Bounds on the dimension of the Brill–Noether schemes of rank two bundles

Abstract

The aim of this note is to find upper bounds on the dimension of Brill–Noether locus inside the moduli space of rank two vector bundles on a smooth algebraic curve. We deduce some consequences of these bounds.

1 Introduction

Let C be a projective smooth algebraic curve of genus g. For nonnegative integers n and d, we denote by U(n, d) the moduli space of stable vector bundles of rank n and degree d, which is an irreducible scheme of dimension $n^2(g-1)+1$. For an integer k with $1\le k\le n+\frac{d}{2}$, the subset

$$\begin{aligned} B^k_{n, d}=\{E\in U(n, d)\mid h^0(E)\ge k \} \end{aligned}$$

of U(n, d) inherits the structure of a closed sub-scheme of U(n, d). With these notations, $B^k_{1, d}$ is the scheme of line bundles of degree d with the space of global sections of dimension at least k, which is denoted commonly in the literature by $W^{k-1}_{d}$. In the case of its non-emptiness, $B^k_{n, d}$ is expected to be of dimension $n^2(g-1)+1-k(k-d+n(g-1))$. As well, for a fixed line bundle ${\mathcal {L}}$ of degree d, we denote the sub-scheme of U(n, d) parameterizing stable bundles $E\in B_{n, d}^k$ with determinant ${\mathcal {L}}$, by $B_{n, {\mathcal {L}}}^k$.

The schemes $B_{n, d}^k$, being as natural generalization of the Brill–Noether spaces of line bundles, as well as the spaces $B_{n, {\mathcal {L}}}^k$, have received wide attention from various authors. However, in contrast with extensive results concerning these schemes, specifically the results on the non-emptiness and existence of components with minimum dimension, there are not, to our knowledge, systematic studies about upper bounds for their dimensions, when $n\ge 2$.

We study this problem for Brill–Noether schemes of rank two bundles, and we obtain upper bounds for $\dim B_{2, d}^k$ and $\dim B_{2, K}^k$, where K denotes the canonical line bundle on C.

The significant point in the rank two case is that a general element in a component of some $B_{2, d}^k$, which violates the upper bound and under some specified circumstances, might be assumed to be globally generated. Under the globally generated assumption, a result of Michael Atiyah is applicable. Based on the mentioned result, a globally generated vector bundle can be represented as an extension of a line bundle by the trivial line bundle. Then, using the structure of tangent spaces of $B_{2, d}^k$, we relate the kernels of the Petri maps of appropriate bundles in suitable exact sequences. See Theorem 1. As a by-product, we obtain a Mumford type classification result. See Corollary 1.

As for the schemes $B_{2, K}^k$, we use an unpublished result of B. Feinberg, which might be considered as a refined version of Atiyah’s result. See proposition 1 and lemma 1.

By proving that for an arbitrary smooth curve C, a specific component $X \subset B^2_{2, d}$ with prescribed circumstances, would be generically smooth of expected dimension; our results push the results of Teixidor [10] and Flamini et al. [5], one step further. See Remark 1(c).

Similar problems, as the problems studied in this paper, have been studied for schemes of Secant Loci in [2,3,4] by the author.

2 Preliminaries

For $E\in B^k_{n, d}{\setminus } B^{k+1}_{n, d}$, the Petri map associated with E controls the tangent vectors of $B^k_{n,d}$ at E. Indeed, the orthogonal of the image of the Petri map

$$\begin{aligned} \mu ^2_{E}: H^0(E)\otimes H^0(K\otimes E^*)\rightarrow H^0(K\otimes E\otimes E^*), \end{aligned}$$

(1)

identifies the tangent space of $B^k_{n, d}$ at E. Similarly, the tangent space for $B^k_{2,K}$ is parameterized by the orthogonal of the image of the symmetric Petri map

$$\begin{aligned} \mu ^0_{s, E}: S^2H^0(E)\rightarrow H^0(S^2E). \end{aligned}$$

(2)

See for example [7].

Assume that $E\in U(2, d)$ and

$$\begin{aligned} 0\longrightarrow G \longrightarrow E \longrightarrow L \rightarrow 0, \end{aligned}$$

(3)

is an exact sequence of bundles, with $G, L\in {\text {Pic}}(C)$. Then, there exists a chain of bundles $S^2(E)\supset E^1\supset E^2\supset 0$, such that

$$\begin{aligned} \frac{S^2(E)}{E^1}=2L, \quad \frac{E^1}{E^2}=G\otimes L, \quad E^2=2G. \end{aligned}$$

(4)

See [8, p. 127]. So, one has two exact sequences

$$\begin{aligned}&0\longrightarrow 2G \longrightarrow S^2E \longrightarrow \frac{S^2E}{2G} \rightarrow 0, \end{aligned}$$

(5)

$$\begin{aligned}&0\longrightarrow G\otimes L \longrightarrow \frac{S^2E}{2G} \longrightarrow 2L \rightarrow 0. \end{aligned}$$

(6)

An unpublished result of B. Feinberg, Lemma 1, is the key tool in the proof of Theorem 4. The lemma is a direct consequence of a characterization result, attributed to B. Feinberg. We quote Teixidor’s statement, [11, Lemma 1.1], of this characterizing result in Proposition 1. The proof we present for proposition 1, is quoted from Feinberg’s unpublished work in [6].

Proposition 1

Denote by F the greatest common divisor of the zeroes of the sections of E. Then, either there is a section of $E(-F)$ without zeroes or all sections of E are sections of a line sub-bundle of E.

Proof

The assertion is an immediate consequence of the following,

Claim 1

Assume that $s_1, \ldots s_{r+1}$ are base point free linearly independent sections of E such that the space $\langle s_1, \ldots , s_{r+1}\rangle$ does not contain a nowhere vanishing section. Then, there exists a line sub-bundle L of E such that $\langle s_1, \ldots , s_{r+1}\rangle$ is contained in $H^0(C, L)$.

Proof of Claim 1

Set $V:=\langle s_1, \ldots , s_{r+1}\rangle$ and consider the evaluation map

$$\begin{aligned} e_V: C\times V \rightarrow E, \quad e_V(p, s):=s(p). \end{aligned}$$

(7)

We show that $\ker (e_V)$ is a vector bundle of rank r and consequently the saturation of the image of $e_V$ is a line bundle. Observe that the hypothesis of being base point free is equivalent to the fact that the dimension of $\ker (e_V)_p$ is at most r for all p in C. If, on the other hand, the rank of $\ker (e_V)$ is generically less than r, then the dimension of the image of $\ker (e_V)$ under the composition:

$$\begin{aligned} \ker (e_V)\rightarrow C\times V \rightarrow V \end{aligned}$$

is at most r. This, however, would imply that V has a nowhere vanishing section, which is a contradiction. Therefore, $\ker (e_V)$ is a vector bundle of rank r and $e_V$ surjectively maps onto a line sub-bundle in E. This completes the proof of Claim 1. $\square$

Lemma 1

Any vector bundle E with $h^0(E)=k\ge 2$ admits an extension as

$$\begin{aligned} 0\longrightarrow {\mathcal {O}}(D) {\mathop {\longrightarrow }\limits ^{i}}E {\mathop {\longrightarrow }\limits ^{\pi }}L \rightarrow 0, \end{aligned}$$

(8)

where D is an effective divisor and either $h^0({\mathcal {O}}(D))=1$ or $h^0({\mathcal {O}}(D))=k$.

Motivated by Lemma 1, two types of bundles with sections are distinguishable.

Definition 1

A vector bundle E with $h^0(E)=k\ge 1$ will be said of first type if it admits an extension as (8) with $h^0({\mathcal {O}}(D))=k$. Otherwise, we call E of second type.

3 Main results

Theorem 1

Let k, d be integers with $3\le d\le 2g-2$, $2\le k\le 2+\frac{d}{2}$. Then,

$$\begin{aligned} \dim B^k_{2, d}\le 2(g-1)+d-2k+1. \end{aligned}$$

(9)

Proof

Observe first that if a general element E of an irreducible component X of $B^k_{2, d}$ satisfies $h^0(E)\ge k+1$, then we can consider X as a component of $B^{k+1}_{2, d}$. Therefore for general $E\in X$ one may assume $h^0(E)=k$. Assume that d is a minimum integer such that for some suitable k there exists a component X of $B^k_{2, d}$ with $\dim X\ge 2(g-1)+d-2k+2$. Then, a general element E in X is globally generated. Indeed otherwise we obtain $\dim B^{k-1}_{2, d-2}\ge 2(g-1)+d-2k+2$, which is impossible by minimality of d. Therefore, by [1, Theorem 2], a general element E in X has a trivial line bundle as its line sub-bundle. Furthermore E admits a representation as

$$\begin{aligned} 0\longrightarrow {\mathcal {O}}_C {\mathop {\longrightarrow }\limits ^{i}}E {\mathop {\longrightarrow }\limits ^{\pi }}L \longrightarrow 0, \end{aligned}$$

(10)

with the property that the sections of L belonging to the image of $H^0(\pi )$ have at most one number of base points. Indeed, if L has the points p, q as its base points, then $h^0(E(-p-q))\ge k-2$. This implies that

$$\begin{aligned} \dim B^{k-2}_{2, d-4}\ge 2(g-1)+d-2k+2, \end{aligned}$$

which is absurd again by minimality of d. Take an extension as (10) and consider the exact sequence

$$\begin{aligned} 0\longrightarrow H^0({\mathcal {O}}_C) {\mathop {\longrightarrow }\limits ^{H^0(i)}}H^0(E) {\mathop {\longrightarrow }\limits ^{H^0(\pi )}}V \longrightarrow 0, \end{aligned}$$

(11)

where V is the image of the map $H^0(\pi ): H^0(E) \longrightarrow H^0(L)$. The exact sequence (11) together with various Petri maps gives rise to a commutative diagram as

(12)

in which the maps $f_{1}$ and $f_{2}$ are injective and $g_{1}$ is surjective. Observe furthermore that the map $\mu$ is an isomorphism. The snake lemma applied to this situation implies that

$$\begin{aligned} \dim \ker \mu ^2_{E}=\dim \ker \mu _{L, V}. \end{aligned}$$

(13)

According to the assumption concerning dimension of X, we obtain

$$\begin{aligned} \dim \ker \mu _{L, V}\ge (k-1)(2g-2-d+k-1). \end{aligned}$$

Assuming $V=\langle v_1, \ldots , v_{k-1} \rangle$ and setting

$$\begin{aligned} V_i:=\langle v_1, \ldots , v_i \rangle , \quad i=2, \ldots , k-1, \end{aligned}$$

we would have $\dim \ker \mu _{L, V_i}-\dim \ker \mu _{L, V_{i-1}}\le h^0(K\otimes E^*).$ These together with the base-point-free pencil trick applied to the map

$$\begin{aligned} \mu _{L, V_2}:V_2\otimes H^0(K\otimes E^*)\rightarrow H^0(K\otimes L\otimes E^*), \end{aligned}$$

implies $h^0(K\otimes E^*\otimes L^*(B))\ge 2(2g-2-d+k)-(k-1)$, where B is the base locus of the sections of $V_2$. Note also that $0\le \deg (B)\le 1$. Therefore,

$$\begin{aligned} h^0(K\otimes E^*\otimes L^*)\ge 2(2g-2-d+k)-k. \end{aligned}$$

(14)

If $\frac{4g-3}{3}\le d$ then, as $\deg (K\otimes E^*\otimes L^*)< 0$ and $K\otimes E^*\otimes L^*$ is stable, one has $h^0(K\otimes E^*\otimes L^*)=0$, which is in contradiction with inequality (14).

Recall that $h^0(E\otimes L)=h^0(K\otimes E^*\otimes L^*)+3d-2(g-1)\ge 2(g-1)+d+k.$ Now if $d\le \frac{4g-4}{3}$, then $\mu (E \otimes L)\le 2g-3$. Observe furthermore that $E \otimes L$ is stable. As a consequence of Propositions 3 and 4 of [9], the Clifford theorem for vector bundles for such a this situation asserts that $h^0(E\otimes L)\le \frac{\deg (E\otimes L)+{\text {rk}}(E\otimes L)}{2}$, by which we obtain $2(g-1)+d+k\le 1+\frac{3d}{2}.$ Consequently, we get $d+k\le 0$, which is absurd. $\square$

Theorem 2

If $g\ge 5$, then

$$\begin{aligned} \dim B^2_{n, d}\le n(n-1)(g-1)+d-3. \end{aligned}$$

(15)

Proof

Assume that X is an irreducible component of $B^2_{n, d}$ and E is a general element of X. Assume moreover, as in theorem 1, that a general element $E\in X$ satisfies $h^0(E)=2$. Observe that, using a diagram as in diagram (12), we can obtain an equality as (13), by which, if E turns out to be of second type, then $\mu ^2_E$ would be injective. So X has to be generically smooth, and it has to have the expected dimension, which is certainly smaller than the claimed bound.

If a general element of X turns to be of first type, then

$$\begin{aligned} \dim X\le n(n-1)(g-1)+d-4. \end{aligned}$$

Indeed, if a general element $E\in X$ admits a presentation as

$$\begin{aligned} 0\rightarrow H \rightarrow E \rightarrow F \rightarrow 0, \end{aligned}$$

where H is a line bundle with $h^0(H)=2, \deg (H)=d_1$ and ${\text {rk}}(F)=n-1$, then since the stable bundles deform to non-stable ones, we can assume in counting that F is stable as well. So the dimension of the set of bundles as F, is bounded by $\dim U(n-1, d-d_1)=(n-1)^2(g-1)+1$. Meanwhile, the line bundles as H would vary in a subset ${\mathcal {H}}$ of $B^2_{1, d_1}$ and the Martens’ theorem asserts that $\dim {\mathcal {H}}\le d_1-2$ ($\dim {\mathcal {H}}$ can be $d_1-2$ if C is hyper-elliptic and $\dim {\mathcal {H}}\le d_1-3$ otherwise). Therefore, the dimension of X would be bounded by

$$\begin{aligned}{}[d_1-2]+[(n-1)^2(g-1)+1]+(h^1(H\otimes F^*)-1). \end{aligned}$$

Observe that $h^1(H\otimes F^*)=(n-1)(g-1)+d-nd_1$ by Riemann–Roch. Moreover, $d_1\ge 2$ and so

$$\begin{aligned} \dim X\le n(n-1)(g-1)+d-2-d_1(n-1)\le n(n-1)(g-1)+d-4, \end{aligned}$$

as required. $\square$

Motivated by [5, Theorem 1.2], one can sharpen the bound in Theorem 1 under some restrictions on the numbers r, d, as

Theorem 3

Let k, d be integers with $3\le d\le 2g-2-\frac{k}{2}$, $2\le k\le 2+\frac{d}{2}$. Then,

if $k\ge 3$, then $\dim B^k_{2, d}\le 2g+d-4k.$ While for $k=2$, the integer d can vary in the set $\{3, \ldots , 2g-5 \}$ with the same bound for $\dim B^2_{2, d}$.

Proof

The argument of proof of Theorem 1 goes through to deduce the result. Notice that the further restriction on d in the case $k=2$ was needed to be imposed, because the quantity $2g+d-4k$ turns out to be smaller than the expected dimension for $2g-4\le d\le 2g-2$. $\square$

3.1 The case of canonical determinant

Theorem 4

For an integer k with $2\le k\le g+1$, any irreducible component X of $B_{2, K}^k$ satisfies

$$\begin{aligned} \dim X\le 3g-2k-2. \end{aligned}$$

(16)

Proof

Let X be an irreducible component of $B_{2, K}^k$ and a general element E of X satisfies $h^0(E)=k$. Assume that a general member $E\in X$ is of second type and set $\gamma :=\dim X$. Then, one has

$$\begin{aligned} 3g-3-(\dim {\mathrm{im\,}}\mu ^0_{s, E})\ge \gamma , \end{aligned}$$

(17)

where $\mu ^0_{s, E}$ is the symmetric Petri map associated with E as in (2). So

$$\begin{aligned} \dim \ker \mu ^0_{s, E}\ge \gamma +\frac{k(k+1)}{2}+3-3g. \end{aligned}$$

(18)

The exact sequence $0\rightarrow {\mathcal {O}}(2D)\rightarrow S^2E\rightarrow \frac{S^2E}{{\mathcal {O}}(2D)} \rightarrow 0,$ arising from the exact sequence (5), gives rise to a commutative diagram as

(19)

Since $S^2H^0({\mathcal {O}}(D))={\mathbb {C}}$, the map $\mu ^0_{s, D}$ turns to be injective. This together with the snake lemma gives an inequality as

$$\begin{aligned} \dim \ker \mu \ge \dim \ker \mu ^0_{s, E}. \end{aligned}$$

(20)

Therefore, using inequality (18), we obtain

$$\begin{aligned} \dim \ker \mu \ge \gamma +\frac{k(k+1)}{2}+3-3g. \end{aligned}$$

(21)

Let V be as in the proof of Theorem 1 and observe by effectiveness of D that the vector space V can be considered as a subspace of $H^0({\mathcal {O}}(D)\otimes L)$. Similar to the previous argument, the exact sequence

$$\begin{aligned} 0\rightarrow {\mathcal {O}}(D)\otimes L\rightarrow \frac{S^2E}{{\mathcal {O}}(2D)}\rightarrow 2L \rightarrow 0, \end{aligned}$$

as well arising from the exact sequence (6), together with the equality

$$\begin{aligned} S^2H^0(E) =S^2V\oplus V\oplus {\mathbb {C}}, \end{aligned}$$

leads to the following commutative diagram of bundles

(22)

where $\mu ^0_{s, V, L}$ is the symmetric Petri map of L restricted to $S^2V$ and $\theta$ is the inclusion map. Once again, as a consequence of the injectivity of $\theta$ and the snake lemma, we obtain

$$\begin{aligned} \dim \ker \mu ^0_{s, V, L} \ge \dim \ker \mu , \end{aligned}$$

(23)

by which together with (21) an inequality as

$$\begin{aligned} \dim \ker \mu ^0_{s, V, L} \ge \gamma +\frac{k(k+1)}{2}+3-3g \end{aligned}$$

(24)

would be obtained. This, in combination with $\dim \ker \mu ^0_{s, V, L} \le \dim S^2V-\dim V$, implies

$$\begin{aligned} \gamma \le 3g-2k-2, \end{aligned}$$

(25)

as required.

Finally, if $\dim X\ge 3g-2k-1$, then a general member E of X fails to be of first type. Indeed otherwise, assume that a general member $E\in X$ admits a presentation as

$$\begin{aligned} 0\longrightarrow {\mathcal {O}}(D) {\mathop {\longrightarrow }\limits ^{i}}E {\mathop {\longrightarrow }\limits ^{\pi }}K\otimes {\mathcal {O}}(-D) \rightarrow 0, \end{aligned}$$

(26)

with $\deg (D)=t$. Then, the stability of E implies that $t\le g-2$, and we would have

$$\begin{aligned} \dim B^{k}_{1,t}+h^1({\mathcal {O}}(D)\otimes L^{-1})-1\ge 3g-2k-1. \end{aligned}$$

This, since $h^0({\mathcal {O}}(D)\otimes L^{-1})=0$ by stability of E, implies that

$$\begin{aligned} \dim B^{k}_{1, t}\ge 2t-2k+3, \end{aligned}$$

which is absurd by Martens’ theorem. $\square$

4 Remarks and Corollaries

Corollary 1

(Mumford’s Theorem for rank two bundles) If C is non-hyper-elliptic of genus $g\ge 19$ and if for some k, d with $0< 2k-2\le d\le 2g-\frac{3}{2}k-\frac{7}{2}$ one had $\dim B_{2, d}^k=2g+d-4k$, then either C is trigonal, or bi-elliptic, or a smooth plane quintic.

Proof

Assume that X is an irreducible component of $B_{2, d}^k$ with $\dim X=2g+d-4k$. If a general element $E\in X$ is of first type and has k number of independent sections, then one has $\dim B_{1, t}^{k}\ge 2(t-2k+1)$ for some integer t with $0<2(k-1)\le t\le g-2$. This, by Mumford’s theorem, might occur only if $t-2k+1=0$ by which the equality $\dim B_{1, 2k-1}^{k}=0$ holds. So ${\mathcal {O}}(D)\in B_{1, 2k-1}^{k}$, which may happen only in the case that either C is trigonal, or bi-elliptic, or a smooth plane quintic.

Claim 2

If E fails to be of first type, then for general points $p_1, \ldots p_{[\frac{k-2}{2}]}\in C$, the stable vector bundle $E(-p_1-\ldots -p_{[\frac{k-2}{2}]})$ would fail to admit an extension of first type.

Proof of Claim 2

Assume first that k is even. If the stable vector bundle $E(-p_1-\ldots -p_{[\frac{k-2}{2}]})$ turns to be of first type, then there exists a set of line bundles H with $h^0(H)\ge 2$ and $\deg {H}\le g-2$. Tensoring H with ${\mathcal {O}}(p_1+\cdots +p_t)$ for general points $p_1+\cdots +p_t$, if necessary, we can assume that $H\in B^2_{1, g-2}$. Therefore we obtain $\dim B^2_{1, g-2}\ge 2g+d-4k+[\frac{k-2}{2}]$. This by Martens’ theorem implies that $7k-8\ge 2g+2d$. On the other hand, the inequalities $2k-2\le d$ and $2k-2\le 2g-\frac{3}{2}k-\frac{7}{2}$ imply $4k\le 2d+4$ and $3k\le \frac{12}{7}g-\frac{9}{7}$, respectively. Summing up all the inequalities we obtain $g\le 18$, which is absurd. If k is an odd number, then the argument goes verbatim to prove the claim by replacing $B^3_{1, g-2}$ with $B^2_{1, g-2}$. So the Claim 2 is established.

If a general bundle $E\in X$ turns to be of second type and if $k=2n$, then the scheme $B_{2, d-2[\frac{k-2}{2}]}^{2}$ contains a subset Y which is at least of dimension $2g+d-4k+[\frac{k-2}{2}]$ and its general member is a vector bundle of second type. According to the work of M. Teixidor in [10], such a subset Y, if non-empty, is of expected dimension and the expected dimension is strictly smaller than $2g+d-4k+[\frac{k-1}{2}]$ for d in the given range. This is a contradiction.

If $k=2n+1$, with similar assumption on E, the scheme $B_{2, d-2[\frac{k-2}{2}]}^{3}$ would contain a subset Y which is at least of dimension $2g+d-4k+[\frac{k-2}{2}]$ and its general member is a vector bundle of second type. This possibility can be excluded by another work of M. Teixidor in [11]. $\square$

Corollary 2

The scheme $B^2_{2, K}$ is reduced and irreducible of dimension $3g-6$.

Proof

The upper bound $3g-6$ on the dimension is obvious by theorem 4. If $E\in B^2_{2, K}{\setminus } B^3_{2, K}$, then the petri map $\mu ^0_{2, K, E}$ turns to be injective. Indeed, if E is a bundle of first type, then using diagram (19), since $\frac{S^2H^0(E)}{S^2H^0({\mathcal {O}}(D))}$ vanishes, the Petri map $\mu ^0_{s, E}$ would be injective. While if E is of second type, since $S^2V$ is one dimensional, then $\mu ^0_{s, V, L}$ is injective and so the map $\mu$ is injective by (23). This together with (20) implies that the Petri map $\mu ^0_{s, E}$ is again injective. So we obtain

$$\begin{aligned} {\mathrm{Sing\,}}B^2_{2, K}=B^3_{2, K}. \end{aligned}$$

(27)

Since $B^2_{2, K}$ is of expected dimension, so it might be reducible only if its singular locus is, by [12], of codimension $\le 1$; i.e., $\dim B^3_{2, K}\ge 3g-7$, by (27). This is a contradiction, because by Theorem (4) the locus $B^3_{2, K}$ is of dimension at most $3g-8$.

Since, again by theorem 4, no irreducible component of $B^2_{2, K}$ is contained entirely in $B^3_{2, K}$, so $B^2_{2, K}$ would be reduced. $\square$

Using Lemma 2, the bound in theorem 4 can be sharpened for odd values of k.

Lemma 2

If ${\mathcal {L}}$ is a globally generated line bundle on C with $h^0({\mathcal {L}})=s+1$, then the set of vector bundles of second type $E\in B^k_{2, d, {\mathcal {L}}}$ ($k=1, 2$), if non-empty, is of dimension at most $s+d-4$, (res. at most of dimension $\frac{d}{2}+2(s-3)$), if $k=2$ (res. if $k=3$).

Proof

For $k=2$, with notations as in proof of [10, Page 124], the dimension of the set of vector bundles $E\in B_{2, d, {\mathcal {L}}}^k$ of second type is bounded by

$$\begin{aligned} \dim \lbrace D \rbrace + h^0({\mathcal {L}}(-D))-1 + \dim \langle \acute{D} \rangle - (h^0(E)-1) \end{aligned}$$

where $\acute{D}$ is a divisor in the linear series $\vert {\mathcal {L}}(-D) \vert$ and $t=\deg (D)$. It is now an easy argument to see that this quantity is bounded by

$$\begin{aligned} t + (s-1) + (d-t-2)-1=d+s-4. \end{aligned}$$

If $k=3$, then a close analysis in the proof of [11, Theorem 2], implies that the dimension of the bundles $E\in \dim B^3_{2, d, {\mathcal {L}}}$ which are of second type, is bounded by the quantity $\dim \lbrace D \rbrace + \dim {\text {Grass}}(2, {\mathcal {L}}(-D)) +\dim \langle \acute{D_1}\cap \acute{D_2}\rangle - (h^0(E)-1) \le \frac{d}{2}+2(s-3),$ as required. $\square$

Corollary 3

If k is odd, then

$$\begin{aligned} \dim B^k_{2, K}\le 3g-2k-3. \end{aligned}$$

(28)

Proof

An irreducible component X of $B_{2, K}^k$ whose general member is a bundle of first type has dimension $\le 3g-2k-3$, because otherwise one obtains $\dim B_{1, t}^{k}\ge 2t-2k+2$ for some k and t with $0<2k-2\le t\le g-2$. This is obviously absurd.

Assume that $\dim X=3g-2k-2$ and set $k-1=2n$.

Claim 3

If a general $E\in X$ fails to be of first type, then for general points $p_1, \ldots p_{i}\in C$ with $1\le i\le \frac{k-1}{2}$ the stable vector bundle $E(-p_1-\ldots -p_i)$ would fail to admit an extension of first type.

The proof of Claim 3 is similar to the proof of Claim 2 in corollary 1.

Lemma (2) together with Claim 3 implies that if a general element of X fails to be of first type then

$$\begin{aligned} 3g-2k-2\le (n-1)+\dim B^3_{2, 2g-2n, K(-2p_1-\ldots -2p_{n-1})}\le 3g-4n-5, \end{aligned}$$

which is absurd. $\square$

Remark 1

• (a) If C is an arbitrary 3-gonal curve, then Theorem 3 together with Theorem [5, Thm. 1.2(b)] imply $\dim B^2_{2, d}=2g+d-8$. Indeed, Theorem [5, Thm. 1.2(b)] establishes this result for a general 3-gonal curve and so for non-general 3-gonal curves, one has $\dim B^2_{2, d}\ge 2g+d-8$. Now, Theorem 3 applied to such a non-generic curve implies the equality for any 3-gonal curve.

• (b) According to Theorem 2, one immediately re-obtains $\dim B^2_{n, n(g-1)}=n^2(g-1)-3$. Meanwhile, by the same theorem, an immediate prediction suggests the quantity $n(n-1)(g-1)+d-2k+1$ as a bound to the dimension of $B^k_{n, d}$ when $n\ge 3, k\ge 3$. A proof to this expectation is unknown to me. Such a bound re-obtains Marten’s bound on the dimension of the Brill–Noether schemes of line bundles.

• (c) The proofs of Theorems 1 and 2 indicate that the Petri map is injective at the bundles $E\in B^2_{n, d}$ which are of second type. Therefore

  $$\begin{aligned} {\mathrm{Sing\,}}B^2_{n, d}\subseteq B^3_{n, d}\cup {\mathcal {E}}_1, \end{aligned}$$

  where ${\mathcal {E}}_1$ denotes the set of bundles $E\in B^2_{n, d}$ of first type. This reproves the generic smoothness of the locus’ introduced by Teixidor [10] and Flamini et al. [5].