# Excess dimensions of Brill–Noether schemes of rank two stable bundles

Ali Bajravani  Department of Mathematics, Faculty of Basic Sciences,
Azarbaijan Shahid Madani University, Tabriz, I. R. Iran., P. O. Box:
53751-71379<EMAIL_ADDRESS>School of Mathematics, Institute for
Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746. Tehran, Iran.

###### Abstract.

We use results of M. Aprodu and E. Sernesi to extend a result by
Fulton–Harris–Lazarsfeld in Brill–Noether theory of line bundles to
Brill–Noether loci of stable bundles inside the moduli space of rank two
stable vector bundles on a smooth projective algebraic curve. As a
consequence; if $B^{k}_{2,d}$ is of expected dimension, then we prove that its
smoothness is equivalent with the smoothness of $B^{k}_{2,d+1}$.

###### Key words and phrases:

Brill–Noether locus, Degeneracy locus, Stable vector bundle, Elementary
transformation

###### 2010 Mathematics Subject Classification:

14H60 (14M12)

##  1\. Introduction

Let $C$ be a smooth projective algebraic curve of genus $g$. For integers $r$
and $d$ with $0\leq 2r\leq d$, the Brill–Noether scheme of line bundles,
$W^{r}_{d}$, parameterizes line bundles of degree $d$ with the space of global
sections of dimension at least $r+1$. Brill–Noether theory (see for example
[13]) computes the expected dimensions of these schemes in terms of $d$, $r$
and $g$, and shows that this dimension is attained for a general curve.
However, the dimension may be strictly larger when $C$ is special in the
scense of moduli. The basic Martens’ theorem bounds the dimension of
$W^{r}_{d}$ in terms of $d$ and $r$, on arbitrary smooth projective curves.
Variations of this basic result have been obtained in the literature, see
[22], [9], [6] and [3]. See also [7, Theorem 3.5] and [7, Theorem 3.7].
Another result concerning dimensions of Brill–Noether schemes of line bundles,
which holds on arbitrary curves, is the excess dimension inequality. William
Fulton, Joe Harris and Rob Lazarsfeld proved in [12] that the difference of
dimensions of $W^{r}_{d}$ and $W_{d}^{r+1}$, (resp., of $W_{d}^{r}$ and
$W_{d}^{r+1}$), are controllable in terms of $r$ (resp., in terms of $r$, $d$
and $g$). A significant consequence of their result is that if $\dim
W^{r}_{d}\geq r+1$, then $W^{r}_{d-1}$ would be non-empty.

Marian Aprodu and Edoardo Sernesi have extended the excess dimension
inequalities to the schemes of secant loci of very ample line bundles. See
[1]. Michael Kemeny used their result in studying the syzygies of curves of
arbitrary gonalities, see [16]. As well, in [5], the author uses Aprodu-
Sernesi’ excess dimension result to obtain a Mumford type theorem for schemes
of secant loci.

Since some bundles appeared in constructing $W^{r}_{d}$’s turns out to be
ample, no non-emptiness prerequisite is needed in Fulton-Harris-Lazarsfeld
framework. But, this positivity property fails to be hold in Aprodu and
Sernesi’ general setting. So they enter a non-emptiness assumption, which is
slightly a weaker hypothesis.

If $X$ is an integral algebraic scheme and

(1.5)

is an arbitrary diagram of bundles on $X$ with
$\mathrm{rk}(\mathbb{E})=m,\mathrm{rk}(\mathbb{F})=n,\mathrm{rk}(\mathbb{F}_{1})=n+1$,
then Aprodu and Sernesi compare dimensions of $D_{k}(\sigma)$,
$D_{k+1}(\sigma)$ and $D_{k}(\pi\sigma)$, where for a morphism
$\sigma:E\rightarrow F$ of vector bundles on $X$, the loci $D_{k}(\sigma)$ is
defined as

$D_{k}(\sigma):=\\{x\in X\mid\mathrm{rk}(\sigma_{x})\leq k\\}.$

Consequently Fulton–Harris–Lazarsfeld’s result is a special case of
Aprodu–Sernesi’ result.

We use Aprodu and Sernesi’ general machinery to study the excess-dimension
problem for Brill–Noether loci of stable vector bundles of rank $2$ and degree
$d$ with the space of global sections at least of dimension $k$ inside the
moduli space of rank two stable vector bundles, $U(2,d)$. The closure of some
suitable loci, denoted by $B^{1}_{2,d+1,p}$, consisting of general dual
elementary transformations of general bundles in $B^{1}_{2,d+1}$ is a divisor
in $U(2,d+1)$ for $1\leq d\leq 2(g-1)-1$. On a suitable dense sub-locus
$\mathcal{U}\subseteq U(2,d+1)$, consisting of general dual elementary
transformations of general bundles in $U(2,d)$, we obtain a diagram of vector
bundles. As Lemma 2.2(i) indicates, there exists a dense subset of $U(r,d+1)$
consisting of general dual elementary transformations of general bundles in
$U(r,d)$. But it is not known if this loci intersects the loci $B^{k}_{r,d}$
for $r\geq 3$. As for the case $r=2$, we observe that such a dense subset has
non-empty intersection with any non-empty component of $B^{k}_{2,d}$. See
lemma 2.2(ii). So comparing dimensions of degeneracy loci of the diagram gives
a right comparision of dimensions of various $B^{k}_{2,d}$’s. The diagram we
use, diagram 2.5, is a “dual”version of Aprodu-Sernesi’ diagram (1.5). We
apply theorem 2.1, by which we obtain our main result. See Theorem (2.6).
Using theorem 2.6 together with a comparision of dimension of tangent spaces
we relate the smoothness of $B^{k}_{2,d}$ to those of $B^{k}_{2,d+1}$. See
corollary 2.8.

### Notation

For a sheaf $F$ on $C$, we abbreviate $H^{i}(C,F)$ and $h^{i}(C,F)$ to
$H^{i}(F)$ and $h^{i}(F)$, respectively. If $\sigma:E\rightarrow F$ is a
morphism of locally free sheaves $E$, $F$ of ranks $\alpha,\beta$ on an
algebraic variety $X$, respectively, then for $p\in X$ the $\alpha\times\beta$
matrix $\sigma\mid_{p}$ acts on an $\alpha$-vector $\nu\in E\mid_{p}$ by
${}^{t}\nu\cdot\sigma\mid_{p}$.

##  2\. Excess dimensions of Brill-Noether loci of rank two stable bundles

###  2.1. A dual version of Aprodu and Sernesi’ result

Let $X$ be an integral algebraic variety and consider a diagram of vector
bundles

(2.5)

######  Theorem 2.1.

Suppose $\mathbb{E},\mathbb{F}_{1}$ and $\mathbb{F}$ are vector bundles of
ranks $m$, $n+1$ and $n$, respectively, which are set in a diagram as 2.5.
Assume moreover that no irreducible component of $D_{k}(\sigma i)$ is
contained in $D_{k}(\sigma)$.

  1. (1)

If $D_{k}(\sigma i)$ is nonempty, then

$\dim D_{k}(\sigma i)\ \geq\dim D_{k+1}(\sigma)-(n-k),$

  2. (2)

If $D_{k}(\sigma)$ is nonempty, then

$\dim D_{k}(\sigma)\geq\dim D_{k}(\sigma i)-(m-k),$ $\dim
D_{k}(\sigma)\geq\dim D_{k+1}(\sigma)-(m+n-2k).$

###### Proof.

For a morphism $\gamma:E\rightarrow F$ of vector bundles on an integral
algebraic scheme $X$, one has $D_{k}(\gamma)=D_{k}(\gamma^{*})$, where
$\gamma^{*}:F^{*}\rightarrow E^{*}$ is the dual homomorphism of $\gamma$. The
theorem follows, because diagram 2.5 is a dual version of the diagram
considered in proposition [1, Th. 3.1]. ∎

###  2.2. A divisorial component in $B^{1}_{r,d+1}$

Assume that $g\geq 2$. Let $U(r,d)$ denote the moduli space of stable vector
bundles of rank $r$ and degree $d$. Then, $U(r,d)$ admits an $\acute{e}$tale
finite cover on which there exists a Poincar$\acute{e}$ bundle. Precisely,
there exists an $\acute{e}$tale finite cover $\pi:\mathcal{U}\rightarrow
U(r,d)$ and a bundle $\mathcal{E}_{d}\rightarrow\mathcal{U}$ such that for
each $E\in\mathcal{U}$ one has $\mathcal{E}_{d\mid\\{E\\}\times
C}\cong\pi(E)$. Furthermore, for any $\lambda\in\mathbb{N}$, $U(r,d)\cong
U(r,d+\lambda r)$ by an isomorphism $\theta$, where $\theta:U(r,d+\lambda
r)\rightarrow U(r,d)$ is given by $E\mapsto E\otimes\mathcal{O}(-D)$ with
$D:=p_{1}+\ldots+p_{\lambda},$ for some fixed $p_{i}\in C$.

For an integer $\delta$ with $\delta:=d+\lambda r>2r(g-1)$, consider the
projections

(2.10)

and set $\mathbb{E}_{\delta}:=(\pi_{2})_{*}(\mathcal{E}_{\delta})$,
$\mathbb{F}_{\delta}:=(\pi_{2})_{*}(\mathcal{E}_{\delta}\otimes\pi_{1}^{*}\mathcal{O}_{D}(D))$.
The bundles $\mathbb{E}_{\delta}$ and $\mathbb{F}_{\delta}$ would be of ranks
$\delta-r(g-1)$ and $\lambda r$, respectively. See [14].

The loci of stable vector bundles $E$, of rank $r$ and degree $d$ with the
space of global sections of dimension at least $k$ is up to the isomorphism
$\theta$ the $(\delta-r(g-1)-k)$-th degeneracy locus of the evaluating
morphism of bundles

(2.11)
$\displaystyle\gamma_{\delta}:\mathbb{E}_{\delta}\rightarrow\mathbb{F}_{\delta}.$

Observe that for $E\in U(r,\delta)$ with $\delta>2r(g-1)$ one has
$h^{0}(E)=\delta-2r(g-1)$ and so the locus $E\in U(r,\delta)$ with
$\delta>2r(g-1)$ and $h^{0}(E)=1$ might be empty. By abuse of notations, we
denote by $B^{1}_{r,\delta}$ the loci $\theta^{-1}(B^{1}_{r,d})$.

Recall that in an extension of vector bundles $0\rightarrow E\rightarrow
F\rightarrow\mathcal{O}_{p}\rightarrow 0$, the bundle $E$ is called an
elementary transformation of $F$ by $p$ and $F$ is called a dual elementary
transformation of $E$ by $p$.

The Segre invariant $s_{1}(E)$ is defined to be

(2.12) $\displaystyle
s_{1}(E):=\min\\{\deg(E)-2\textperiodcentered\deg(L):L\subset\mbox{E a line
subbundle}\\}.$

Then E is stable if and only if $s_{1}(E)\geq 1$.

######  Lemma 2.2.

(i) Assume that $1\leq d\leq n(g-1)-1$ and $p\in C$ is generic. Then, a
general extension as $0\rightarrow E\rightarrow
F\rightarrow\mathcal{O}_{p}\rightarrow 0,$ with $E$ a general element of
$B^{i}_{r,d}$, $0\leq i\leq 1$, is stable.

(ii) Suppose $E$ is an arbitrary stable rank two vector bundle and $p\in C$
generic. Then, a general dual elementary transformation of $E$ by $p$ is
stable.

(iii) Assume $p\in C$ is generic and $E$ an arbitrary stable rank two vector
bundle. Then, a general elementary transformation of $E$ by $p$ is stable.

###### Proof.

(i) Observe that for $0\leq d\leq r(g-1)$, the loci $B^{1}_{r,d}$ and
$B^{0}_{r,d}:=U(r,d)$ are irreducible by [23, Theorem II.3.1]. So, according
to the openness of stability, it is enough to prove that there exists one
special extension in each of them with the stated property.

We prove the case $i=1$. The other case is similar. Set $d=r\alpha+t$ and
observe then that $\alpha\leq g-1$. Take line bundles $L_{1},\ldots,L_{r-1}$
in the loci $B^{0}_{1,\alpha}\setminus B^{1}_{1,\alpha}$ and a line bundle
$H\in B^{1}_{1,\alpha}\setminus B^{2}_{1,\alpha}$ with the property that no
two of these line bundles are isomorphic. For generic points
$p_{1},\ldots,p_{t}$ in $C$, a general extension as

(2.13) $\displaystyle
0\rightarrow\left(\bigoplus_{i=1}^{r-1}L_{i}\right)\oplus H\rightarrow
F\rightarrow\left(\bigoplus_{i=1}^{t}\mathcal{O}_{p_{i}}\right)\oplus\mathcal{O}_{p}\rightarrow
0,$

is stable by the Theorem of Mercat. See [20, p. 76]. A general extension $F$
as in 2.13 is mapped by $\mu$ to a general element of
$\mathrm{Ext}^{1}\left((\bigoplus_{i=1}^{t}\mathcal{O}_{p_{i}}),(\bigoplus_{i=1}^{r-1}L_{i})\oplus
H\right)$, where

$\mu:\mathrm{Ext}^{1}\left((\bigoplus_{i=1}^{t}\mathcal{O}_{p_{i}})\oplus\mathcal{O}_{p},(\bigoplus_{i=1}^{r-1}L_{i})\oplus
H\right)\longrightarrow\mathrm{Ext}^{1}\left((\bigoplus_{i=1}^{t}\mathcal{O}_{p_{i}}),(\bigoplus_{i=1}^{r-1}L_{i})\oplus
H\right)$

is the pull back morphism of the inclusion of sheaves
$\bigoplus_{i=1}^{t}\mathcal{O}_{p_{i}}\rightarrow\left(\bigoplus_{i=1}^{t}\mathcal{O}_{p_{i}}\right)\oplus\mathcal{O}_{p}$.
Observe that $\mu$ is the second connecting homomorphism in the long exact
cohomology sequence associated to the exact sequence

$0\rightarrow\bigoplus_{i=1}^{t}\mathcal{O}_{p_{i}}\rightarrow\left(\bigoplus_{i=1}^{t}\mathcal{O}_{p_{i}}\right)\oplus\mathcal{O}_{p}\rightarrow\mathcal{O}_{p}\rightarrow
0.$

So
$\mathrm{Coker}(\mu)\subseteq\mathrm{Ext}^{2}\left((\bigoplus_{i=1}^{t}\mathcal{O}_{p_{i}}),(\bigoplus_{i=1}^{r-1}L_{i})\oplus
H\right)$ and it has to vanish. Consequently $\nu$ would be surjective.
Therefore,
$\dim\mathrm{Im}\,\mu=\dim\mathrm{Ext}^{1}\left((\bigoplus_{i=1}^{t}\mathcal{O}_{p_{i}}),(\bigoplus_{i=1}^{r-1}L_{i})\oplus
H\right)$.

Since, again by Mercat’s theorem, a general element of
$\mathrm{Ext}^{1}\left((\bigoplus_{i=1}^{t}\mathcal{O}_{p_{i}}),(\bigoplus_{i=1}^{r-1}L_{i})\oplus
H\right)$ is stable, we obtain an extension as $0\rightarrow E\rightarrow
F\rightarrow\mathcal{O}_{p}\rightarrow 0,$ with $E$ and $F$ stable.

The genericity of $p_{i},$ $p$ with $1\leq i\leq t$ and the genericity of $E$
together with lemma [21, 2.5] implies that
$h^{0}(E)=h^{0}\left((\bigoplus_{i=1}^{r-1}L_{i})\oplus H\right)=h^{0}(F)$. So
$h^{0}(E)=h^{0}(F)=1$, as required.

(ii) Either the Segre invariant $s_{1}(E)\geq 2$ or $s_{1}(E)=1$. If
$s_{1}(E)\geq 2$, then for any $L\subset E$ we have

$(\deg(E)+1)-(2\textperiodcentered\deg(L)-2)\geq 1;$

that is,

$\deg(F)-2\textperiodcentered\deg(L(p))=s_{1}(F)\geq 1,$

for any elementary transformation $0\rightarrow E\rightarrow
F\rightarrow\mathcal{O}_{p}\rightarrow 0$. Thus $F$ is stable.

If $s_{1}(E)=1$, then by [17, Prop. 4.2] the number of maximal line subbundles
of $E$ is finite. Let $0\rightarrow E\rightarrow
F\rightarrow\mathcal{O}_{p}\rightarrow 0$ be an elementary transformation with
$\mathrm{Ker}(E_{\mid_{p}}\rightarrow F_{\mid_{p}})\neq L{\mid_{p}}$ for any
maximal subbundle $L$. So if $M\subset E$ is a line subbundle with
$M_{\mid_{p}}=\Lambda$, then $\deg(E)-2\textperiodcentered\deg(M)\geq 3$. As
above, $(\deg(E)+1)-(2\textperiodcentered\deg(M)-2)\geq 1;$ that is,

$\deg(F)-2\textperiodcentered\deg(M(p))=s_{1}(F)\geq 1.$

Also, if $L$ is a maximal subbundle of $E$, then by the choice of $\Lambda$,
also $L$ is a subbundle of $F$, and

$\deg(F)-(2\textperiodcentered\deg(L))\geq 2.$

Thus $s_{1}(F)\geq 1$. Hence a general elementary transformation $0\rightarrow
E\rightarrow F\rightarrow\mathcal{O}_{p}\rightarrow 0$ is a stable vector
bundle.

(iii) This is immediate by dualizing (ii). ∎

Motivated by Lemma 2.2, for a generic point $p\in C$ and $i=0,1$, we set

(2.14) $\displaystyle B^{i}_{r,d+1,p}:=\\{F\in B^{i}_{r,d+1}\mid
F\\!\\!\\!\\!\\!\quad\mbox{is a general extension}\\!\\!\\!\\!\\!\quad
0\rightarrow E\rightarrow F\rightarrow\mathcal{O}_{p}\rightarrow
0,\quad\\!\\!\\!\\!\\!E\in B^{i}_{r,d}\\},$

where we have considered $B^{1}_{r,d}$ the same as its preimage via
$\acute{e}$tale map $\mathcal{U}\rightarrow U(r,d)$.

Furthermore, we set

(2.15) $\displaystyle
B^{0}_{r,\delta+1,p}:=\theta^{-1}(B^{0}_{r,d+1,p})\quad,\quad
B^{1}_{r,\delta+1,p}:=\theta^{-1}(B^{1}_{r,d+1,p}).\qquad\qquad\qquad$

Observe that the superscript numbers $0$ and $1$ in $B^{0}_{r,\delta+1,p}$ and
$B^{1}_{r,\delta+1,p}$, respectively, don’t refer to the number of linearly
independent global sections of the bundles therein.

Taking Lemma 2.2 into account one obtains
$\overline{B^{0}_{r,\delta+1,p}}=U(r,\delta+1)$. As for
$\overline{B^{1}_{r,\delta+1,p}}$ we have the following

######  Theorem 2.3.

Suppose $1\leq d\leq n(g-1)-1$. Then, the loci
$\overline{B^{1}_{r,\delta+1,p}}$ is an irreducible closed subscheme of
$B^{1}_{r,\delta+1}$ of codimension $1$.

###### Proof.

Since $B^{1}_{r,\delta+1,p}=\theta^{-1}(B^{1}_{r,d+1,p})$ by our convention in
2.14 and 2.15, Lemma 2.2 implies that $B^{1}_{r,\delta+1,p}$ is a non-empty
subset of $B^{1}_{r,\delta+1}$, so $\overline{B^{1}_{r,\delta+1,p}}$ would be
non-empty as well. Restrict the Poincare bundle $\mathcal{E}_{\delta}$ to
$B^{1}_{r,\delta}\times C$ and denote its restriction by the same letter. Set

(2.16)
$\displaystyle\mathbb{V}_{\delta}:=R^{1}(\pi_{1})_{*}\left(\mathcal{E}xt^{1}\left((\pi_{2,\delta})^{*}\mathcal{O}_{p},\mathbb{E}_{\delta}\right)\right),$

where $\mathcal{E}xt^{i}(F,G)$ denotes the $i-th$ “ext”sheaves of $F$ and $G$.
The sheaf $\mathbb{V}_{\delta}$ is a vector bundle on $B^{1}_{r,\delta}$ of
rank $r$, by [2, pp. 166-167]; and the projective bundle

(2.17)
$\displaystyle\alpha_{\delta}:\mathbb{P}(\mathbb{V}_{\delta})\rightarrow
B^{1}_{r,\delta}$

is the family of $\mathrm{Ext}^{1}(\mathcal{O}_{p},E)$’s as $E$ varies in an
open subset of $B^{1}_{r,\delta}$. Then, one has

$\overline{B^{1}_{r,\delta+1,p}}=\mathrm{Im}\,(\nu_{\delta}),$

where the morphism $\nu_{\delta}:\mathbb{P}(\mathbb{V}_{\delta})\rightarrow
B^{1}_{r,\delta+1}$ assigns the stable bundle $F$ to each extension
$0\rightarrow E\rightarrow F\rightarrow\mathcal{O}_{p}\rightarrow 0$. Now, in
order to prove that $\overline{B^{1}_{r,\delta+1,p}}$ is of codimension $1$ in
$B^{1}_{r,\delta+1}$, observe that the fibers of both morphisms
$\alpha_{\delta}$ and $\nu_{\delta}$ are $r$-dimensional. Indeed, for an
extension $e:0\rightarrow E\rightarrow F\rightarrow\mathcal{O}_{p}\rightarrow
0$ and for $E\in B^{1}_{r,\delta}$, we have

$\dim\nu_{\delta}^{-1}(e)=\dim\mathrm{Ext}^{1}(\mathcal{O}_{p},F^{*})=r\quad,\quad\dim\alpha_{\delta}^{-1}(E)=\dim\mathrm{Ext}^{1}(\mathcal{O}_{p},E)=r.$

Therefore

$\dim\overline{B^{1}_{r,\delta+1,p}}=\dim B^{1}_{r,\delta}=\dim
B^{1}_{r,d}=\dim B^{1}_{r,d+1}-1=\dim B^{1}_{r,\delta+1}-1,$

by [23, Theorem II.3.1], as required.

As for irreducibility of $B^{1}_{r,\delta+1,p}$, it suffices to prove the
irreduciblity of $B^{1}_{r,d,p}$. If $U\subset B^{1}_{r,d}$ is the set of
$E\in B^{1}_{r,d}$ in which their general dual elementary transformations are
stable, then $U$ is irreducible, as well. Since the fibers of
$\alpha_{\delta}$ are irreducible, so
$B^{1}_{r,\delta+1,p}=\nu_{\delta}(\alpha_{\delta}^{-1}(U))$ would be
irreducible, as desired. ∎

######  Remark 2.4.

The content of Lemma 2.2 may already exist in the literature, but we were
unable to locate a suitable reference. A similar situation is studied in [8,
Lemma 1, Lemma 2].

###  2.3. Excess dimension inequality

In this subsection, we apply the results of subsection 2.1 to Brill-Noether
loci of stable rank two vector bundles.

For a general $p\in C$, consider $p$ as an effective divisor of degree $1$ on
$C$ and set $\mathcal{L}:=\mathcal{O}(\pi_{1}^{*}p)$. Then
$\mathbb{L}:=(\pi_{2})_{*}(\mathcal{L})$ is a line bundle on the sub-locus
$\mathcal{U}:=B^{0}_{2,\delta+1,p}$. As parts (ii) and (iii) of lemma 2.2
show, the dense sub-locus $\mathcal{U}$ has non-empty intersection with any
irreducible component $X\subset B^{k}_{2,\delta+1}$.

Let

(2.18) $\displaystyle
0\rightarrow[(\nu_{\delta})_{*}(\alpha_{\delta})^{*}(\mathbb{E}_{\delta})]\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\mathbb{F}\rightarrow\mathbb{L}\rightarrow
0,$

be a general extension of $\mathbb{L}$ by
$(\nu_{\delta})_{*}(\alpha_{\delta})^{*}(\mathbb{E}_{\delta})$ and observe
that $\mathbb{F}$ is not necessarily a Poincar$\acute{e}$ bundle neither on
$U(2,\delta+1)$ nor on $\mathcal{U}$. But it can be used to define a dense
sub-locus of B-N loci on $\mathcal{U}$, on which we can compare the dimensions
of $B^{k}_{2,\delta}$ and $B^{k}_{2,\delta+1}$.

Now we consider a diagram of vector bundles on $\mathcal{U}$ as

(2.23)

where
$\tilde{\gamma}_{\delta+1}:=((\gamma_{\delta+1})_{\mid\mathcal{U}})\circ\alpha$.

######  Lemma 2.5.

For $k$ in the allowable range, we have

(2.24) $\displaystyle\dim D_{\delta-2(g-1)-k}(\tilde{\gamma}_{\delta+1})=\dim
B^{k}_{2,d}\quad,\quad\dim
D_{\delta-2(g-1)-k}((\gamma_{\delta+1})_{\mid\mathcal{U}})=\dim
B^{k}_{2,d+1}.$

###### Proof.

At a point $e:0\rightarrow E\longrightarrow
F\rightarrow\mathcal{O}_{p}\rightarrow 0\in\mathcal{U}$ the stalks of
$[(\nu_{\delta})_{*}(\alpha_{\delta})^{*}(\mathbb{E}_{\delta})]_{\mid\mathcal{U}}$,
$(\mathbb{F})_{\mid\mathcal{U}}$ and
$[\mathbb{F}\otimes(\pi_{2})_{*}\pi_{1}^{*}(\mathcal{O}_{D}(D))]_{\mid\mathcal{U}}$
are $H^{0}(E)$, $H^{0}(F)$ and $H^{0}(F_{D})$, respectively. Also
$\ker(\tilde{\gamma}_{\delta+1})_{e}=H^{0}(E(-D))$. So
$\dim\theta^{-1}(D_{\delta-2(g-1)-k}(\tilde{\gamma}_{\delta+1}))=\dim
B^{k}_{2,d}$. This gives the first equality and the proof for another equality
is the same. ∎

Now we apply Theorem 2.1 to this situation, by which we obtain:

######  Theorem 2.6.

If $B^{k}_{2,d}$ is nonempty, then

  1. (a)

$\dim B^{k}_{2,d}\ \geq\dim B^{k}_{2,d+1}-k$,

  2. (b)

$\dim B^{k}_{2,d}\geq\dim B^{k-1}_{2,d}-(2(g-1)-d+k),$

  3. (c)

$\dim B^{k}_{2,d+1}\geq\dim B^{k-1}_{2,d}-(2(g-1)-d+2k).$

In particular, if $B^{k}_{2,d}$ is non-empty, then the scheme $B^{k}_{2,d+1}$
is of expected dimension if and only if $B^{k}_{2,d}$ is of expected
dimension.

###### Proof.

(a) Taking the inequality ((a)) of theorem (2.1) into account, it suffices to
prove that no irreducible component of $B^{k}_{2,d}$ is contained in
$B^{k+1}_{2,d}$, which is immediate by [10, Remark 2.3] or by [19, Prop. 1.6].

Part (b) is a direct consequence of part (a) together with the Riemann-Roch
theorem and part (c) is a combination of (a) and (b). ∎

Example 1: There are cases in which the inequalities in theorem 2.6 get to be
sharp. On general curves of genus $g\geq 1$ and for $d$ with $3\leq d\leq
2(g-1)$, M. Teixidor i Bigas computes the dimension of $B^{2}_{2,d}$ to be
$2d-3$ and $B^{3}_{2,d}$ to be either empty or of dimension $3d-2g-6$. See
[24] and [25, Theorem 2]. So, the equality holds for such $B^{2}_{2,d}$,
$B^{2}_{2,d+1}$, $B^{3}_{2,d}$ and $B^{3}_{2,d+1}$, in the allowable range.

Also inequality can hold in theorem 2.6 strictly. Let $C$ be a general curve
of genus $6$. Then, the scheme $B^{4}_{2,10}$ is non-empty, because the loci

$B^{4}_{2,K}:=\\{E\in B^{4}_{2,10}\mid\det(E)=K\\},$

is non-empty. Furthermore, $B^{4}_{2,10}$ would have superabundant components
of dimension at least $6$, provided that $B^{4}_{2,K}$ is strictly contained
in $B^{4}_{2,10}$. Assuming this to hold, we would have $\dim B^{4}_{2,10}\geq
6$. Once again, using [24] and [25, Theorem 2], we obtain $\dim
B^{3}_{2,9}=9$. So 2.6(c) holds strictly. Furthermore $\dim B^{4}_{2,11}=\dim
B^{3}_{2,9}=9$, by residuation. So 2.6(b) holds strictly.

It remains only to prove that not any vector bundle $E\in B^{4}_{2,10}$
belongs to $B^{4}_{2,K}$, to which it suffices to represent a bundle $E\in
B^{4}_{2,10}$ with $\det(E)\neq K$. In order to see this, take line bundles
$L_{1},L_{2}\in W^{1}_{5}\setminus W^{2}_{5}$ with $L_{1}+L_{2}\neq K$ and
consider the extensions as

$0\rightarrow L_{1}\rightarrow E\rightarrow L_{2}\rightarrow 0.$

Such an extension, belonging to $B^{4}_{2,10}\setminus B^{4}_{2,K}$, exists
according to [17, Proposition 3.1] and [18, Lemma 2.18].

Example 2: This example shows that Lemma 2.2(ii) fails if the genericity of
the dual elementary transformation was removed. Suppose $E$ is a rank two
bundle with $\deg(E)=1$ and has a line sub bundle of degree $0$. Take a
generic point $p\in C$ and let $F$ be the dual elementary transformation of
$E$ by $p$ satisfying $\mathrm{Ker}(E_{p}\rightarrow F_{p})=L_{\mid p}$. Then
for $p_{1},\cdots,p_{t}\in C$ with $p_{i}\neq p$ we have a diagram as

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(p_{1}+\cdots+p_{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(p_{1}+\cdots+p_{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L(p_{1}+\cdots+p_{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L(p_{1}+\cdots+p_{t}+p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{0.}$

Then, $\deg L(p_{1}+\cdots+p_{t}+p)\geq\mu(F(p_{1}+\cdots+p_{t}))$, so $F$ is
strictly semi-stable.

###  2.4. A byproduct:

As a consequence, we relate the smoothness of rank two Brill-Noether schemes
with different degrees. In order to do this, we first make a comparison among
the dimension of the tangent spaces of these schemes. See also [4, Th. 3.1],
[11] and [15].

######  Lemma 2.7.

Let $p\in C$ be a general point, $E\in B^{k}_{r,d}\setminus B^{k+1}_{r,d}$ and
$F$ be a general stable dual elementary transformation of $E$ by $p$. Then,
for $1\leq d\leq r(g-1)$ and $1\leq k\leq\frac{d}{2}+1$ we have

$\dim T_{E}(B^{k}_{r,d})\geq\dim T_{F}(B^{k}_{r,d+1})-k.$

###### Proof.

Recall that

${\small T_{E}(B^{k}_{r,d})=[\mathrm{Im}\,\\{\mu_{E}:H^{0}(E)\otimes
H^{0}(K\otimes E^{*})\rightarrow H^{0}(K\otimes E\otimes E^{*})\\}]^{\perp},}$

where for a sub-vector space $W\leq V$, by $W^{\perp}$ we mean the annihilator
vector space of $W$ inside the dual of $V$ and $\mu_{E}$ denotes the Petri map
of $E$.

Notice that, under the hypothesis, the bundle $E$ is special. So lemma [21,
2.5] can be applied to obtain $H^{0}(E)=H^{0}(F)$, therefore $F\in
B^{k}_{r,d+1}\setminus B^{k+1}_{r,d+1}$. There exists now a diagram as

$\textstyle{H^{0}(E)\otimes H^{0}(K\otimes
E^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{E}}$$\textstyle{H^{0}(K\otimes
E\otimes
E^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{H^{0}(F)\otimes
H^{0}(K\otimes
F^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{F}}$$\scriptstyle{i}$$\textstyle{H^{0}(K\otimes
F\otimes
F^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{H^{0}(K\otimes
F\otimes E^{*}),}$

where the morphisms $i$, $\alpha$ and $\beta$ are all injective. Therefore, we
will have $\dim\ker(\mu_{E})\geq\dim\ker(\mu_{F})$. The Riemann-Roch theorem
implies that $h^{0}(K\otimes F^{*})=h^{0}(K\otimes E^{*})-1$. Therefore,

$kh^{0}(K\otimes E^{*})-\dim\ker(\mu_{E})\leq kh^{0}(K\otimes
F^{*})-\dim\ker(\mu_{F})+k,$

implying $\dim\mathrm{Im}\,(\mu_{E})\leq\dim\mathrm{Im}\,(\mu_{F})+k$, as
required. ∎

######  Corollary 2.8.

Let $E\in B^{k}_{2,d}$ and $F$ be a general dual elementary transformation of
$E$ by a general point $p\in C$. Assume moreover that $B^{k}_{2,d}$ is of
expected dimension. Then, for $1\leq d\leq 2(g-1)$ and $1\leq
k\leq\frac{d}{2}+1$ the scheme $B^{k}_{2,d}$ is smooth at $E$ if and only if
$B^{k}_{2,d+1}$ is smooth at $F$.

###### Proof.

Since $B^{k}_{2,d}$ is of expected dimension so is the scheme $B^{k}_{2,d+1}$,
and vice versa; by theorem 2.6. The smoothness of $B^{k}_{2,d}$ at $E\in
B^{k}_{2,d}$ implies that $\dim B^{k}_{2,d}=\dim T_{E}B^{k}_{2,d}$. This, by
Lemma 2.7 together with the fact that $B^{k}_{2,d+1}$ is of expected
dimension, gives rise to the assertion. ∎

Acknowledgement: George H. Hitching read a previous version of this paper
carefully and made various useful comments, I express my gratitude to him. I
am also deeply grateful to Edoardo Ballico and Peter Newstead for their useful
hints and responses to my questions. This research was in part supported by a
grant from IPM (No. 1400140038).

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