,d is irreducible. Proof. By [NR75, Proposition 2.6], there exists a smooth irreducible variety M admitting a Poincaré family V → M × C such that every stable bundle of rank r and degree d over C is represented in M. Throughout, we will writeṼ for a point of M lying over V ∈ U (r, d).
Set e = 0 and let P → Pic 0 (C) × C be a Poincaré bundle. We consider the locus G k (V, P) → M × Pic 0 (C) defined in §3.4. Define By Theorem 2.1 (3), (4) and by hypothesis, this is non-empty. By Proposition 3.11 (2), it is precisely the smooth locus of G k (V, P).
Let t : G k (V, P) → U (r, d) be the morphism given by t(Ṽ , L, Λ) = V ⊗ L. By definition, t(G k ) is the locus (4.1). By Brill-Noether theory, t(G k ) has the expected dimension, so is a union of components of B k r,d . Thus it will suffice to show that G k is irreducible. Let p be the projection map G k (V, P) → M × Pic 0 (C) → M. By Theorem 2.1 (1), this is surjective. We claim that there is a unique irreducible component of G k which dominates M. Suppose that X 1 and X 2 are components of G k such that p(X 1 ) and p(X 2 ) are both dense in M. LetṼ be a general point of p(X 1 ) ∩ p(X 2 ). The fibre p −1 (Ṽ ) is identified with here viewingṼ as a singleton family. By hypothesis and by Theorem 3.8, we may assume V is Petri general. Thus, by the proof of Theorem 4.1, the locus G k (Ṽ , P) is irreducible. Hence by semicontinuity of fibre dimension, p −1 (Ṽ ) is generically contained in both X 1 and X 2 . In particular, X 1 ∩ X 2 is nonempty. Since G k is smooth, the only possibility is that X 1 = X 2 . Therefore, to conclude, it will suffice to show that the restriction of p to any component X of G k is dominant. To see this: Let (Ṽ , L, Λ) be a point of X. By Proposition 3.11 (2) applied to the locus (4.2), we have 1)).
On the other hand, again by Proposition 3.11 (2), since X is smooth, we have Thus dim(p(X)) ≥ dim(M). As M is irreducible, p(X) is dense in M. This completes the proof.
5. Nonemptiness of B k n,e (V ) andB k n,e (V ) We now prove Theorem 1.1 using the method of [Mer99]. This is very straightforward; the necessary ingredients already exist by Theorem 2.1 and the results in [Mer99] on stability of elementary transformations. We note that Mercat's construction was used in a similar way in [BBPN08, §6] to show the nonemptiness of certain moduli spaces of coherent systems.
Proof of Theorem 1.1. By hypothesis, B k 0 1,e 0 (V ) is of positive dimension. Thus we can find mutually nonisomorphic line bundles L 1 , . . . , L n of degree e 0 such that be a general elementary transformation. We have the cohomology sequence

then it follows easily from [Mer99, Théorème A.5] that
E is a stable vector bundle, and the result follows for e > ne 0 . If τ = 0 then E = ⊕ n i=1 L i gives an element ofB nk 0 n,ne 0 , and the result follows also in the case e = ne 0 .
In the next sections, we will refine this statement in some cases.

Generatedness of Petri general bundles
To prove the existence of components of B k n,e (V ) which are generically smooth and of the expected dimension, it will emerge to show the existence of bundles W of rank r ≥ 2 and degree d satisfying the following conditions: (1) W is Petri trace injective. Equivalently, K C ⊗ W * is Petri trace injective.

Tensoring both sides with
where T is a torsion sheaf of degree (r − 1)l + m with 0 ≤ m ≤ l − 1. To ease notation, we write t := (r − 1)l + m. The set of such W is parametrised by the Quot scheme Quot 0,t (G), an irreducible variety of dimension rt.
Proof. Since the surjectivity condition is open, it is sufficient to prove the existence of one bundle W with the required property. Suppose first that t = 1 and let T = K p , where p is a point at which some section of G is nonzero. Then we can find a surjection G → T such that H 0 (G) → H 0 (T ) is surjective. Repeating this argument, we obtain the result by induction on t.
We want one more generality condition on W . For 1 ≤ i ≤ r, writeĜ i := j =i M j , and consider the sheaf Lemma 6.4. If W is sufficiently general in Quot 0,t (G), then h 0 (W ∩Ĝ i ) = 0.
Proof. Since the condition h 0 (W ∩Ĝ i ) is open, it is sufficient to find one example of an elementary transformation (6.3) for which this property holds. For this, consider elementary where T is as in (6.3). The same argument as for Lemma 6.3 shows that, for general W 1 , we have h 0 (W 1 ) = max{0, −m} = 0. Now take W = W 1 ⊕ M i .
Proposition 6.5. A general elementary transformation W ∈ Quot 0,t (G) is stable and Petri trace injective, and has h 0 (K C ⊗ W * ) generically generated.
Proof. As the M i are mutually non-isomorphic by Lemma 6.2 (3), by [Mer99, Théorème A.5] the bundle W is stable for general T . By Lemma 6.3, we have H 0 (K C ⊗W * ) ∼ = H 0 (K C ⊗G * ). From Lemma 6.2 (2), it then follows that K C ⊗ W * is generically generated. We describe H 0 (K C ⊗ W * ) more explicitly. For 1 ≤ i ≤ r, let t ′ i be a generator of H 0 (K C M −1 i ), and write t i for the image of t ′ i in H 0 (K C ⊗ W * ). Thus we obtain a splitting H 0 (K C ⊗W * ) = r i=1 K·t i . Therefore, we can write any element of H 0 (W )⊗H 0 (K C ⊗W * ) in the form where σ i = (s i,1 , s i,2 , . . . , s i,r ) is a section of G belonging to W , that is, lying in the kernel of H 0 (G) → H 0 (T ). The Petri trace is then given bȳ To analyse this, note that, by Lemma 6.2 (2), the homomorphism where the lefthand vertical map is an isomorphism by Lemma 6.2 (1) and (2). Since h 0 (K C ⊗ N −1 ) = 1 by Lemma 6.2 (2), µ N is injective. By commutativity, the composed This means that a tensor of the form (6.4) has trace zero only if s i,i = 0 for all i. Thus σ i belongs to the subsheaf W ∩Ĝ i . But by Lemma 6.4, for general W ∈ Quot 0,t (G), h 0 (W ∩Ĝ i ) = 0 amd σ i = 0 for all i. This completes the proof.
Corollary 6.6. Let g, r, l = g r and r 0 be as above. For 0 ≤ m ≤ l − 1, there exists a Petri trace injective bundle W of rank r and degree r(g − 2) + l − m with h 0 (W ) = l − m and such that K C ⊗ W * is generically generated. It is therefore sufficient to find one example of a bundle W with the stated property. For this, take W as in Corollary 6.6. The statement (6.5) is then equivalent to the injectivity of the coboundary map in the sequence By Serre duality, the coboundary map is injective if and only if the evaluation map (6.6) H 0 (K C ⊗ W * ) → K C ⊗ W * | p is surjective. By Corollary 6.6, this is true for general p.

Smoothness of twisted Brill-Noether loci
In this section, we prove our main result Theorem 1.2. The major part of the section is concerned with proving the following technical proposition.
Proposition 7.1. Let C be a general curve of genus g. Suppose e 0 and k 0 are integers satisfying and furthermore that there exists a bundle W ∈ B k 0 r,d+re 0 PTI such that for general p in C we have h 0 (W ) = h 0 (W (p)) = k 0 . Write e = ne 0 + e 1 where 1 ≤ e 1 ≤ n. Then for general V ∈ U (r, d) and for re + nd − rn(g − 1) ≤ k ≤ nk 0 , the twisted Brill-Noether locus B k n,e (V ) has a component which is nonempty, generically smooth and of the expected dimension.
We begin with a lemma which has applications to coherent systems (twisted or untwisted) as well as to twisted Brill-Noether loci. For this, let V be any bundle of rank r and degree d and consider the Grassmannian bundle G k 0 (V, P e 0 ), where P is a Poincaré family on Pic e 0 (C) × C, which parametrises pairs (L, Λ 0 ) with L a line bundle of degree e 0 and Λ 0 ⊂ H 0 (V ⊗L) a linear subspace of dimension k 0 . Suppose further that X is an irreducible component of G k 0 (V, P e 0 ) which is generically smooth of dimension Λ 1 ), . . . , (L n , Λ n ) be points of X, and write F := n i=1 L i and Λ := n i=1 Λ i . We consider elementary transformations with τ a torsion sheaf of length e 1 .
Then the restricted Petri E-trace map is injective.
Proof. Consider first the restricted Petri F -trace map of V , given by Noting that , we see that (7.4) is the direct sum of the trace maps which is a non-empty open subset of X by semicontinuity. We can assume that (L i , Λ i ) ∈ U for all i. Now let p be a point of C. For each (L, Λ 0 ) ∈ U , we have a commutative diagram where, in the second line, Λ 0 is regarded as a subspace of H 0 (V ⊗ L(p)) and the horizontal arrows are trace maps. Since X is smooth at (L, Λ 0 ), µ 0 is injective. Hence so is µ ′ 0 . Next, let A be the direct image sheaf over U × U whose fibre at ((L, Λ 0 ), (N, Since H 0 (K C ⊗ N −1 ) is constant on U , this is locally free. Furthermore, let B be the direct image sheaf over U whose fibre at ((L, Λ 0 ), (N, Λ ′ 0 )) is H 0 (K C ⊗N −1 ⊗L(p)). This is locally free of rank g.
Writeμ : A → B for the globalised Petri trace map whose restriction to ((L, Λ 0 ), (N, Λ ′ 0 )) is the trace map We have already seen that µ F is injective. Hence first b, then c, then µ E = d • c are all injective.
Proof of Proposition 7.1. Suppose that the hypotheses of Proposition 7.1 hold and let V ∈ U (r, d) be general. By Proposition 3.12, it suffices to exhibit a stable bundle E ∈ U (n, e) with h 0 (V ⊗ E) = nk 0 and such that V is Petri E-trace injective. For this, we use the construction of (7.2), where we now assume that this elementary transformation is general and the bundles L i are all distinct. It then follows from [Mer99, Théorème A.5] that E is stable.
Note next that, since h 0 (W (p)) = k 0 , we must have k 0 > d+re 0 −r(g−1), so ρ k 0 1,e 0 (V ) < g. By Theorem 2.1, it follows that the locus B k 0 1,e 0 (V ) is non-empty and irreducible of dimension ρ k 0 1,e 0 (V ) ≥ 1 (by (7.1)), and (7.6) h 0 (V ⊗ L) = k 0 for general L ∈ B k 0 1,e 0 (V ). By Theorem 3.8, we may assume also that V is Petri general. By Proposition 4.3, V ⊗ L is a general point of B k 0 r,d+re 0 PTI . It now follows from the hypotheses of Proposition 7.1 that Now let L 1 , . . . , L n be general points of B k 0 1,e 0 (V ), and write F := n i=1 L i . We can assume that h 0 (V ⊗ L i ) = k 0 for all i, so that h 0 (V ⊗ F ) = nk 0 . Now note that the condition h 0 (V ⊗ E) = nk 0 is an open condition, so it is sufficient to exhibit a single elementary transformation (7.2) satisfying this condition. In fact, by (7.7), for general p i ∈ C, we can take Finally, we note that B k 0 1,e 0 (V ) is irreducible of the expected dimension by Theorem 2.1, and moreover B k 0 +1 1,e 0 (V ) is of the expected dimension. It follows that G k 0 (V, P e 0 ) is also irreducible of the expected dimension. Now apply Lemma 7.2 with X = G k 0 (V, P e 0 ) and Λ i = H 0 (V ⊗ L i ) for all i. It follows that V is Petri E-trace injective. Now we can prove our main result on smoothness and dimension of twisted Brill-Noether loci.
Proof of Theorem 1.2. A straightforward computation shows that the numerical hypotheses and Corollary 6.7 imply that the hypotheses of Proposition 7.1 are satisfied.
Remark 7.3. By [CMTiB11, Theorem 2.10], if V is general in U (r, d) then for any E ∈ B k n,e (V ) the bundle V ⊗ E is stable. Remark 7.4.
(1) According to [RTiB99, Theorem 0.3], if 0 ≤ ρ 1 n,e (V ) ≤ n 2 (g − 1) + 1, then every component of B k n,e (V ) has dimension ρ 1 n,e (V ). The authors do not require the more stringent numerical conditions of our Theorem 1.2. Our result can be seen as a partial generalisation of [RTiB99, Theorem 0.3], although we do not show that every component of B k n,e (V ) has the expected dimension. (2) It seems reasonable to conjecture that the hypotheses of Proposition 7.1 are satisfied in more cases than those covered in Theorem 1.2. The main obstacle to generalising the theorem is to show the generic generatedness of a general bundle in B k 0 r,d+re 0 PTI in more cases. (Theoretical bounds can be deduced from (6.1).) Remark 7.5. For general C, the bundle O is Petri general and Petri trace injective. The above proofs are therefore valid and we recover [CMTiB18, Theorem 1.1] (see also [Tei91]).

A non-empty twisted Brill-Noether locus with negative expected dimension
It is well known that higher-rank Brill-Noether loci B k n,e can exhibit more complicated behaviour than their rank one counterparts. Here we give an example of a nonempty twisted Brill-Noether locus with negative Brill-Noether number, where the curve C and the bundle V are general. Firstly, we recall some facts about maximal line subbundles of vector bundles (see [Oxb00] for more general and detailed information): Suppose r|(g − 1), and set e 0 := (r − 1) g−1 r . Let V be a general bundle of rank r and degree zero. A computation shows that ρ 1 1,e 0 (V ) = 0. As V is general, by Theorem 2.1 (3) the locus B 1 1,e 0 (V ) is of dimension zero. Furthermore, for e < e 0 or k 0 > 1 we check that ρ k 0 1,e 0 (V ) < 0. Hence, for all L ∈ B 1 1,e 0 (V ), we have h 0 (V ⊗ L) = 1 and L −1 is a line subbundle of maximal degree in V . By [Oxb00, Proposition 1.4 and Lemma 2.2], B 1 1,e 0 (V ) consists of r g points (of multiplicity 1).
Proposition 8.1. Let r, g and e 0 be as above and let V be a general bundle of rank r and degree 0. Let n be an integer satisfying r < n ≤ r g . Then the twisted Brill-Noether locus B n n,ne 0 +1 (V ) has negative expected dimension n(r − n) + 1 but contains a component of dimension at least 1.
Proof. By the previous paragraph we can choose mutually nonisomorphic L 1 , . . . , L n ∈ B 1 1,e 0 (V ). Let be a general elementary transformation, where K p is the skyscraper sheaf of degree 1 supported at p ∈ C. Then E is stable by [Mer99, Théorème A.5], and h 0 (V ⊗ E) ≥ n.
The Quot scheme parametrising the elementary transformations E has dimension n; after acting by Aut (⊕ n i=1 L i ), we see that there is precisely one stable E for any given p. Thus dim B n ne 0 +1 (V ) ≥ 1. On the other hand, we compute easily that ρ n n,ne 0 +1 (V ) = n(r − n) + 1. Since by hypothesis n > r ≥ 2, this is negative. The result follows.
In exactly the same way, one can prove Proposition 8.2. Let C be a curve and V a bundle of rank r ≥ 2 and degree d over C. Suppose k 0 ≥ 1 and e 0 are integers satisfying ρ k 0 1,e 0 (V ) = 0. Assume that # B k 0 1,e 0 (V ) > k 0 r.
This example shows that even for general stable V , the twisted Brill-Noether loci can exhibit pathologies.