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 h0(C,V⊗E)≥k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^0 (C, V \otimes E) \ge k$$\end{document}. We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the 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 describe the tangent cones to the twisted 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 of characteristic zero. A fundamental feature of the geometry of C and Pic d (C) is the Brill-Noether locus These objects have been much studied. The expected dimension of W r d (C) is the Brill-Noether number (g, d, r) = g − (r + 1)(g − d + r) where g is the genus of C; every (1.1) W r d (C) = L ∈ Pic d (C) ∶ h 0 (C, L) ≥ r + 1 .

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irreducible component has dimension ≥ (g, d, r) , and a great deal is known about these loci (for details, see Sect. 2). A natural generalisation of (1.1) to vector bundles of higher rank is given as follows. We denote by U s (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 (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 [36,Sect. 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 The space U s (n, e) is an open subset of the moduli space U(n, e) of S-equivalence classes of semistable bundles of rank n and degree e. We write [E] for the S-equivalence class of a semistable E and grE for the graded bundle associated to E; grE depends only on [E]. The definition of B k n,e (V) is extended to include semistable bundles by setting As outlined in [35,Sect. 1], the construction of B k n,e in [13,Sect. 2] is easily generalised to B k n,e (V) , substituting a vector bundle V for O C in the appropriate places. (In Sect. 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 Provided that this number is less than dim U s (n, e) , every irreducible component of B k n,e (V) has dimension at least equal to k n,e (V) . If k n,e (V) ≥ dim U s (n, e) then B k n,e (V) = U s (n, e). In the case k = 1 with V of integral slope h = d∕r , we have 1 1,g−1−h (V) = 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 [2] for an overview. See also [8] for results on the singular loci of Θ V . It can also happen for special V that B 1 1,g−1−h (V) fails to be a divisor; see [28,29,31] for some examples. If V does not have integral slope, then the theta divisor of V, if it exists, belongs to U s (n, e) for some n ≥ 2 . See [30] 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) given by E ↦ E * . In particular, when n = 1 and e is maximal, this is a question of maximal line subbundles. In the case r = 2 , these have k n,e (V) ∶= dim U s (n, e) − k(k − (C, V ⊗ E)) = n 2 (g − 1) + 1 − k(k − re − nd + rn(g − 1)).

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been studied for a long time, dating back to [19]; for more recent work and all r, see [27]. For n > 1 , see [16,20,32, Theorem 0.3] and [7]. 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 [14]. Our purpose in this article is to study the case n > 1.
In Sect. 2, we give more details on some of the background material mentioned in the introduction. In Sect. 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 [1] and the moduli spaces of -stable coherent systems, although we do not discuss stability or moduli.
In Sect. 4 we give two applications of the machinery set up in Sect. 3. In Theorem 4.1, we generalise Theorem 2.1(5) to families of vector bundles which are general in the sense of [35]. 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 Sect. 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 ≥ 2 and V any vector bundle of rank r and degree d over C. Let e 0 and k 0 be integers satisfying 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 k n,e (V) (resp., B k n,e (V) ) is nonempty.
This directly generalises both the main result and the construction of [24]. 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 Sect. 6 we construct such bundles for some values of r, g and d. We then prove in Sect. 7 our main result: Theorem 1.2 Let C be a general curve of genus g ≥ 2 and r, l, m integers with l ∶= Suppose that e and k are integers satisfying Then, for general V ∈ U s (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.
Recall here that for V ∈ U s (r, d) and E ∈ U s (n, e) , the Euler characteristic (V ⊗ E) = re + nd − rn(g − 1) and moreover, if k ≤ (V ⊗ E) , then B k n,e (V) = U s (n, e) . The hypotheses of Theorem 1.2 may look rather restrictive. In fact, (1.2) is analogous to the conditions of [24,33] for the case r = 1 . However, the restriction on d + re 0 is strong and can probably be relaxed.
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 Sect. 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 [10,[33][34][35]. Although we do not use limit linear series directly, several of our proofs rely on the main result of [35].
In Sect. 8, we consider the tangent cones of B k n,e (V) , which can be studied using the same techniques as in [1,9]. Using Theorem 1.2, we describe the tangent cones as determinantal varieties and compute their degrees. We also give a geometric description of the tangent cones for large values of h 0 (C, V ⊗ E) , generalising [1, VI, Theorem 1.6 (i)] on secant varieties of canonical curves.
Finally, in Sect. 9, we describe some twisted Brill-Noether loci which are nonempty but have negative expected dimension. These are closely connected with varieties of maximal subbundles, and we exploit results of [27] in discussing them. This gives another motivation for studying twisted Brill-Noether loci: as these examples arise for general C and V (with prescribed numerical properties), twisted Brill-Noether loci give a way of systematically obtaining determinantal varieties of larger than expected dimension. This line of research will be further pursued in the future.

Notation
We work over an algebraically closed field of characteristic zero. We denote a locally free sheaf and the corresponding vector bundle by the same letter.
. 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 g ≥ 2 and 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 [1,Chapter V]): (i) Existence theorem: For any curve, Note that if (g, d, r) ≥ g , then W r d (C) = Pic d (C). Many of the basic questions on nonemptiness of B k n,e were answered in [24,33], and more detailed results have been obtained in several cases. However, analogues of statements (i)-(v) may be false in higher rank. See [13] 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 [4,5]. See also [26] 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 This containment may, however, be strict. See [9] for a detailed discussion of singular 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 (by "general", we mean stable and general in the moduli space): 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.
Proof Statement (1) was proven in [12] for general V, and for all V in [22, (2.6)]. Part (2) is [22, (2.7)]. Parts (3) and (4) follow from [35]. Lastly, (5) is [14,Théorème 1.2]. ◻ 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.

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 B k+1 n,e (V) ⊆ Sing B k n,e (V) .
Then over S × T × C , we have the short exact sequence Pushing down to S × T , we obtain a complex ∶ K 0 → K 1 of locally free sheaves satisfying In particular (see [1,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 Sect. 3.5.

Definition 3.2 Let
The following is straightforward to check:

Proposition 3.3 Let V be any bundle of rank r and degree d over C.
( is nonempty only if re + nd > 0 or (n, e) = (r, −d) . In the latter case, is nonempty only if re + nd > 0 or re + nd = 0 and r ≥ n. (4) For any line bundle L of degree , there is a canonical isomorphism given by E ↦ L −1 ⊗ E. n,e,r,d ≥ 0 (see, for example, [6,23]). It seems to be much harder to find examples for r > 1 , but Proposition 3.3 does provide some. Suppose that re + nd = 0 and r < n ; then B 1 n,e (V) = � by Proposition 3.3 (2). On the other hand, in this case, Furthermore, it was noted in [6] 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 s (r, d) is a stable bundle, n > r and re + nd = 0 . Then, by Proposition 3.

The tangent spaces of B k (V, E)
We now recall some standard facts on deformations of bundles and sections. Suppose 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 inspecting Čech cocycles, we see that More generally, let us consider again the families where is the Kodaira-Spencer map.

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 Thus for any bundle E, tensoring tr ∶ End V → O C by End E , we obtain a linear map which is dual to c E , and an induced map By Serre duality and the above discussion, tr E is dual to We can now formulate a dual version of Proposition 3.5.
The most important corollary of this proposition is:  This motivates another definition.

Remark 3.11
A curve C is Petri in the usual sense if and only if O C is a Petri general vector bundle. It is well known that the general curve C is a Petri curve.

A partial desingularisation of B k (V, E)
Here we generalise the construction G r d (C) of [1, IV.4] to twisted Brill-Noether loci. For families V and E , let us fix the effective divisor D in Sect. 3.1 and recall the complex We define 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).
) . Let us describe the Zariski tangent spaces of G k (V, E) at (s, t).

Proposition 3.12
(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 is injective.

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 3.13
Suppose S × T is smooth at (s, t) and for some k ′ ≥ 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 3.6. Now suppose ≤ k < k ′ and that the proposition holds for B k+1 (V, E) . Then, there exists Then, by Proposition 3.12, G k (V, E) is smooth of the expected dimension at (V, E, Λ) . Since k ≥ , it follows from (3.1) that every component of G k (V, E) has dimension greater than the dimension of −1 (B k+1 (V, E)) at (V, E). Hence, there exists a point (V 1 , 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.

Two irreducibility results
Here we will give some applications of the machinery assembled in the previous section.

Rank one twisted Brill-Noether loci
If C is a Petri curve, B k 1,e = W k−1 e (C) is irreducible whenever k 1,e = g − k(k − e + g − 1) ≥ 1. Let P → Pic e (C) × C be a Poincaré bundle. Proof The fibre of B k (V, P) over each s ∈ S is exactly B k 1,e (V s ) . Since 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 3.12(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 3.6. ◻

Irreducibility of Petri trace injective loci
Here we give an application to "nontwisted" Brill-Noether loci.  Let t ∶ G k (V, P) → U s (r, d) be the morphism given by t( � V, L, Λ) = V ⊗ L . By definition, t(G k ) is locus (4.1). Thus it will suffice to show that G k is irreducible.

Theorem 4.3 Suppose C is general and
Since the projection p of (4.2) is surjective, there is at least one irreducible component of G k which dominates M . We claim first that this component is unique. Suppose that X 1 and X 2 were components of G k such that p(X 1 ) and p(X 2 ) are both dense in M . Let Ṽ be a general point of p(X 1 ) ∩ p(X 2 ) . The fibre p −1 (Ṽ) is identified with here viewing Ṽ as a singleton family. By Theorem 3.9, we may assume V is Petri general. Since k 1,0,r,d ≥ 1 by hypothesis, it follows from Theorem 4.1 that G k (Ṽ, P) is irreducible. Since X 1 ∩ G k (Ṽ, P) and X 2 ∩ G k (Ṽ, P) are both components of G k ∩ G k (Ṽ, P) , it follows that X 1 ∩ G k (Ṽ, P) = X 2 ∩ G k (Ṽ, P) and, 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 3.12 (2) applied to locus (4.4), we have

On the other hand, by (4.3), we have
Thus dim(p(X)) ≥ dim(M) . As M is irreducible, p(X) is dense in M . This completes the proof. ◻

Nonemptiness of B k n,e (V) and B k n,e (V)
We now prove Theorem 1.1 using the method of [24]. This is very straightforward; the necessary ingredients already exist by  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, the need 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.
Remark 6.1 Note that the above conditions give strong bounds on d and k. By (1) and (2) and Serre duality, we have Values of d satisfying (6.1) exist if and only if k ≤ g r , which is in any case a necessary condition for (1) and (3) to hold.

The construction
Suppose g ≥ r and write g = rl + r 0 where l, r 0 are integers with 0 ≤ r 0 < r . Let D 0 be an effective divisor of degree r 0 such that (3) The bundles M 1 , … , M r are mutually nonisomorphic. (1) We have inclusions H 0 (M i ) ↪ H 0 (N) for all i. It is easy to see that

Proof
Since the dimensions agree, we obtain (1). (2) Calculating values for degrees and h 0 , this follows from Riemann-Roch.
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) . By for example the proof of [11,Lemma 4.2], this is an irreducible variety of dimension rt. 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 K C ⊗ W * generically generated.

Proof
. From Lemma 6.2(2) and since G = ⊕ r i=1 M i , it then follows that K C ⊗ W * is generically generated.
We describe H 0 (K C ⊗ W * ) more explicitly.
⋅ t i . Therefore, we can write any element of H 0 (W) ⊗ H 0 (K C ⊗ W * ) in the form 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 by To analyse this, note that, by Lemma 6.2(2), the homomorphism M i → N induces an isomorphism H 0 (K C ⊗ N −1 ) → H 0 (K C ⊗ M −1 ) . It follows that there is a commutative diagram where the left-hand 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), N is injective. By commutativity, the composed map 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. 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 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 s (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.
Note again that for V ∈ U s (r, d) and E ∈ U s (n, e) , the Euler characteristic (V ⊗ E) = re + nd − rn(g − 1) . 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 space G k 0 (V, P e 0 ) , where P e 0 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 Let (L 1 , 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 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 , we see that (7.4) is the direct sum of the trace maps 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 ) , 0 is injective. Hence so is ′ 0 . Next, let A be the direct image sheaf over U × U whose fibre at ((L, Λ 0 ), (N, 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, ) . This is locally free of rank g.
To see that E is injective, we note that K C ⊗ E * ⊗ V * ⊂ K C ⊗ F * ⊗ V * and consider the diagram of cohomology spaces We have already seen that F is injective. Hence first b, then c, then E = d•c are all injective. ◻ U ∶= (L, Λ 0 ) ∈ X ∶ X smooth at (L, Λ 0 ), h 0 (V ⊗ L) takes its minimum value , Proof of Proposition 7.1 Suppose that the hypotheses of Proposition 7.1 hold and let V ∈ U s (r, d) be general. By Proposition 3.13, it suffices to exhibit a stable bundle E ∈ U s (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 [24,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 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 Now we can prove our main result on smoothness and dimension of twisted Brill-Noether loci.

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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 7.4
For general C, the bundle O C is Petri general and Petri trace injective. Moreover, when r = 1 , Corollary 6.7 holds without the restrictive numerical conditions. The above proofs are therefore valid, and we recover [10, Theorem 1.1] (see also [33]).

Tangent cones of twisted Brill-Noether loci
Suppose Y ⊂ X are varieties, and let x ∈ Y be a smooth point of X. Recall that the tangent cone x Y to Y at x, set-theoretically, is Generalising theorems of Kempf [18] and Laszlo [21], in [9] the theory of determinantal varieties is used to describe the tangent cones to B k r,d at points where the appropriate Petri maps are injective. In [9,Remark 2.8], it is noted that the same approach can be used to describe E B k n,e (V) , which is a subvariety of T E U s (n, e) = H 1 (End E) . In the following proposition, we follow up this remark and use Theorem 1.2 to give some situations in which it applies. , respectively, then the ideal of E B k n,e (V) is generated by the minors of size As before, let E →Ũ s (n, e) × C be a Poincaré bundle, where Ũs (n, e) → U s (n, e) is a suitable étale cover. Fix a point in Ũs (n, e) lying over E, and, abusing notation, denote {v ∈ T x X ∶ v is tangent to a smooth arc in Y}.
it again by E. Recall the map ∶ G k (V, E) =∶ G k n,e (V) →Ũ s (n, e) . By hypothesis, E | Λ⊗H 0 (K C ⊗E * ⊗V * ) is injective for all Λ ∈ Gr(k, H 0 (V ⊗ E)) . By Proposition 3.12(2), therefore, G k n,e (V) is smooth and of dimension k n,e (V) in a neighbourhood of −1 (E) , and −1 (E) is a smooth scheme. Moreover, by Proposition 3.13 the component of B k n,e (V) containing E also has dimension k n,e (V) . As the fibres of are connected, being Grassmannians, is birational in a neighbourhood of −1 (E).
Thus hold for general V and general E ∈ B k n,e (V) 0 . In particular, we can describe some tangent cones of generalised theta divisors at well-behaved singular points. Suppose g ≥ r 2 and 1 ≤ d ≤ r − 1 . Then g r ≥ r , so we may set k 0 = d . We write r � ∶= r gcd(r,d) and d � ∶= d gcd(r,d) , and set e 0 = g − 2 . Then for any positive integer , the values satisfy both the hypotheses of Theorem 1.2 and the equation re + nd = rn(g − 1) . (Here e 1 ∶= e − ne 0 = ⋅ (r � − d � ) . Note also that necessarily n ≥ 2).
By Theorem 1.2, for 1 ≤ k ≤ nk 0 there exists a component X k of B k n,e (V) ⊂ B 1 n,e (V) upon which the Petri maps E are injective for general E ∈ X k . By Proposition 8.1(3), for each such E we have mult E B 1 n,e (V) = h 0 (V ⊗ E).

Geometry of the tangent cones
n = r � and e = r � e 0 + (r � − d � ) = ⋅ (r � (g − 2) + r � − d � ) which are furthermore globally generated (similar as in Sect. 6) was used in [15] to prove the generic injectivity of the theta map.

A nonempty 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 [27] for more general and detailed information): Let C be a general curve of genus g ≥ 3 . 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 over C. A computation shows that Proposition 9.1 Let C, r, g and e 0 be as above, and let V be a general bundle of rank r ≥ 2 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.

Remark 9.2
The construction below does not require C and V to be general. The hypothesis of generality is made so that the "expected dimension" makes sense.
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 p is the skyscraper sheaf of degree 1 supported at p ∈ C . Then E is stable by [24,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 Since by hypothesis n > r ≥ 2 , this is negative. The result follows.