# Higher rank Brill-Noether theory on $\mathbb{P}^{2}$

Ben Gould, Yeqin Liu, and Woohyung Lee

###### Abstract.

Let $M_{\mathbb{P}^{2}}(\mathbf{v})$ be a moduli space of semistable sheaves
on $\mathbb{P}^{2}$, and let $B^{k}(\mathbf{v})\subseteq
M_{\mathbb{P}^{2}}(\mathbf{v})$ be the Brill-Noether locus of sheaves $E$ with
$h^{0}(\mathbb{P}^{2},E)\geq k$. In this paper we develop the foundational
properties of Brill-Noether loci on $\mathbb{P}^{2}$. Set $r=r(E)$ to be the
rank and $c_{1},c_{2}$ the Chern classes. The Brill-Noether loci have natural
determinantal scheme structures and expected dimensions $\dim
B^{k}(\mathbf{v})=\dim M_{\mathbb{P}^{2}}(\mathbf{v})-k(k-\chi(E))$. When
$c_{1}>0$, we show that the Brill-Noether locus $B^{r}(\mathbf{v})$ is
nonempty. When $c_{1}=1$, we show all of the Brill-Noether loci are
irreducible and of the expected dimension. We show that when $\mu=c_{1}/r>1/2$
is not an integer and $c_{2}\gg 0$, the Brill-Noether loci are reducible and
describe distinct irreducible components of both expected and unexpected
dimension.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Extension parametrizations of moduli spaces
  4. 4 Emptiness & non-emptiness of Brill-Noether loci
  5. 5 Irreducibility & reducibility of Brill-Noether loci
  6. 6 References

##  1\. Introduction

On a polarized variety $(X,H)$ the moduli spaces $M_{X,H}(\mathbf{v})$ of
$H$-semistable sheaves of numerical type $\mathbf{v}$ carry Brill-Noether loci
$B^{k}(\mathbf{v})\subseteq M_{X,H}(\mathbf{v})$ whose members $E$ satisfy
$h^{0}(X,E)\geq k$. In this paper we develop the foundational properties of
Brill-Noether loci on $X=\mathbb{P}^{2}$. Set $r=\text{ch}_{0}(\mathbf{v})$,
$\mu=\mu(\mathbf{v})$ and $\Delta=\Delta(\mathbf{v})$ to be the rank, slope,
and discriminant of $\mathbf{v}$. The Brill-Noether loci have natural
determinantal scheme structures and expected dimensions $\dim
B^{k}(\mathbf{v})=\dim M_{\mathbb{P}^{2}}(\mathbf{v})-k(k-\chi(E))$. When
$\mu>0$ we show that $B^{r}(\mathbf{v})$ is nonempty. When
$\text{ch}_{1}(\mathbf{v})=1$, we show each Brill-Noether locus
$B^{k}(\mathbf{v})$ is irreducible and of the expected dimension as a
determinantal variety, $\dim B^{k}(\mathbf{v})=\dim
M_{\mathbb{P}^{2}}(\mathbf{v})-k(k-\chi(\mathbf{v}))$. We prove that when
$r,\text{ch}_{1}>0$ and $\mu>1/2$ with $\Delta\gg 0$, then $B^{r}(\mathbf{v})$
is reducible, and describe distinct irreducible components.

When $X=C$ is a smooth curve and the rank is 1,
$M_{X}(\mathbf{v})=\text{Pic}^{d}(C)$ is the space of line bundles of a given
degree $d$, and the Brill-Noether loci $B^{k}(\mathbf{v})=W^{k}_{d}(C)$ have
been studied for over a century [ACGH85]. Much is known about their geometry
when $C$ is general: when the expected dimension of $W^{k}_{d}(C)$ as a
determinantal variety is positive, it is nonempty of that dimension and
irreducible. The general $L\in\text{Pic}^{d}(C)$ has $h^{0}(L)=\chi(L)=d-g+1$
when this is non-negative. In higher rank the general bundle still has
$h^{0}(E)=\chi(E)$ when this is non-negative, and the Brill-Noether loci have
been studied in detail (for a survey see, e.g., [New21] and its bibliography).

On algebraic surfaces, however, much less is known about Brill-Noether theory.
The basic theory has begun to be worked out on Hirzebruch surfaces (see
[CMR10]) and in rank 2 on $\mathbb{P}^{2}$ (see [RLTLZ21]). On
$\mathbb{P}^{2}$, the moduli spaces $M_{\mathbb{P}^{2}}(\mathbf{v})$ of
semistable bundles of any rank are irreducible, and a well-known theorem of
Göttsche-Hirschowitz describes the global sections of a general bundle $E\in
M_{\mathbb{P}^{2}}(\mathbf{v})$ of rank at least two: if the slope $\mu(E)$ is
positive, then $h^{0}(E)=\max\\{0,\chi(E)\\}$, which is determined by
$\mathbf{v}$. More generally, the general bundle $E$ of any slope has at most
one nonzero cohomology group, and by semicontinuity there is an open dense
subset of $M_{\mathbb{P}^{2}}(\mathbf{v})$ of bundles with this cohomology.
The Brill-Noether loci $B^{k}(\mathbf{v})$ with $k>\chi(\mathbf{v})$ form its
complement. These foundational results make the study of the Brill-Noether
loci $B^{k}(\mathbf{v})$ on $\mathbb{P}^{2}$ approachable.

###  1.1. Geometry of Brill-Noether loci

The moduli spaces $M(\mathbf{v})=M_{\mathbb{P}^{2}}(\mathbf{v})$ of semistable
sheaves on $\mathbb{P}^{2}$ with Chern character $\mathbf{v}$ are irreducible,
projective algebraic varieties. We call a Chern character $\mathbf{v}\in
K(\mathbb{P}^{2})$ stable if it is the Chern character of a stable sheaf, so
that $M(\mathbf{v})\neq\emptyset$. We are interested in the Brill-Noether loci
$B^{k}(\mathbf{v})\subseteq M(\mathbf{v})$. As in the case of line bundles on
curves, the Brill-Noether loci are constructed as determinantal varieties, so
each has an expected dimension, which is a lower bound for the dimension of an
irreducible component of $B^{k}(\mathbf{v})$; it is

$\text{expdim}B^{k}(\mathbf{v})=\dim
M_{\mathbb{P}^{2}}(\mathbf{v})-k(k-\chi(\mathbf{v})).$

Our main qualitative result is the following.

######  Theorem 1.1.

Let $\mathbf{v}\in K(\mathbb{P}^{2})$ be a stable Chern character and
$M(\mathbf{v})$ the associated moduli space of semistable sheaves. Suppose
that $\mu(\mathbf{v})>0$ and set $r=\text{ch}_{0}(\mathbf{v})$.

  1. (1)

(Emptiness) For any $E\in M(\mathbf{v})$,

(1)
$h^{0}(E)\leq\max\left\\{\frac{1}{2}c_{1}(E)^{2}+\frac{3}{2}c_{1}(E)+1,r(E)\right\\}.$

When $r(E)\geq\frac{1}{2}c_{1}(E)^{2}+\frac{3}{2}c_{1}(E)+1$, this bound is
sharp, and $B^{r}(\mathbf{v})$ contains a component of the expected dimension
(Theorem 4.1 and Theorem 4.6).

  2. (2)

(Nonemptiness) $B^{r}(\mathbf{v})$ is nonempty (Theorem 4.7).

  3. (3)

(Irreducibility) If $\text{ch}_{1}(\mathbf{v})=1$, all of the nonempty Brill-
Noether loci on $M(\mathbf{v})$ are irreducible and of the expected dimension
(Theorem 5.1).

  4. (4)

(Reducibility) Suppose $\text{ch}_{1}(\mathbf{v})>1$. If $\mu(\mathbf{v})>1/2$
is not an integer and $\Delta(\mathbf{v})\gg 0$, then $B^{r}(\mathbf{v})$ is
reducible and contains components of both expected and unexpected dimensions
(Theorem 5.7).

Since the Brill-Noether loci are nested, Theorem 1.1 (2) implies the Brill-
Noether loci $B^{k}(\mathbf{v})$ are nonempty for all $0\leq k\leq r$. The
bound (1) is also sharp for the line bundles $\mathcal{O}(d)$ and the
Lazarsfeld-Mukai-type bundles
$M_{d}=\text{coker}(\mathcal{O}(-d)\rightarrow\mathcal{O}^{h^{0}(\mathcal{O}(d))})$,
including $M_{1}=T_{\mathbb{P}^{2}}(-1)$.

Our main technical tools for studying Brill-Noether loci are parametrizations
of certain moduli spaces $M(\mathbf{v})$ and Brill-Noether loci
$B^{k}(\mathbf{v})$ by projective bundles of extension spaces. The
applications to Brill-Noether loci are often of the following form. A sheaf
$E$ on $\mathbb{P}^{2}$ has an evaluation map on global sections

$\text{ev}_{E}:H^{0}(\mathbb{P}^{2},E)\otimes\mathcal{O}_{\mathbb{P}^{2}}\rightarrow
E.$

Assume that the general $E\in M(\mathbf{v})$ has $m$ global sections, i.e.,
$h^{0}(\mathbb{P}^{2},E)=m$, and $m\leq r(E)$. When $\text{ev}_{E}$ has full
rank $E$ sits as an extension class

$[0\rightarrow
H^{0}(\mathbb{P}^{2},E)\otimes\mathcal{O}_{\mathbb{P}^{2}}\stackrel{{\scriptstyle\text{ev}_{E}}}{{\rightarrow}}E\rightarrow
E^{\prime}\rightarrow
0]\in\text{Ext}^{1}(E^{\prime},H^{0}(\mathbb{P}^{2},E^{\prime})\otimes\mathcal{O}_{\mathbb{P}^{2}}).$

If $E^{\prime}$ is semistable with Chern character $\mathbf{v}^{\prime}\in
K(\mathbb{P}^{2})$, we may form the projective bundle $\mathbb{P}$ over
$M(\mathbf{v}^{\prime})$ whose fiber over $E^{\prime}$ is the projective space
$\mathbb{P}\text{Ext}^{1}(E^{\prime},H^{0}(\mathbb{P}^{2},E^{\prime})\otimes\mathcal{O}_{\mathbb{P}^{2}})$.
If the general such extension is stable, we obtain a classifying rational map
$\phi:\mathbb{P}\dashrightarrow M(\mathbf{v})$, which we call the extension
parametrization of $M(\mathbf{v})$.

The critical technical challenge in constructing such parametrizations is in
proving the stability of extension sheaves. When the rank of $E^{\prime}$ is
0, so $\chi(E)=r(E)$, we accomplish this by studying smooth curves in
$\mathbb{P}^{2}$ realized as the support of $E^{\prime}$ and the moduli spaces
$\text{Pic}^{d}_{\mathcal{C}/U}$ of these sheaves (Sections 4 and 5). The
central inputs to our results are the classification of stable Chern
characters on $\mathbb{P}^{2}$ by Drézet-Le Potier ([DLP85, Théorème C]) and
the computation of the cohomology of the general stable bundle in
$M(\mathbf{v})$ by Göttsche-Hirschowitz (Theorem 2.2). Our methods likely
extend to other surfaces for which these two notions are understood, e.g.,
Hirzebruch surfaces and K3 surfaces of Picard rank 1 (see [CH21] and [CNY21]
respectively).

###  1.2. Organization of the paper

In Section 2 we recall basic facts about moduli spaces of sheaves on
$\mathbb{P}^{2}$. In Section 3 we introduce extension parametrizations, which
are the main tools used in further sections.

In Section 4 we prove emptiness and non-emptiness statements for Brill-Noether
loci. In Section 5 we prove irreducibility and reducibility statements for
Brill-Noether loci.

###  1.3. Acknowledgements

We are happy to thank Izzet Coskun for very many helpful conversations and his
support while this work was carried out. The first author was supported by NSF
RTG no. 1246844.

The figures in this paper were generated on Mathematica and graffitikz.
graffitikz is an open-source 2D vector shape editor that supports TikZ output,
created by Wenyu Jin. It can be found online at
https://github.com/wyjin/graffitikz.

##  2\. Preliminaries

###  2.1. Stability on $\mathbb{P}^{2}$

In this section we collect basic facts about stability for coherent sheaves on
$\mathbb{P}^{2}$. We refer the reader to the books by Le Potier [LP97] and
Huybrechts-Lehn [HL10] for further details.

####  2.1.1. Numerical invariants & stability

All sheaves in this paper will be coherent, but not necessarily torsion-free.
Let $E$ be a coherent sheaf on $\mathbb{P}^{2}$. The Hilbert polynomial
$P_{E}(m)=\chi(E(m))$ is of the form

$P_{E}(m)=\alpha_{d}\frac{m^{d}}{d!}+O(m^{d-1}),$

and we define the reduced Hilbert polynomial $p_{E}(m)$ to be

$p_{E}(m)=P_{E}(m)/\alpha_{d}.$

A pure-dimensional coherent sheaf $E$ on $\mathbb{P}^{2}$ is (semi)stable if
for all nontrivial subsheaves $F\hookrightarrow E$ we have
$p_{F}<(\leq)p_{E}$, where polynomials are compared for sufficiently large
$m$. We will assume throughout this paper that a semistable sheaf has positive
rank.

Given a character $\mathbf{v}\in K(\mathbb{P}^{2})$, we define, respectively,
the slope and the discriminant of $\mathbf{v}$ by

$\mu(\mathbf{v})=\frac{\text{ch}_{1}(\mathbf{v})}{\text{ch}_{0}(\mathbf{v})},\quad\Delta(\mathbf{v})=\frac{1}{2}\mu(\mathbf{v})^{2}-\frac{\text{ch}_{2}(\mathbf{v})}{\text{ch}_{0}(\mathbf{v})}.$

On $\mathbb{P}^{2}$ these classes may be considered as rational numbers. When
$\mathbf{v}=\text{ch}(E)$ for a coherent sheaf $E$ we set
$\mu(E)=\mu(\mathbf{v})$ and $\Delta(E)=\Delta(\mathbf{v})$. We have

$\mu(E\otimes F)=\mu(E)+\mu(F),\quad\Delta(E\otimes F)=\Delta(E)+\Delta(F)$

for coherent sheaves $E,F$ on $\mathbb{P}^{2}$.

In terms of these invariants, the Riemann-Roch theorem takes the form

$\chi(E)=r(E)(p(\mu(E))-\Delta(E)),$

where

$p(x)=p_{\mathcal{O}_{\mathbb{P}^{2}}}(x)=\frac{1}{2}(x^{2}+3x+2).$

More generally for sheaves $E,F$ on $\mathbb{P}^{2}$, we set
$\text{ext}^{i}(E,F)=\dim\text{Ext}^{i}(E,F)$. The relative Riemann-Roch
theorem states

$\chi(E,F)=\sum(-1)^{i}\text{ext}^{i}(E,F)=r(E)r(F)(p(\mu(E)-\mu(F))-\Delta(E)-\Delta(F)),$

and similarly for Chern characters.

Additionally, we may define slope-stability for sheaves on $\mathbb{P}^{2}$:
$E$ is slope-(semi)stable if for all subsheaves $F\hookrightarrow E$ of
smaller rank, we have $\mu(F)<(\leq)\mu(E)$. Slope-stability implies
stability, and is often easier to check.

####  2.1.2. Classification of stable bundles

The Chern characters of stable bundles on $\mathbb{P}^{2}$ have been
classified (see, e.g., [DLP85], [LP97]). The classification is important for
our results, and we sketch it in this subsection.

Recall that we say a character $\mathbf{v}\in K(\mathbb{P}^{2})$ is stable if
there is a stable sheaf of Chern character $\mathbf{v}$, and that we assume
stable characters have $\text{ch}_{0}(\mathbf{v})>0$. Stability imposes strong
conditions on the numerical data of $\mathbf{v}$, which are determined by
exceptional bundles on $\mathbb{P}^{2}$: a sheaf $E$ on $\mathbb{P}^{2}$ is
exceptional if $\text{Ext}^{i}(E,E)=0$ for $i>0$.

By a theorem of Drézet [Dré86], every exceptional sheaf on $\mathbb{P}^{2}$ is
a stable vector bundle. A stable bundle $E$ is exceptional if and only if
$\Delta(E)<1/2$, by Riemann-Roch [LP97, Proposition 16.1.1]. By definition,
exceptional bundles are rigid and their moduli spaces consist of a single
reduced point [LP97, Corollary 16.1.5]. Exceptional bundles on
$\mathbb{P}^{2}$ include the line bundles $\mathcal{O}_{\mathbb{P}^{2}}(k)$
and the tangent bundle $T_{\mathbb{P}^{2}}$.

The slopes of exceptional bundles can be completely described [LP97, §16.3];
set $\mathcal{E}$ to be the set of slopes of exceptional bundles. For an
exceptional bundle $E$ of slope $\alpha$ we write $E=E_{\alpha}$ and
$\Delta(E)=\Delta_{\alpha}$. We define the Drézet-Le Potier curve to be the
following piecewise-polynomial curve in the $(\mu,\Delta)$-plane:

$\delta(\mu)=\sup_{\alpha\in\mathcal{E}:|\mu-\alpha|<3}(p(-|\mu-\alpha|)-\Delta_{\alpha}).$

A plot of an approximation of the Drézet-Le Potier curve between 0 and 1 is
shown in Figure 1. The precise relationship of exceptional bundles to stable
characters is due to Drézet and Le Potier: a character $\mathbf{v}\in
K(\mathbb{P}^{2})$ is stable if and only if $c_{1}:=r\mu\in\mathbb{Z}$,
$\chi:=r(P(\mu)-\Delta)\in\mathbb{Z}$, and
$\Delta(\mathbf{v})\geq\delta(\mu(\mathbf{v}))$, or $\mathbf{v}$ is
exceptional ([DLP85, Théorème C], [LP97]). A character satisfying the first
two conditions in the theorem is called integral. Each exceptional bundle
$E_{\alpha}$ of slope $\alpha$ determines two “branches” of the Drézet-Le
Potier curve, on the left and right sides of the line $\mu=\alpha$. The
characters $\mathbf{v}$ on the branch to the right of $\mu=\alpha$ satisfy
$\chi(\mathbf{v},E_{\alpha})=0$, and we call this branch the
${}^{\perp}E_{\alpha}$-branch of the curve, and characters $\mathbf{v}$ on the
branch to the left of $\mu=\alpha$ satisfy $\chi(E_{\alpha},\mathbf{v})=0$ and
we call this the $E_{\alpha}^{\perp}$-branch.

Figure 1. The Drézet-Le Potier curve $\Delta=\delta(\mu)$

Let $\mathbf{v}\in K(\mathbb{P}^{2})$ be a character and set
$\text{ch}_{0}(\mathbf{v})=r$, $\mu(\mathbf{v})=\mu$, and
$\Delta(\mathbf{v})=\Delta$. When $\mathbf{v}$ is stable with $r>0$, Drézet
and Le Potier showed the moduli space of stable sheaves with Chern character
$\mathbf{v}$ is a normal, irreducible, factorial projective variety of
dimension

$\dim M(\mathbf{v})=r^{2}(2\Delta-1)+1$

([LP97, Theorem 17.0.1]). When $\mathbf{v}$ is a non-exceptional stable
character of rank $>1$, the general bundle $E\in M(\mathbf{v})$ is slope-
stable ([DLP85, Corollaire 4.12]). Slope-stability allows us to use elementary
modifications:

######  Remark 2.1.

If $\Delta_{0}$ is the discriminant of a stable non-exceptional sheaf $E$,
then by integrality of the Euler characteristic one can see that the
discriminant of another stable sheaf of slope $\mu(E)$ and rank $r(E)$ differs
from $\Delta_{0}$ by an integral multiple of $\frac{1}{r(E)}$. One can in fact
obtain any such discriminant larger than or equal to $\delta(\mu(E))$ by
taking elementary modifications: choosing a point $p\in\mathbb{P}^{2}$ and a
surjection $E\rightarrow\mathcal{O}_{p}$ one forms the exact sequence

$0\rightarrow E^{\prime}\rightarrow E\rightarrow\mathcal{O}_{p}\rightarrow 0.$

The sheaf $E^{\prime}$ is not locally free, but it has $r(E^{\prime})=r(E)$,
$\mu(E^{\prime})=\mu(E)$, and $\Delta(E^{\prime})=\Delta(E)+1/r(E)$. One can
check that when $E$ is slope stable, so is $E^{\prime}$ (but this is not true
for stability). In this way one can construct slope-stable bundles of
discriminant $\Delta_{0}+k/r(E)$ for any non-negative integer $k$. See [CH20,
Lemma 2.7] for details.

###  2.2. Brill-Noether loci

We now define the Brill-Noether loci and endow them with a natural
determinantal structure, which leads to a lower bound on the dimension of
their components.

The expected value of $h^{0}(E)$ when $E$ is stable is provided by the
following well-known theorem, due to Göttsche-Hirschowitz.

######  Theorem 2.2 ([GH98]).

When $\text{ch}_{0}(\mathbf{v})\geq 2$, the general sheaf $E\in M(\mathbf{v})$
has at most one nonzero cohomology group:

  1. (1)

if $\chi(E)\geq 0$ and $\mu(E)>-3$, then $h^{1}(E)=h^{2}(E)=0$;

  2. (2)

if $\chi(E)\geq 0$ and $\mu(E)<-3$, then $h^{0}(E)=h^{1}(E)=0$;

  3. (3)

if $\chi(E)<0$, $h^{0}(E)=h^{2}(E)=0$.

When a sheaf $F\in M(\mathbf{v})$ has cohomology groups as prescribed in the
Theorem, we will say that it is cohomologically general or has general
cohomology. By semicontinuity, when $r\geq 2$ the locus of sheaves with
general cohomology forms an open subset of the moduli space; the Brill-Noether
loci $B^{k}(\mathbf{v})$ with $k>\chi(\mathbf{v})$ make up the complement of
this open set. Let $M^{s}(\mathbf{v})\subseteq M(\mathbf{v})$ denote the locus
of strictly stable sheaves.

######  Definition 2.3.

For a stable Chern character $\mathbf{v}\in K(\mathbb{P}^{2})$, the $k$th
Brill-Noether locus $B^{k}(\mathbf{v})\subseteq M(\mathbf{v})$ is defined as a
set to be the closure

$B^{k}(\mathbf{v})=\overline{\\{E\in M^{s}(\mathbf{v}):h^{0}(E)\geq
k\\}}\subseteq M(\mathbf{v}).$

$B^{k}(\mathbf{v})$ is the locus where $h^{0}$ jumps by at least
$k-\chi(\mathbf{v})$.

We restrict to stable sheaves in the definition because the number of global
sections is not constant in $S$-equivalence classes, so $h^{0}([E])$ is not
defined for strictly semistable points $[E]\in M(\mathbf{v})$. The Brill-
Noether loci are clearly nested: $B_{k}(\mathbf{v})\supseteq
B_{k+1}(\mathbf{v})$ for all $k$.

Note that for a stable sheaf $F$ with $\mu(F)<0$, any map
$\mathcal{O}\rightarrow F$ corresponding to $0\neq s\in H^{0}(F)$ would be
destabilizing, so $H^{0}(F)=0$ and there are no Brill-Noether loci in the
associated moduli space. Thus in what follows we assume the slope is
nonnegative. In these cases, we expect $h^{0}(F)=\max\\{0,\chi(F)\\}$.

We endow $B^{k}(\mathbf{v})$ with a determinantal scheme structure, as follows
(cf. [CHW17, Proposition 2.6] and [CMR10, §2]). Let $\mathscr{E}/S$ be a
proper flat family of stable sheaves on $\mathbb{P}^{2}$ of Chern character
$\mathbf{v}$. We will examine the relative Brill-Noether locus
$B^{k}_{S}(\mathbf{v})$; when $M^{s}(\mathbf{v})$ admits a universal family
$\mathscr{U}$, this will give the appropriate scheme structure on
$B^{k}(\mathbf{v})$. In general, one can work on the stack
$\mathcal{M}^{s}(\mathbf{v})$ and take the image in the coarse moduli space
$M^{s}(\mathbf{v})$ (see [Alp13, Theorem 4.16 and Example 8.7]).

Choose $s\in S$ and let $d\gg 0$ be sufficiently large and let
$C\subseteq\mathbb{P}^{2}$ be a general-enough curve of degree $d$, chosen so
that the singularities of $\mathscr{E}_{s}$ do not meet $C$. After replacing
$S$ by an open subset, we can assume $C$ does not meet the singularities of
any member of $\mathscr{E}$. Let $p:S\times\mathbb{P}^{2}\rightarrow S$ and
$q:S\times\mathbb{P}^{2}\rightarrow\mathbb{P}^{2}$ be the projections. Form
the exact sequence

$0\rightarrow\mathscr{E}\rightarrow\mathscr{E}\otimes
q^{*}\mathcal{O}(d)\rightarrow\mathscr{E}\otimes(q^{*}\mathcal{O}(d)|_{C})\rightarrow
0$

of sheaves on $S\times\mathbb{P}^{2}$.

Choosing $d$ large enough so that each of the cohomology groups
$H^{1}(\mathscr{E}_{s}\otimes q^{*}\mathcal{O}(d))$ vanish, we conclude
$R^{1}p_{*}(\mathscr{E}\otimes q^{*}\mathcal{O}(d))=0$ and we obtain the exact
sequence

$0\rightarrow p_{*}\mathscr{E}\rightarrow p_{*}(\mathscr{E}\otimes
q^{*}\mathcal{O}(d))\stackrel{{\scriptstyle\phi}}{{\rightarrow}}p_{*}(\mathscr{E}\otimes(q^{*}\mathcal{O}(d)|_{C}))\rightarrow
R^{1}p_{*}(\mathscr{E})\rightarrow 0$

of sheaves on $S$. When $d$ is sufficiently large the map $\phi$ is a map
between bundles of the same rank, and the $k$th determinantal variety
associated to $\phi$ is supported on those $s\in S$ such that
$h^{0}(\mathscr{E}_{s})\geq k$, or equivalently $h^{1}(\mathscr{E}_{s})\geq
k-\chi(\mathbf{v})$. Thus scheme-theoretically we define $B^{k}(\mathbf{v})$
to be this determinantal variety, and observe that its support agrees with the
set described above.

The construction above does not depend on the choice of $d$ or $C$; precisely,
over an open subset $U$ of $S$ over which the source and target of $\phi$ are
trivial, the Fitting ideal $\text{Fitt}_{0}(R^{1}p_{*}\mathcal{E})$ is
generated by $\det\phi$, viewed as a section of $\mathcal{O}_{U}$. See [Eis13,
Chapter 20]. The Fitting ideal is the ideal sheaf of $B^{k}(\mathbf{v})$, and
gives the desired scheme structure.

Being a determinantal variety, the codimension of a component $Z$ of
$B^{k}(\mathbf{v})$ is bounded:

$\text{codim}_{M(\mathbf{v})}(Z)\leq k(k-\chi(\mathbf{v})).$

Thus since

$\dim(M(\mathbf{v}))=\text{ext}^{1}(\mathbf{v},\mathbf{v})=r(\mathbf{v})^{2}(2\Delta(\mathbf{v})-1)+1$

the dimension of a component $Z$ of $B^{k}(\mathbf{v})$ is bounded below:

$\dim(Z)\geq
r(\mathbf{v})^{2}(2\Delta(\mathbf{v})-1)+1-k(k-\chi(\mathbf{v})).$

######  Definition 2.4.

The expected codimension of the $k$th Brill-Noether locus
$B^{k}(\mathbf{v})\subseteq M(\mathbf{v})$ is

$\text{expcodim}^{k}(\mathbf{v})=k(k-\chi(\mathbf{v})).$

##  3\. Extension parametrizations of moduli spaces

In this section we study the stability of extension sheaves. When extension
sheaves are semistable, we obtain families of sheaves over loci in the
associated moduli space.

######  Definition 3.1.

Suppose that $\mathbf{v},\mathbf{v}^{\prime},\mathbf{v}^{\prime\prime}\in
K(\mathbb{P}^{2})$ are stable characters with
$\mathbf{v}=\mathbf{v}^{\prime}+\mathbf{v}^{\prime\prime}$. Let
$\mathscr{E}^{\prime}/S^{\prime}$ and
$\mathscr{E}^{\prime\prime}/S^{\prime\prime}$ be families of semistable
sheaves of Chern characters $\mathbf{v}^{\prime}$ and
$\mathbf{v}^{\prime\prime}$, respectively. If the general extension $E$

$0\rightarrow E^{\prime}_{s^{\prime}}\rightarrow E\rightarrow
E^{\prime\prime}_{s^{\prime\prime}}\rightarrow 0$

with $s^{\prime}\in S^{\prime}$ and $s^{\prime\prime}\in S^{\prime\prime}$ is
semi-stable, then the induced rational map

${\mathbb{P}\text{Ext}^{1}(E^{\prime\prime},E^{\prime})}$${M(\mathbf{v})}$${S^{\prime}\times
S^{\prime\prime}}$

defined on a locus where
$\text{Ext}^{1}(E^{\prime\prime}_{s^{\prime\prime}},E^{\prime}_{s^{\prime}})$
has constant rank, is called an extension parametrization associated to
$M(\mathbf{v})$.

Stability of these extensions is the central technical challenge of this
paper. However, there is a class called the extremal extensions whose
stability is not hard to show, which we now recall.

###  3.1. Extremal extensions

Stability of extension sheaves is known when the slope of the subsheaf is
extremal with respect to the extension sheaf. The following definition is
derived from [CH16, Definition 4.1], but we make some slight modifications for
our purposes.

######  Definition 3.2.

A Chern character $\mathbf{v}^{\prime}$ on $\mathbb{P}^{2}$ is called extremal
for $\mathbf{v}$ if $\mathbf{v}^{\prime}$ satisfies:

  1. (D1)

$\text{ch}_{0}(\mathbf{v}^{\prime})<\text{ch}_{0}(\mathbf{v})$;

  2. (D2)

$\mu(\mathbf{v}^{\prime})\leq\mu(\mathbf{v})$;

  3. (D3)

$\mathbf{v}^{\prime}$ is stable;

and furthermore, for any $\mathbf{w}\in K(\mathbb{P}^{2})$ satisfying
(D1)-(D3):

  1. (E1)

$\mu(\mathbf{w})\leq\mu(\mathbf{v}^{\prime})$;

  2. (E2)

If $\mu(\mathbf{w})=\mu(\mathbf{v}^{\prime})$, then
$\Delta(\mathbf{w})\geq\Delta(\mathbf{v}^{\prime})$;

  3. (E3)

If $\mu(\mathbf{w})=\mu(\mathbf{v}^{\prime})$ and
$\Delta(\mathbf{w})=\Delta(\mathbf{v}^{\prime})$, then
$\text{ch}_{0}(\mathbf{v}^{\prime})\leq\text{ch}_{0}(\mathbf{w})$.

A triple $\Xi=(\mathbf{v}^{\prime},\mathbf{v},\mathbf{v}^{\prime\prime})$ of
Chern characters on $\mathbb{P}^{2}$ is called extremal if
$\mathbf{v}^{\prime}+\mathbf{v}^{\prime\prime}=\mathbf{v}$, where
$\mathbf{v}^{\prime}$ is the extremal character for $\mathbf{v}$, and
$\mathbf{v}^{\prime\prime}$ is stable.

It is not obvious a priori that extremal decompositions exist for any
$\mathbf{v}\in K(\mathbb{P}^{2})$ since $\mathbf{v}^{\prime\prime}$ may not be
stable, but at least it is clear when $\mathbf{v}$ has large discriminant, see
e.g. [CH16, Lemma 4.3], or simply observe that
$\Delta(\mathbf{v}^{\prime\prime})\gg 0$ when $\Delta(\mathbf{v})\gg 0$.

Our main use of Definition 3.2 will be to construct extension parametrizations
associated to moduli spaces. The following simple observation is crucial:

######  Proposition 3.3.

Suppose $\Xi=(\mathbf{v}^{\prime},\mathbf{v},\mathbf{v}^{\prime\prime})$ is an
extremal triple, and we have a nonsplit short exact sequence

$0\rightarrow E^{\prime}\rightarrow E\rightarrow E^{\prime\prime}\rightarrow
0$

with $E^{\prime}\in M(\mathbf{v}^{\prime})$ and $E^{\prime\prime}\in
M(\mathbf{v}^{\prime\prime})$. Then $E$ is semistable.

###### Proof.

Suppose that $E\rightarrow Q$ is a destabilizing quotient of $E$, so that
$\mu(Q)\leq\mu(E)$ with $r(Q)<r(E)$. After perhaps passing to a further
quotient of $Q$, we may assume $Q$ is stable. Slope-closeness (E1) implies
that $\mu(Q)\leq\mu(E^{\prime})$. If $\mu(Q)<\mu(E^{\prime})$, the composition
$E^{\prime}\rightarrow E\rightarrow Q$ is zero by stability, so there is an
induced map $E^{\prime\prime}\rightarrow Q$, however
$\mu(\mathbf{v}^{\prime\prime})\geq\mu(\mathbf{v})>\mu(Q)$, hence the induced
map is zero, which implies that $E\rightarrow Q$ vanishes.

We now have $\mu(Q)=\mu(E^{\prime})$. Discriminant-minimality (E2) implies
that $\Delta(Q)\geq\Delta(E^{\prime})$. If $\Delta(Q)>\Delta(E^{\prime})$, we
get a contradiction as in the preceding paragraph.

Hence $\mu(Q)=\mu(E^{\prime})$ and $\Delta(Q)=\Delta(E^{\prime})$. By (E3),
$E^{\prime},Q$ are stable. Since $E$ is a non-split extension,
$E^{\prime}\rightarrow Q$ is zero, inducing a map $E^{\prime\prime}\rightarrow
Q$. Now by (D1), (D2) this map can be non-zero only if
$\mu(\mathbf{v}^{\prime\prime})=\mu(\mathbf{v})=\mu(\mathbf{v}^{\prime})$ and
$\Delta(\mathbf{v}^{\prime\prime})=\Delta(\mathbf{v})=\Delta(\mathbf{v}^{\prime})$.
Hence $E$ is semi-stable. ∎

######  Corollary 3.4.

Let $\mathbf{v}\in K(\mathbb{P}^{2})$ be a stable character which admits an
extremal decomposition
$(\mathbf{v}^{\prime},\mathbf{v},\mathbf{v}^{\prime\prime})$, and let
$\mathscr{E}^{\prime}/S^{\prime}$ and
$\mathscr{E}^{\prime\prime}/S^{\prime\prime}$ be families of semistable
sheaves of characters $\mathbf{v}^{\prime}$ and $\mathbf{v}^{\prime\prime}$
respectively. Then the induced extension parametrization

${\mathbb{P}\text{Ext}^{1}(E^{\prime\prime}_{s^{\prime\prime}},E^{\prime}_{s^{\prime}})}$${M(\mathbf{v})}$${S^{\prime}\times
S^{\prime\prime}}$

defined over the locus where
$\text{Ext}^{1}(E^{\prime\prime}_{s^{\prime\prime}},E^{\prime}_{s^{\prime}})$
is of constant rank, exists, where $s^{\prime}\in S^{\prime}$ and
$s^{\prime\prime}\in S^{\prime\prime}$.

###### Proof.

By Proposition 3.3, for any $s^{\prime}\in S^{\prime}$ and
$s^{\prime\prime}\in S^{\prime\prime}$, a nonsplit extension

$0\rightarrow E^{\prime}_{s^{\prime}}\rightarrow E\rightarrow
E^{\prime\prime}_{s^{\prime\prime}}\rightarrow 0$

is semistable of Chern character $\mathbf{v}$. This produces a rational map
$\mathbb{P}\dashrightarrow M(\mathbf{v})$, where $\mathbb{P}$ is the
projective bundle over $S^{\prime}\times S^{\prime\prime}$ with fiber
$\mathbb{P}\text{Ext}^{1}(E^{\prime\prime}_{s^{\prime\prime}},E^{\prime}_{s^{\prime}})$
over $(s^{\prime},s^{\prime\prime})\in S^{\prime}\times S^{\prime\prime}$. ∎

When the subsheaf is not extremal for the extension sheaf, in general
additional conditions are required to prove generic stability, see for
instance Lemma 4.5 and Lemma 5.8.

##  4\. Emptiness & non-emptiness of Brill-Noether loci

For a stable Chern character $\mathbf{v}\in K(\mathbb{P}^{2})$ there are only
finitely many values of $k\geq 0$ such that $B^{k}(\mathbf{v})\subseteq
M(\mathbf{v})$ is nonempty. That is, there are only finitely many values of
$h^{0}(E)$ for a stable sheaf $E\in M(\mathbf{v})$. We first give a bound on
this quantity. Later on in this section we will analyze which such values are
achieved.

######  Theorem 4.1.

If $E\in M(\mathbf{v})$ is a semistable sheaf, then

$h^{0}(E)\leq\max\\{p(\text{ch}_{1}(\mathbf{v})),\text{ch}_{0}(\mathbf{v})\\},$

where $p(x)=p_{\mathcal{O}_{\mathbb{P}^{2}}}(x)=\frac{x^{2}+3x+2}{2}$.

In the proof we will consider partial evaluation maps
$V\otimes\mathcal{O}\rightarrow E$ for a subspace $V\subseteq H^{0}(E)$. For
fixed $V$, let $F\subseteq E$ be the image sheaf of the evaluation map, and
let $0=F_{0}\subseteq F_{1}\subseteq\cdots\subseteq F_{m}=F$ be the Harder-
Narasimhan filtration of $F$, with graded pieces
$\text{gr}_{i}=F_{i}/F_{i-1}$.

We claim that $c_{1}(\text{gr}_{i})<c_{1}(E)$ for each $i$ when $r(F)<r(E)$.
If not, then since $r(\text{gr}_{i})\leq r(F)<r(E)$,

$\mu(E)=\frac{c_{1}(E)}{r(E)}\leq\frac{c_{1}(\text{gr}_{i})}{r(E)}<\frac{c_{1}(\text{gr}_{i})}{r(\text{gr}_{i})}=\mu(\text{gr}_{i}).$

The inclusions $\text{gr}_{1}=F_{1}\hookrightarrow F\hookrightarrow E$ and
semistability of $E$ imply

$\mu(\text{gr}_{1})\leq\mu(E)<\mu(\text{gr}_{i})$

contradicting that the slopes of the $\text{gr}_{i}$ are non-increasing. We
conclude that $c_{1}(\text{gr}_{i})<c_{1}(E)$ for each $i$.

We observe for use in the cases below that the function $p(x)/x$ is increasing
for integers $x\geq 2$, with $p(1)=p(2)/2$.

###### Proof.

First, since we have $E\subseteq E^{\vee\vee}$, we have $h^{0}(E)\leq
h^{0}(E^{\vee\vee})$. Because $r(E^{\vee\vee})=r(E)$ and
$c_{1}(E^{\vee\vee})=c_{1}(E)$, it suffices to prove the theorem when $E$ is
locally free.

Case I. In this case we assume $r(E)=\text{ch}_{0}(\mathbf{v})\geq
p(\text{ch}_{1}(\mathbf{v}))$, and proceed by induction on $r(E)$. In this
case when $r(E)=1$ we have $c_{1}(E)\leq 0$, and when $c_{1}(E)<0$ stability
implies $h^{0}(E)=0$, as desired. When $c_{1}(E)=0$, it follows that $E$ is
the ideal sheaf of a finite length subscheme $Z\subseteq\mathbb{P}^{2}$. Thus
$h^{0}(E)\leq 1$, and equality holds if and only if $Z$ is empty.

We first show that any $r(E)$-dimensional subspace $V\subseteq H^{0}(E)$ spans
a full rank subsheaf of $E$. That is, the evaluation map
$V\otimes\mathcal{O}\rightarrow E$ is full rank. Suppose not, so that
$r(F)<r(E)$. To show this we first claim that each $\text{gr}_{i}$ is in this
case as well, so by induction on the rank we have $h^{0}(\text{gr}_{i})\leq
r(\text{gr}_{i})$. It follows that

$h^{0}(F)\leq\sum h^{0}(\text{gr}_{i})\leq\sum r(\text{gr}_{i})=r(F)<r(E).$

Since by definition $h^{0}(F)\geq r(E)$, we conclude from this contradiction
that any partial evaluation map of any $r(E)$ independent sections in this
case is of full rank.

To check the claim, assume otherwise that
$r(\text{gr}_{i})<p(c_{1}(\text{gr}_{i}))$. It is enough to produce a
contradiction for those $\text{gr}_{i}$ with $c_{1}(\text{gr}_{i})>0$, since
when $c_{1}(\text{gr}_{i})\leq 0$ we have $h^{0}(\text{gr}_{i})\leq
r(\text{gr}_{i})$. Then

$\mu(E)=\frac{c_{1}(E)}{r(E)}\leq\frac{c_{1}(E)}{p(c_{1}(E))}\leq\frac{c_{1}(\text{gr}_{i})}{p(c_{1}(\text{gr}_{i}))}<\frac{c_{1}(\text{gr}_{i})}{r(\text{gr}_{i})}=\mu(\text{gr}_{i}),$

since $x/p(x)$ is non-increasing when $x\geq 1$. The slopes
$\mu(\text{gr}_{i})$ in the Harder-Narasimhan filtration are decreasing and
$E$ is semistable, so we have

$\mu(\text{gr}_{i})\leq\mu(\text{gr}_{1})\leq\mu(E)$

so this contradiction proves the claim. We conclude that any $r(E)$ sections
span a full rank subsheaf of $E$.

Therefore an $r(E)$-dimensional subspace $V\subseteq H^{0}(E)$ determines a
nonzero section of $H^{0}(\mathcal{O}(c_{1}(E))$ via the natural map

$\wedge^{r(E)}H^{0}(E)\rightarrow
H^{0}(\wedge^{r(E)}E)=H^{0}(\mathcal{O}(c_{1}(E))),$

where we have $\wedge^{r}E=\mathcal{O}(c_{1}(E))$ because $E$ is locally free.
In this way we obtain a morphism

$f:G(r(E),H^{0}(E))\rightarrow\mathbb{P}H^{0}(\mathcal{O}(c_{1}(E))).$

When $h^{0}(E)\geq r(E)+1$, it is easy to see that $f$ is nonconsant. Indeed,
any $r(E)$-dimensional subspace $V\subseteq H^{0}(E)$ sits in a short exact
sequence

$0\rightarrow V\otimes\mathcal{O}\rightarrow E\rightarrow Q\rightarrow 0$

because the evaluation map is of full rank. Choose $s\in
H^{0}(E)\smallsetminus V$, and consider $p\in\text{Supp}(Q)$ a general point.
Then $0\neq s_{p}\in H^{0}(Q_{p})$ does not lie in $V_{p}\subseteq E_{p}$.
Choosing a generating set $s_{1},...,s_{r(E)}$ for the span $\langle
s_{p},V_{p}\rangle\subseteq E_{p}$; then the $s_{i}$ span a rank $r(E)$
subspace $V^{\prime}\subseteq H^{0}(E)$ such that the induced section
$\wedge^{r(E)}V^{\prime}\in H^{0}(\wedge^{r(E)}E)$ is independent from
$\wedge^{r(E)}V$. For instance, the multiplicity of $\wedge^{r(E)}V^{\prime}$
at $p$ is lower than that of $\wedge^{r(E)}V$.

However if $h^{0}(E)\geq r(E)+1$, we have

$\dim G(r(E),H^{0}(E))\geq r(E)\geq p(c_{1}(E))$

since $E$ has $r(E)\geq p(c_{1}(E))$, and
$\dim\mathbb{P}H^{0}(\mathcal{O}(c_{1}(E)))=p(c_{1}(E))-1$. Since the Picard
rank of a Grassmannian is 1, there is no non-constant map from a Grassmannian
to a lower-dimensional projective variety. We conclude that $h^{0}(E)\leq
r(E)$, as desired.

Case II. We are now in the situation that
$r(E)<p(c_{1}(E))=\frac{c_{1}(E)^{2}+3c_{1}(E)+2}{2}$, and we want to prove
$h^{0}(E)\leq p(c_{1}(E))$. We will use induction on $c_{1}(E)$. If
$c_{1}(E)=0$, then $p(c_{1}(E))=1$ and there is nothing to prove.

We first show that any $p(c_{1}(E))$-dimensional subspace $V\subseteq
H^{0}(E)$ spans a full rank subsheaf of $E$. Since $p(x)/x$ is non-decreasing,
when $c_{1}(\text{gr}_{i})\geq 1$ we have

$p(c_{1}(\text{gr}_{i}))=c_{1}(\text{gr}_{i})\frac{p(c_{1}(\text{gr}_{i}))}{c_{1}(\text{gr}_{i})}\leq
c_{1}(\text{gr}_{i})\frac{p(c_{1}(E))}{c_{1}(E)}\leq
p(c_{1}(E))\frac{r(\text{gr}_{i})}{r(E)}.$

The $\text{gr}_{i}$ may fall into either Case I or Case II. When
$\text{gr}_{i}$ is in Case II, we obtain

$h^{0}(\text{gr}_{i})\leq p(c_{1}(\text{gr}_{i}))\leq
p(c_{1}(E))\frac{r(\text{gr}_{i})}{r(E)}$

by induction on $c_{1}$. When $c_{1}(\text{gr}_{i})<1$, we have

$h^{0}(\text{gr}_{i})\leq
r(\text{gr}_{i})=r(E)\frac{r(\text{gr}_{i})}{r(E)}\leq
p(c_{1}(E))\frac{r(\text{gr}_{i})}{r(E)}$

by the assumption that $E$ is in Case II. For those $\text{gr}_{i}$ in Case I
with $r(\text{gr}_{i})\geq p(c_{1}(\text{gr}_{i}))$, we also have

$h^{0}(\text{gr}_{i})\leq
r(\text{gr}_{i})=r(E)\frac{r(\text{gr}_{i})}{r(E)}<p(c_{1}(E))\frac{r(\text{gr}_{i})}{r(E)}.$

We now have

(2) $h^{0}(F)\leq\sum h^{0}(\text{gr}_{i})\leq\sum
p(c_{1}(E))\frac{r(\text{gr}_{i})}{r(E)}=p(c_{1}(E))\frac{r(F)}{r(E)}<p(c_{1}(E)).$

By definition we have $h^{0}(F)\geq p(c_{1}(E))$, and this contradiction
proves that any $p(c_{1}(E))$-dimensional subspace $V$ spans a full rank
subsheaf of $E$.

We can additionally assume that the dependency locus of $V$ is zero-
dimensional. If not, consider the image $F\subseteq E$ of the evaluation map
$V\otimes\mathcal{O}\rightarrow E$ and its Harder-Narasimhan filtration. Then
$c_{1}(F)<c_{1}(E)$. By (2), we know that $h^{0}(F)\leq p(c_{1}(E))$ (only the
final inequality in (2) is no longer strict), and equality holds only if for
every $i$, $h^{0}(\text{gr}_{i})=p(c_{1}(E))\frac{r(\text{gr}_{i})}{r(E)}$ for
every graded piece. This is true only if $c_{1}(\text{gr}_{i})>0$,
$r(\text{gr}_{i})<p(c_{1}(\text{gr}_{i}))$ and
$c_{1}(\text{gr}_{i})=1,c_{1}(E)=2$. In this case, $c_{1}(F)<c_{1}(E)=2$,
$c_{1}(F)\leq 1$. If $F$ is unstable, then there is at least one Harder-
Narasimhan factor $\text{gr}_{i}$ whose degree is non-positive, which is not
the case. If $F$ is stable, it satisfies $r(F)<p(c_{1}(F))$, then by
induction, $h^{0}(F)\leq p(c_{1}(F))<p(c_{1}(E))$. In any case, we have
$h^{0}(F)<p(c_{1}(E))$, a contradiction.

We now return to the bound on $h^{0}(E)$. We first prove $h^{0}(E)\leq
p(c_{1}(E))$ when $0\leq\mu(E)<1$. We have just shown that any $p(c_{1}(E))$
sections of $E$ span a full rank subsheaf, and their dependency locus is zero-
dimensional. Let $L\subseteq\mathbb{P}^{2}$ be a general line. Consider

$0\rightarrow E(-1)\rightarrow E\rightarrow E|_{L}\rightarrow 0$

and write $E|_{L}=\bigoplus_{i=1}^{r(E)}\mathcal{O}_{L}(d_{i})$. Since
$\mu(E)<1$, there is at least one $d_{i}\leq 0$, say $d_{1}$. Taking $H^{0}$
we obtain
$\pi:H^{0}(E)\rightarrow\bigoplus_{i=1}^{r(E)}H^{0}(\mathcal{O}_{L}(d_{i}))$;
let $W\subseteq\bigoplus_{i=1}^{r(E)}H^{0}(\mathcal{O}_{L}(d_{i}))$ be the
image of $\pi$. If
$W\subseteq\bigoplus_{i=2}^{r(E)}H^{0}(\mathcal{O}_{L}(d_{i}))$, then sections
of $E$ are dependent along $L$, hence $h^{0}(E)<p(c_{1})$. If
$W\not\subseteq\bigoplus_{i=2}^{r(E)}H^{0}(\mathcal{O}_{L}(d_{i}))$, then
$d_{1}=0$, and since
$\pi^{-1}(W\cap\bigoplus_{i=2}^{r(E)}H^{0}(\mathcal{O}_{L}(d_{i})))$ has
codimension 1, there are $h^{0}(E)-1$ sections dependent along $L$, hence
$h^{0}(E)-1<p(c_{1})$. That is, $h^{0}(E)\leq p(c_{1}(E))$, as desired.

Now if $n\leq\mu<n+1$, we induct on $n$. We have just shown the statement is
true for $n=0$; now assume $n\geq 1$. Again we use that any $p(c_{1}(E))$
sections span a full rank subsheaf, and the dependency locus is supported in
dimension zero. Consider again the restriction to a line:

$0\rightarrow E(-1)\rightarrow E\rightarrow E|_{L}\rightarrow 0.$

Taking $H^{0}$ we get $h^{0}(E)\leq h^{0}(E(-1))+h^{0}(E|_{L})$. By induction,
$h^{0}(E(-1))\leq p(c_{1}(E)-r(E))$. Consider
$E|_{L}=\bigoplus_{i=1}^{r(E)}\mathcal{O}_{L}(d_{i})$. If some $d_{i}<0$, say
$d_{1}$, then the span of sections in $W$ is contained in $\bigoplus_{i\geq
2}\mathcal{O}_{L}(d_{i})$, and it follows that $h^{0}(E)<p(c_{1}(E))$. Hence
we assume $d_{i}\geq 0$ for all $i$. Then
$h^{0}(E|_{L})=\sum_{i=1}^{r(E)}h^{0}(\mathcal{O}_{L}(d_{i}+1))=c_{1}(E)+r(E)$.
Now since $n\geq 1$, $c_{1}(E)\geq r(E)\geq 1$ and we obtain

$\displaystyle r(E)(2c_{1}(E)-r(E)+1)-2c_{1}(E)$
$\displaystyle=(r(E)-1)2c_{1}(E)-r(E)^{2}+r(E)$
$\displaystyle\geq(r(E)-1)2r(E)-r(E)^{2}+r(E)$ $\displaystyle=r(E)(r(E)-1)\geq
0$

hence $r(E)(2c_{1}(E)-r(E)+1)\geq 2c_{1}(E)$, and

$\displaystyle p(c_{1}(E))-p(c_{1}(E)-r(E))$
$\displaystyle=\frac{2c_{1}(E)r(E)-r(E)^{2}+3r(E)}{2}$
$\displaystyle=r(E)+\frac{r(E)(2c_{1}(E)-r(E)+1)}{2}$ $\displaystyle\geq
r(E)+c_{1}(E).$

In particular we have

$h^{0}(E)\leq h^{0}(E(-1))+h^{0}(E|_{L})\leq
p(c_{1}(E)-r(E))+(r(E)+c_{1}(E))\leq p(c_{1}(E)).$

This finishes the proof. ∎

We will later need the following corollary, which is an immediate consequence
of the proposition.

######  Corollary 4.2.

If $E$ is a semistable sheaf on $\mathbb{P}^{2}$ with $c_{1}(E)=1$ and
$r(E)\geq 3$, then $h^{0}(E)\leq r(E)$.

######  Remark 4.3.

We can describe the behavior of $E$ with $c_{1}(E)=1$ also when the rank is
small. If the rank is 1, then $E$ is a twist of an ideal sheaf $E\simeq
I_{Z}(1)$. Then $h^{0}(E)\neq 0$ if and only if $Z$ is colinear, in which case

  * •

$h^{0}(E)=1$ if $\ell(Z)\geq 2$,

  * •

$h^{0}(E)=2$ if $\ell(Z)=1$, and

  * •

$h^{0}(E)=3$ if $\ell(Z)=0$.

If the rank is 2, suppose $h^{0}(E)\geq 3$ and consider the partial evaluation
map $V\otimes\mathcal{O}\rightarrow E$ for $V\subseteq H^{0}(E)$ a rank 3
subspace. Remark 4.4 implies that the image sheaf $F$ of this map is
semistable of slope $1/2$; this induces a short exact sequence

$0\rightarrow K\rightarrow V\otimes\mathcal{O}\rightarrow F\rightarrow 0$

with $r(K)=1$, $c_{1}(K)=-1$. It follows that $K\simeq I_{Z}(-1)$ for some
$Z\subseteq\mathbb{P}^{2}$ a codimension two subscheme. In particular,
$\text{ch}_{2}(K)=\frac{1}{2}-\ell(Z)$. Thus

$\Delta(F)=\frac{1}{2}\left(\frac{1}{2}\right)^{2}-\frac{\ell(Z)-1/2}{2}=\frac{3}{8}-\frac{\ell(Z)}{2}.$

Semistability additionally implies that $\Delta(F)=\frac{3}{8}+\frac{k}{2}$
for some $k\geq 0$ (see Remark 2.1), from which we find $k=-\ell(Z)\leq 0$, so
$k=0$ and $F\simeq E=T_{\mathbb{P}^{2}}(-1)$. In all other cases, we again
have $h^{0}(E)\leq r(E)$.

######  Remark 4.4.

Let $E$ be a sheaf on a variety $X$ and $F$ the image and $K$ the kernel of
the evaluation map on global sections for $E$. Then $h^{0}(K)=0$, as can be
seen by taking cohomology of the short exact sequence

$0\rightarrow K\rightarrow
H^{0}(E)\otimes\mathcal{O}\stackrel{{\scriptstyle\text{ev}_{E}}}{{\rightarrow}}F\rightarrow
0$

and noting that the induced map on global sections
$H^{0}(\text{ev}_{E}):H^{0}(E)\rightarrow H^{0}(F)$ is an isomorphism.

###  4.1. The Brill-Noether loci $B^{r}(\mathbf{v})$

In this section we show that $B^{k}(\mathbf{v})\neq\emptyset$ whenever
$k\leq\text{ch}_{0}(\mathbf{v})$. We can say a great deal about the loci of
sheaves $E$ with $h^{0}(E)\geq r=r(E)$, i.e., the Brill-Noether loci
$B^{r}(\mathbf{v})$. We will need the following statements to study the Brill-
Noether loci when $\text{ch}_{0}(\mathbf{v})\geq
p(\text{ch}_{1}(\mathbf{v}))$.

######  Lemma 4.5.

Let $\mathbf{v}\in K(\mathbb{P}^{2})$ be a stable Chern character with
$\text{ch}_{0}(\mathbf{v})\geq p(\text{ch}_{1}(\mathbf{v}))$. Let $E\in
B^{r}(\mathbf{v})$ be a general sheaf.

  1. (1)

The evaluation map on global sections $\text{ev}_{E}$ sits in an exact
sequence

$0\rightarrow\mathcal{O}^{r}\rightarrow E\rightarrow Q_{E}\rightarrow 0,$

where $Q_{E}$ is a pure sheaf supported on a 1-dimensional subscheme of
$\mathbb{P}^{2}$.

  2. (2)

If $Q_{E}$ is supported on a smooth curve $C$ of degree $c_{1}(E)$, then
$Q_{E}$ is the pushforward of a line bundle on $C$. Writing
$Q_{E}\simeq\mathcal{O}_{C}(D)$, we have

$\deg_{C}D\leq\frac{c_{1}(E)^{2}+3c_{1}(E)}{2}-r(E)\leq-1.$

  3. (3)

Conversely, let $\mathcal{O}_{C}(D)$ be the pushforward of a line bundle on a
smooth curve $C\subseteq\mathbb{P}^{2}$, where
$\deg_{C}D\leq\frac{1}{2}(\deg(C)^{2}+3\deg(C))-r\leq-1$. Then a general
extension sheaf

$0\rightarrow\mathcal{O}^{r}\rightarrow
E\rightarrow\mathcal{O}_{C}(D)\rightarrow 0$

is stable.

###### Proof.

(1) First, the proof of Theorem 4.1, Case I, implies that the evaluation map
has full rank. It follows that $Q_{E}$ is a torsion sheaf. Consider a
sufficiently negative twist

$0\rightarrow\mathcal{O}(-m)^{r}\rightarrow E(-m)\rightarrow
Q_{E}(-m)\rightarrow 0$

of the above exact sequence, where $m\gg 0$. If $Q_{E}$ is has torsion
$T\subseteq Q_{E}$ supported in codimension two, then $T(-m)\simeq T$, so in
particular $T(-m)$ and $Q_{E}(-m)$ have global sections. The associated
cohomology exact sequence

$H^{0}(E(-m))\rightarrow H^{0}(Q_{E}(-m))\rightarrow
H^{1}(\mathcal{O}(-m)^{r})$

has outer terms which vanish and an inner term which does not. This
contradiction proves that $Q_{E}$ is supported in pure codimension one.

(2) There is a filtration $0=F_{k}Q_{E}\subseteq
F_{k-1}Q_{E}\subseteq\cdots\subseteq F_{1}Q_{E}\subseteq Q_{E}$, where each
graded piece is the pushforward of a sheaf on $C$ [Dré08]. If $F_{i}Q_{E}\neq
0$ then its support is $C$, otherwise it would be a torsion subsheaf of
$Q_{E}$ supported on zero dimensional scheme, contradicting the purity of
$Q_{E}$ from (1). Now if $F_{1}Q_{E}\neq 0$, then $c_{1}(F_{1}Q_{E})\geq
c_{1}(E)$, but $Q_{E}/F_{1}Q_{E}$ has support $C$, whose first Chern class is
$\geq c_{1}(E)$, hence $c_{1}(Q_{E})\geq 2c_{1}(E)$, a contradiction.

For the degree, note that since $r(E)\geq p(c_{1}(E))$, we have

$\mu(E)\leq\frac{c_{1}(E)}{p(c_{1}(E))}\leq 1/3$

since $x/p(x)\leq 1/3$ for integers $x$. Thus $E$ lies directly above the
${}^{\perp}\mathcal{O}$-branch of the Drézet-Le Potier curve, which gives
$\chi(E,\mathcal{O})\leq 0$. Thus

$\chi(\mathcal{O}_{C}(D),\mathcal{O}_{\mathbb{P}^{2}})=\chi(E,\mathcal{O}_{\mathbb{P}^{2}})-\chi(\mathcal{O}_{\mathbb{P}^{2}}^{r},\mathcal{O}_{\mathbb{P}^{2}})\leq-r.$

On the other hand,

$\chi(\mathcal{O}_{C}(D),\mathcal{O}_{\mathbb{P}^{2}})=\chi(\mathcal{O}_{C},\mathcal{O}_{C}(D-3H))=\deg_{C}(D-3H)+1-g(C)$

so

$\deg_{C}D-3c_{1}(E)+1-\frac{c_{1}(E)^{2}-3c_{1}(E)+2}{2}\leq-r$

i.e., $\deg_{C}D\leq\frac{1}{2}(c_{1}(E)^{2}+3c_{1}(E))-r$. The last
inequality follows from the assumption $r(E)\geq p(c_{1}(E))$.

(3) Let $F$ be a possible destabilizing semistable quotient sheaf of $E$. Now
since $F$ is torsion-free, there is no nonzero map from
$\mathcal{O}_{C}(D)\rightarrow F$, hence the composition
$\mathcal{O}^{r}\rightarrow E\rightarrow F$ is nonzero. The sheaves
$\mathcal{O}^{r}$ and $F$ are both semistable, which gives $\mu(F)\geq 0$. On
the other hand, since $F$ has smaller rank than $E$, $c_{1}(F)<c_{1}(E)$,
hence the assumption on $\deg(D)$ implies $F$ also has $r(F)\geq p(c_{1}(F))$,
and the quotient $\mathcal{O}^{r}\rightarrow F$ is full rank. But
$h^{0}(F)\leq r(F)$, hence $\mathcal{O}^{r}\rightarrow F$ gives only $r(F)$
sections, i.e., it factors through $\mathcal{O}^{r(F)}\rightarrow F$. By part
(1), we have a short exact sequence

$0\rightarrow\mathcal{O}^{r(F)}\rightarrow F\rightarrow Q_{F}\rightarrow 0$

where $Q_{F}$ is supported on a curve $C^{\prime}$, and $\deg C^{\prime}\leq
c_{1}(F)<c_{1}(E)=\deg C$. Now we have a diagram

${0}$${\mathcal{O}^{r}}$${E}$${\mathcal{O}_{C}(D)}$${0}$${0}$${\mathcal{O}^{r(F)}}$${F}$${Q_{F}}$${0}$

where the vertical maps are surjective. The map $\mathcal{O}_{C}(D)\rightarrow
Q_{F}$ induces an inclusion $\text{Supp}Q_{F}\subseteq C$. $\text{Supp}Q_{F}$
has strictly smaller degree. Since $C$ is irreducible, this is impossible. We
conclude $Q_{F}=0$, so $F$ is a slope 0 semistable sheaf with $r(F)$ sections.
Hence $F=\mathcal{O}^{r(F)}$, and the exact sequence defining $E$ is partially
split.

We count dimensions to show such extensions form a closed locus in the
associated extension space. If $\deg
D\leq\frac{1}{2}(\deg(C)^{2}+3\deg(C))-r$, then
$\text{ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}_{\mathbb{P}^{2}})\geq r$. If
$E=\mathcal{O}_{\mathbb{P}^{2}}^{q}\oplus E^{\prime}$ where
$E^{\prime}\in\text{Ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}_{\mathbb{P}^{2}}^{r-q})$,
then the dimension of the locus of such
$E\in\text{Ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}_{\mathbb{P}^{2}}^{r})$ is
no larger than

$\dim
G(q,r)+\text{ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}^{r-q})=q(r-q)+(r-q)e,$

and $\text{ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}_{\mathbb{P}^{2}}^{r})=re$.
Now $q(r-q)+(r-q)e-re=(r-e-q)q$, so we see that when $e\geq r$ and $q>0$,

$\dim
G(q,r)+\text{ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}_{\mathbb{P}^{2}}^{r-q})<\text{ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}_{\mathbb{P}^{2}}^{r}).$

By openness of stability, the lemma follows. ∎

######  Theorem 4.6.

Suppose that $\mathbf{v}\in K(\mathbb{P}^{2})$ is stable and
$\text{ch}_{0}(\mathbf{v})\geq p(\text{ch}_{1}(\mathbf{v}))$,
$\text{ch}_{1}(\mathbf{v})>0$. Then $B^{r}(\mathbf{v})$ is nonempty and
contains a component of the expected dimension.

###### Proof.

By Lemma 4.5 (2), we get a morphism
$\psi:B^{r}(\mathbf{v})\rightarrow\mathbb{P}H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(\text{ch}_{1}(\mathbf{v})))=\mathbb{P}V$,
by sending $E\in M(\mathbf{v})$ to $s_{1}\wedge\cdots\wedge
s_{r}\in\mathbb{P}V$, and by Lemma 4.5 (3), $\psi$ is surjective. Denoting by
$U=\mathbb{P}V\smallsetminus\Gamma$ the locus of smooth curves, we see that
$\psi$ factors through the relative Picard scheme
$\text{Pic}^{d}_{\mathcal{C}/U}$ over the universal curve
$\mathcal{C}\rightarrow U$, where $d=\deg(D)$. The induced map
$\phi:B^{r}(\mathbf{v})\rightarrow\text{Pic}^{d}_{\mathcal{C}/U}$ is
surjective, and since $E$ uniquely determines $Q_{E}=\mathcal{O}_{C}(D)$ its
fibers are quotients of open subsets of the extension spaces
$\text{Ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}_{\mathbb{P}^{2}}^{r})$. It
follows in particular that $\psi^{-1}(U)\subseteq B^{r}(\mathbf{v})$ is
irreducible and open, so its closure forms a nonempty irreducible component.

We now compute its dimension. Write $r=\text{ch}_{0}(\mathbf{v})$,
$c_{1}=\text{ch}_{1}(\mathbf{v})$, $\mu=c_{1}/r$, and
$\Delta(\mathbf{v})=\Delta_{0}+k/r$ where $\Delta_{0}$ is the minimal
discriminant of a stable sheaf with $\mu=\mu(\mathbf{v})$ and
$r=\text{ch}_{0}(\mathbf{v})$. Note in particular that $\mathbf{v}$ lies
directly above the ${}^{\perp}\mathcal{O}$-branch of the Drézet-Le Potier
curve, which is given by

$\Delta=p(-\mu)=\frac{1}{2}\mu^{2}-\frac{3}{2}\mu+1.$

In particular, the characters $(\mu,\delta(\mu))$ are integral, so
$\delta(\mu)=\Delta_{0}$ for these slopes. Then $M(\mathbf{v})$ is irreducible
of dimension

$\dim
M(\mathbf{v})=r^{2}(2(\Delta_{0}+k/r)-1)+1=1+c_{1}^{2}-3c_{1}r+r^{2}+2kr.$

The expected codimension of the Brill-Noether locus $B^{m}(\mathbf{v})$ is
$m(m-\chi(\mathbf{v}))$. Now

$\chi(\mathbf{v})=r\left(p\left(\mu\right)-\Delta_{0}-k/r\right)=3c_{1}-k,$

so the expected codimension of $B^{r}(\mathbf{v})$ is $r(r-3c_{1}+k)$, and its
expected dimension is

$\text{expdim}B^{r}(\mathbf{v})=(1+c_{1}^{2}-3c_{1}r+r^{2}+2kr)-r(r-3c_{1}+k)=1+c_{1}^{2}+kr.$

The dimension of the locus of interest is equal to
$\dim\phi^{-1}\text{Pic}^{d}(\mathcal{C}/U)$. The general
$E\in\text{Ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}^{r})$ admits precisely one
map $\mathcal{O}^{r}\rightarrow E$, and automorphisms of $\mathcal{O}^{r}$
determine distinct extension classes but isomorphic extension sheaves. Thus

$\dim\phi^{-1}\text{Pic}^{d}_{\mathcal{C}/U}=\dim
U+g(C)+\text{ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}^{r})-r^{2}+1.$

We have
$\text{ext}^{2}(\mathcal{O}_{C}(D),\mathcal{O})=h^{0}(\mathcal{O}_{C}(D-3H))=0$,
and clearly $\hom(\mathcal{O}_{C}(D),\mathcal{O})=0$, so
$\text{ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O})=-\chi(\mathcal{O}_{C}(D),\mathcal{O})$.
From the exact sequence

$0\rightarrow\mathcal{O}^{r}\rightarrow
E\rightarrow\mathcal{O}_{C}(D)\rightarrow 0$

we obtain

$\chi(\mathcal{O}_{C}(D),\mathcal{O})=\chi(E,\mathcal{O})-\chi(\mathcal{O}^{r},\mathcal{O})=-k-r.$

Hence

$\dim\phi^{-1}(\mathcal{O}_{C}(D))=r(k+r)-r^{2}=kr.$

As $\dim
U=\dim\mathbb{P}H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(c_{1}))=\frac{1}{2}(c_{1}^{2}+3c_{1})$,
we have

$\dim\text{Pic}^{d}_{\mathcal{C}/U}=\dim
U+g(C)=(c_{1}^{2}+3c_{1})/2+(c_{1}^{2}-3c_{1}+2)/2=c_{1}^{2}+1,$

hence

$\dim\psi^{-1}(U)=c_{1}^{2}+1+kr,$

which equals the expected dimension. ∎

When we have instead $\text{ch}_{0}(\mathbf{v})<p(\text{ch}_{1}(\mathbf{v}))$,
we prove a weaker statement: that the Brill-Noether locus $B^{r}(\mathbf{v})$
is nonempty.

Note that the condition $\chi(\mathbf{v})=\text{ch}_{0}(\mathbf{v})$ can be
expressed via Riemann-Roch as the polynomial condition

$\Delta(\mathbf{v})=p(\mu(\mathbf{v}))-1=\frac{1}{2}\mu(\mathbf{v})^{2}+\frac{3}{2}\mu(\mathbf{v}).$

This defines a parabola $\xi_{r}$ in the $(\mu,\Delta)$-plane. ($\xi_{r}$ does
not depend on the rank.)

In the region of the $(\mu,\Delta)$-plane with $\mu>0$, the parabola $\xi_{r}$
meets the Drézet-Le Potier curve only at the ${}^{\perp}\mathcal{O}$ branch,
at the point $(\mu,\Delta)=(1/3,5/9)$. The Chern characters $\mathbf{v}\in
K(\mathbb{P}^{2})$ lying on $\xi_{r}$ satisfy
$\chi(\mathbf{v})=\text{ch}_{0}(\mathbf{v})$, and the general sheaf $E\in
M(\mathbf{v})$ has $\chi(E)=r(E)$.

######  Theorem 4.7.

Let $\mathbf{v}\in K(\mathbb{P}^{2})$ be a stable Chern character with
$\mu(\mathbf{v})>0$. The Brill-Noether locus $B^{r}(\mathbf{v})$ is nonempty.

###### Proof.

When $r=1$, $c_{1}\geq 1$, we may take $I_{Z}(c_{1})$ where $Z$ lies on a
curve of degree $c_{1}$. When $c_{1}=1$, by Theorem 4.6 the theorem is true
for $r\geq 3$. Hence in the following we assume $r>1$, $c_{1}>1$ or
$c_{1}=1,r=2$.

We separate into two cases. In the first case we assume $0<\mu(\mathbf{v})\leq
1/3$; in particular, $\mathbf{v}$ lies above the
${}^{\perp}\mathcal{O}$-branch of the Drézet-Le Potier curve. We induct on the
rank for a stronger statement: there exists $E\in M(\mathbf{v})$ such that
$h^{0}(E)=\text{ch}_{0}(E)$. When the rank is 1 or 2 the statement is vacuous.
When $r=3$, $c_{1}=1$, by Corollary 4.2 the theorem is true. Let
$\mathbf{v}^{\prime}$ be the associated extremal character, and
$\mathbf{v}^{\prime\prime}=\mathbf{v}-\mathbf{v}^{\prime}$ the quotient
character. Then $0\leq\mu(\mathbf{v}^{\prime})\leq\mu(\mathbf{v})\leq 1/3$.
$\mu(\mathbf{v}^{\prime})=0$ if and only if $c_{1}=1$, where the only
possiblity for the theorem to be false is $r=2$, which is not considered in
this case, hence $0<\mu(\mathbf{v}^{\prime})\leq 1/3$.

The same is true for $\mathbf{v}^{\prime\prime}$; perhaps the easiest way to
see this is to use an extraneous set of coordinates. On the
$(\text{ch}_{0},\text{ch}_{1})$-plane, we plot the point corresponding to
$\mathbf{v}$ and the point
$\mathbf{w}=(\text{ch}_{0}(\mathbf{v})-3,\text{ch}_{1}(\mathbf{v})-1)$; the
slope of the segment between them is 1/3. The fact that
$\frac{\text{ch}_{1}(\mathbf{v})-1}{\text{ch}_{0}(\mathbf{v})}\leq\mu(\mathbf{v}^{\prime})\leq\mu(\mathbf{v})$
and $\text{ch}_{1}(\mathbf{v}^{\prime})\leq\text{ch}_{1}(\mathbf{v})-1$ imply
that the point corresponding to $\mathbf{v}^{\prime}$ does not lie outside the
triangle whose vertices are $\mathbf{v}$, $\mathbf{w}$, and the origin. The
slope of the segment between $\mathbf{v}$ and $\mathbf{v}^{\prime}$ is
$\mu(\mathbf{v}^{\prime\prime})$, and this is no larger than the slope of the
segment between $\mathbf{v}$ and $\mathbf{w}$, which is 1/3. See Figure 2(a).

$\mathbf{v}$$\mathbf{w}$$\mathbf{v}^{\prime}$$\text{ch}_{0}$$\text{ch}_{1}$
(a) $\mathbf{v}$ and $\mathbf{v}^{\prime}$ in the
$(\text{ch}_{0},\text{ch}_{1})$-plane.

(b) The regions verifying $\xi_{r}(\mu)-\delta(\mu)>1/r$.

Figure 2. Figures for Theorem 4.7.

An immediate consequence is that the extremal decomposition exists in this
case. Denote the Drézet-Le Potier curve by $\Delta=\delta(\mu)$. If
$\mu(\mathbf{v}^{\prime})<\mu(\mathbf{v})$, then
$\Delta(\mathbf{v}^{\prime})\leq\delta(\mu(\mathbf{v}^{\prime}))$ since
$(r(\mathbf{v}^{\prime}),\mu(\mathbf{v}^{\prime}),\delta(\mu(\mathbf{v}^{\prime})))$
is integral ($\mathbf{v}^{\prime}$ is either on the Drézet-Le Potier curve or
semi-exceptional). Then $\chi(\mathbf{v}^{\prime},\mathcal{O})\geq 0$,
$\chi(\mathbf{v},\mathcal{O})\leq 0$, hence
$\chi(\mathbf{v}^{\prime\prime},\mathcal{O})=\chi(\mathbf{v},\mathcal{O})-\chi(\mathbf{v}^{\prime},\mathcal{O})\leq
0$, so $\mathbf{v}^{\prime\prime}$ lies on or above the Drézet-Le Potier curve
since $0<\mu(\mathbf{v}^{\prime\prime})\leq 1/3$. If
$\mu(\mathbf{v}^{\prime})=\mu(\mathbf{v})$, we may write
$\mathbf{v}=(mr_{0},\mu,\delta(\mu)+\frac{k}{mr})$ where
$\text{ch}_{1}(\mathbf{v})=md_{0}$, $\mu=d/r_{0}$, and $r_{0},d_{0}$ are
coprime. Let $\mathbf{v}^{\prime}_{0}:=(r_{0},\mu,\delta(\mu)+\frac{k}{r})$,
$\mathbf{v}^{\prime\prime}_{0}:=((m-1)r_{0},\mu,\delta(\mu))$, one may check
that $\mathbf{v}^{\prime}_{0}$ and $\mathbf{v}^{\prime\prime}_{0}$ are indeed
integral. Then
$\mathbf{v}=\mathbf{v}^{\prime}_{0}+\mathbf{v}^{\prime\prime}_{0}$. Now
clearly $\Delta(\mathbf{v}^{\prime}_{0})\geq\Delta(\mathbf{v})$, hence by
definition of extremal character,
$\Delta(\mathbf{v}^{\prime})\leq\Delta(\mathbf{v}^{\prime}_{0})$, hence
$\Delta(\mathbf{v}^{\prime\prime})\geq\Delta(\mathbf{v}^{\prime\prime}_{0})=\delta(\mu)$,
so $\mathbf{v}^{\prime\prime}$ also lies on or above the Drézet-Le Potier
curve.

By assumption $r\geq 2$, both $\mathbf{v}^{\prime}$ and
$\mathbf{v}^{\prime\prime}$ have smaller rank than $\mathbf{v}$, thus by
induction we can assume that there are $E^{\prime}\in M(\mathbf{v}^{\prime})$
and $E^{\prime\prime}\in M(\mathbf{v}^{\prime\prime})$ satisfying
$h^{0}(E^{\prime})=r(E^{\prime})$ and
$h^{0}(E^{\prime\prime})=r(E^{\prime\prime})$, and by Proposition 3.3 any
nontrivial extension sheaf

$0\rightarrow E^{\prime}\rightarrow E\rightarrow E^{\prime\prime}\rightarrow
0$

is semistable. To show $h^{0}(E)=r(E)=r(E^{\prime})+r(E^{\prime\prime})$, it
is enough to show that we can find such an extension such that the associated
connecting homomorphism $H^{0}(E^{\prime\prime})\rightarrow H^{1}(E^{\prime})$
vanishes.

Indeed, the association $e\mapsto\delta_{e}$ determines a linear map

$\text{Ext}^{1}(E^{\prime\prime},E^{\prime})\rightarrow\text{Hom}(H^{0}(E^{\prime\prime}),H^{1}(E^{\prime})).$

We need to show that it has nonzero kernel; in fact, the dimension of the
source exceeds the dimension of the target. By stability and Serre duality,
$\text{ext}^{2}(E^{\prime\prime},E^{\prime})=\hom(E^{\prime},E^{\prime\prime}(-3))=0$,
so

$\text{ext}^{1}(E^{\prime\prime},E^{\prime})\geq-\chi(E^{\prime\prime},E^{\prime})=-r^{\prime}r^{\prime\prime}(p(\mu^{\prime}-\mu^{\prime\prime})-\Delta^{\prime}-\Delta^{\prime\prime}),$

and we know already that $h^{0}(E^{\prime\prime})=r^{\prime\prime}$, and

$h^{1}(E^{\prime})=h^{0}(E^{\prime})-\chi(E^{\prime})=r^{\prime}-r^{\prime}(p(\mu^{\prime})-\Delta^{\prime}).$

Since $\mathbf{v}^{\prime\prime}$ lies above the
${}^{\perp}\mathcal{O}$-branch of the Drézet-Le Potier curve,
$\chi(E^{\prime\prime},\mathcal{O})\leq 0$. It follows in particular from
Riemann-Roch that $\Delta^{\prime\prime}\geq p(-\mu^{\prime\prime})$. We then
have:

$\displaystyle\text{ext}^{1}(E^{\prime\prime},E)-\hom(H^{0}(E^{\prime\prime}),H^{1}(E^{\prime}))$
$\displaystyle\geq-r^{\prime}r^{\prime\prime}(p(\mu^{\prime}-\mu^{\prime\prime})-\Delta^{\prime}-\Delta^{\prime\prime})$
$\displaystyle\qquad-r^{\prime\prime}(r^{\prime}-r^{\prime}(p(\mu^{\prime})-\Delta^{\prime}))$
$\displaystyle=-r^{\prime}r^{\prime\prime}(p(\mu^{\prime}-\mu^{\prime\prime})-p(\mu^{\prime})-\Delta^{\prime\prime}+1)$
$\displaystyle\geq-r^{\prime}r^{\prime\prime}(p(\mu^{\prime}-\mu^{\prime\prime})-p(\mu^{\prime})-p(-\mu^{\prime\prime})+1)$
$\displaystyle=r^{\prime}r^{\prime\prime}\mu^{\prime}\mu^{\prime\prime}=c_{1}^{\prime}c_{1}^{\prime\prime}>0$

as desired. We conclude there is $E\in M(\mathbf{v})$ with

$h^{0}(E)=h^{0}(E^{\prime})+h^{0}(E^{\prime\prime})=r(E^{\prime})+r(E^{\prime\prime})=r(E),$

i.e., $B^{r}(\mathbf{v})\neq\emptyset$.

In the second case, we assume $\mu(\mathbf{v})>1/3$. Let $\mathbf{v}\in
K(\mathbb{P}^{2})$ be a stable Chern character of slope $\mu=\mu(\mathbf{v})$
and rank $r=\text{ch}_{0}(\mathbf{v})$, and let $\Delta_{0}$ be the minimum
discriminant of a stable character of slope $\mu$ and rank $r$. Then we may
write $\Delta(\mathbf{v})=\Delta(\mathbf{v}_{0})+k/r$ for some $k\geq 0$. For
every rational number $\mu^{\prime}=c_{1}^{\prime}/r^{\prime}$, the character
$\mathbf{v}^{\prime}=(r^{\prime},\mu^{\prime},\xi_{r}(\mu))$ is integral since
$\chi(\mathbf{v}^{\prime})=r^{\prime}$ is an integer.

We claim that the minimal-discriminant character
$\mathbf{v}_{0}=(\mu(\mathbf{v}),\Delta_{0})$ lies below the parabola
$\xi_{r}$ in the $(\mu,\Delta)$-plane, so $\chi(\mathbf{v}_{0})>r$. To check
this we will show that $\xi_{r}(\mu)-\delta(\mu)>1/r$, so that any character
with discriminant larger than $\xi_{r}(\mu)$ is obtained from one below
$\xi_{r}$ by an elementary modification, so is not minimal. We divide the
$(\mu,\Delta)$-plane into smaller regions, see Figure 2(b): first, it is
enough to check this for $1/3<\mu<1/2$, since the Drézet-Le Potier curve is
symmetric along the line $\mu=1/2$. For slopes $\mu$ lying directly above the
${}^{\perp}\mathcal{O}$-branch of the Drézet-Le Potier curve, one can easily
check that the points $(\mu,\delta(\mu))$ are integral, so the claim is true
here. This branch terminates at $\mu=(3-\sqrt{5})/2$ (see [CHW17, §2.3]); the
branch $T_{\mathbb{P}^{2}}(-1)^{\perp}$ to the left of the line $\mu=1/2$
terminates at $\mu=1/2-(3-\sqrt{8})/2$. In this region we have the coarse
bounds

$\displaystyle\xi_{r}(\mu)-\delta(\mu)$
$\displaystyle\geq\xi_{r}(\mu)-(\text{peak of DLP curve above }E_{2/5})$
$\displaystyle=\xi_{r}(\mu)-\frac{13}{25}\geq\xi_{r}\left(\frac{3-\sqrt{5}}{2}\right)-\frac{13}{25}\approx.12589$

Thus if $\xi_{r}(\mu)-\delta(\mu)\leq 1/r$ then $r<9$, and the few slopes
$\mu=c_{1}/r$ with $r<9$ in this region can be easily checked to have minimal
discriminants lying below the parabola $\xi_{r}$.

Now for slopes $\mu$ lying directly above the
$T_{\mathbb{P}^{2}}(-1)^{\perp}$-branch, we have

$\xi_{r}(\mu)-\delta(\mu)=\xi_{r}(\mu)-\left(\frac{1}{2}\left(\mu-\frac{1}{2}\right)^{2}+\frac{3}{2}\left(\mu-\frac{1}{2}\right)+1\right)=\frac{\mu}{2}$

and $\mu/2\leq 1/r$ only when $c_{1}\leq 2$. All statements are clear when
$c_{1}=1$, and when $r\geq 5$, $\mu=2/r$ is to the left of this region, and
for smaller ranks the minimal discriminant is easily computed to be below
$\xi_{r}(\mu)$.

By generic slope-stability ([DLP85, Corollaire 4.12]), the general member of
$M(\mathbf{v}_{0})$ is slope-stable. Choose such a $E_{0}\in
M(\mathbf{v}_{0})$ with $r$ independent sections $s_{1},...,s_{r}\in
H^{0}(E_{0})$; choose $Z$ a codimension two subscheme of length $k$. Then the
elementary modification $E$ defined by the short exact sequence

$0\rightarrow E\rightarrow E_{0}\rightarrow\mathcal{O}_{Z}\rightarrow 0$

has Chern character $\mathbf{v}$. Since $E_{0}$ is slope-stable, so is $E$
(see Remark 2.1). Furthermore we can choose the map
$E_{0}\rightarrow\mathcal{O}_{Z}$ such that the map $H^{0}(E_{0})\rightarrow
H^{0}(\mathcal{O}_{Z})$ vanishes, since the dependency locus of
$s_{1},\cdots,s_{r}$ has dimension at least 1. So $h^{0}(E)\geq r$ for all $E$
constructed in this way. ∎

##  5\. Irreducibility & reducibility of Brill-Noether loci

In this section we address the reducibility of the Brill-Noether loci
$B^{k}(\mathbf{v})$. We show that when $\text{ch}_{1}(\mathbf{v})=1$, all
Brill-Noether loci are irreducible and of the expected dimension. When
$\text{ch}_{1}(\mathbf{v})>1$, we give examples where the Brill-Noether loci
are reducible, and describe their components.

######  Theorem 5.1.

Suppose $\mathbf{v}\in K(\mathbb{P}^{2})$ is a stable Chern character with
$\text{ch}_{1}(\mathbf{v})=1$. Then all of the Brill-Noether loci
$B^{k}(\mathbf{v})$ are irreducible and of the expected dimension.

Each Brill-Noether locus $B^{k}(\mathbf{v})$ is the quotient of a projective
bundle over some $M(\mathbf{w})$ where $\text{ch}_{1}(\mathbf{w})=1$ as well,
by a free action of a projective linear group.

###### Proof.

Choose $E\in M(\mathbf{v})$ and set $r=r(E)=\text{ch}_{0}(\mathbf{v})$.
Consider the image sheaf $0\neq F\subseteq E$ of the evaluation map on global
sections for $E$. The proof of Theorem 4.1, Case I implies that when $r\geq
3$, $h^{0}(E)\leq r$, and that the evaluation map in this case is a sheaf-
theoretic embedding. (In rank 2, see Remark 4.3. The only sheaf whose
evaluation map on global sections is not an embedding is
$T_{\mathbb{P}^{2}}(-1)$, and here the statements are vacuous.) Thus every
$E\in M(\mathbf{v})$ is given by an extension

(3) $0\rightarrow\mathcal{O}^{r-s}\rightarrow E\rightarrow
E^{\prime}\rightarrow 0$

where $h^{0}(E)=r-s$, $s\geq 0$, and $E^{\prime}$ has
$c_{1}(E^{\prime})=c_{1}(E)=1$ and $r(E^{\prime})=s$. In fact when $s\geq 0$
all such $E^{\prime}$ are semistable, and when $s>0$, torsion-free. To check
torsion-freeness, consider the torsion part $T$ of $E^{\prime}$, which sits in
an exact sequence

$0\rightarrow T\rightarrow E^{\prime}\rightarrow E^{\prime}/T\rightarrow 0.$

Then arguing as in the proof of Lemma 4.5 (1), we see that $T$ is supported in
dimension 1. Then $c_{1}(E^{\prime}/T)=c_{1}(E^{\prime})-c_{1}(T)\leq 0$, and
the quotient $E\rightarrow E^{\prime}/T$ destabilizes $E$. Now for stability,
if $s\geq 1$ then $c_{1}(E^{\prime})$ and $r(E^{\prime})=s$ are coprime, any
destablizing quotient $Q$ of $E^{\prime}$ would have $\mu(Q)<\mu(E^{\prime})$,
but since $c_{1}(E^{\prime})=1$, we have $\mu(Q)\leq 0$, so $Q$ would be a
destablizing quotient of $E$. When $s=0$, by Lemma 4.5 (2), $E^{\prime}$ is of
the form $\mathcal{O}_{L}(a)$ for some $a\in\mathbb{Z}$ and
$L\subseteq\mathbb{P}^{2}$ a line, hence stable. Set
$\mathbf{w}=\text{ch}(E^{\prime})=\text{ch}(E)-(r-s)\text{ch}(\mathcal{O})$.

We form extension parametrizations $\phi_{r-s}:\mathbb{P}_{r-s}\dashrightarrow
M(\mathbf{v})$ whose image lies in $B_{r-s-\chi(\mathbf{v})}(\mathbf{v})$ as
follows. The expected extensions (3) are determined by classes in
$\text{Ext}^{1}(E^{\prime},\mathcal{O}^{r-s})$. Since
$\text{Hom}(E^{\prime},\mathcal{O}^{r-s})=\text{Ext}^{2}(E^{\prime},\mathcal{O}^{r-s})=0$,
$\text{ext}^{1}(E^{\prime},\mathcal{O}^{r-s})=-\chi(E^{\prime},\mathcal{O}^{r-s})$
is constant, so the extension parametrization is defined over the whole moduli
space $M(\mathbf{w})$. $\mathbb{P}_{r-s}$ maps rationally to
$B_{r-s-\chi(\mathbf{v})}(\mathbf{v})\subseteq M(\mathbf{v})$.

When $s>0$, Proposition 3.3 implies that for any $F^{\prime}\in M(\mathbf{w})$
any nonzero extension
$0\neq[F]\in\text{Ext}^{1}(F^{\prime},\mathcal{O}^{h^{0}(E)})$ is stable. When
$s=0$, Lemma 4.5 (3) implies the same statement. In particular, each of the
extension parametrizations $\phi_{k}:\mathbb{P}_{k}\rightarrow M(\mathbf{v})$
is a morphism. Since each $E^{\prime}$ as above is stable, it follows that
$\phi_{k}$ surjects onto $B^{k}(\mathbf{v})$. The moduli spaces
$M(\mathbf{v}^{\prime})$ are irreducible, so each $\mathbb{P}_{k}$ is as well.
It follows that so is each $B^{k}(\mathbf{v})$. We now show that they are of
the expected dimension.

We will need to know the other numerical invariants of $E$ and $E^{\prime}$;
they are determined as follows. Set
$\text{ch}_{2}(E)=a-1/2=\text{ch}_{2}(E^{\prime})$ (see Remark 5.2). This
gives

$\Delta(E)=\frac{1}{2r^{2}}-\frac{a}{r}+\frac{1}{2r},\quad\chi(E)=r(p(1/r)-\Delta(E))=r+a+1.$

Similarly,

$\Delta(E^{\prime})=\frac{1}{2s^{2}}-\frac{a}{s}+\frac{1}{2s}.$

The dimension of $\mathbb{P}$ is

$\dim\phi(\mathbb{P})=\dim
M(\mathbf{w})+\text{ext}^{1}(E^{\prime},\mathcal{O}^{r-s})-1.$

The latter quantity is, by Serre duality, Theorem 2.2, and Riemann-Roch,

$\displaystyle\text{ext}^{1}(E^{\prime},\mathcal{O}^{r-s})$
$\displaystyle=(r-s)\cdot h^{1}(E^{\prime}(-3))$
$\displaystyle=-(r-s)\cdot\chi(E^{\prime}(-3))$ $\displaystyle=-(r-s)\cdot
s(P(1/s-3)-\Delta(E^{\prime}))$ $\displaystyle=-(r-s)\cdot(s+a-2).$

The dimensions of the moduli spaces are

$\dim M(\mathbf{v})=r^{2}(2\Delta(E)-1)+1=-r^{2}+(1-2a)r+2$

and

$\dim M(\mathbf{w})=s^{2}(2\Delta(E^{\prime})-1)+1=-s^{2}+(1-2a)s+2.$

The fibers of $\phi$ are of dimension
$\text{Aut}(\mathcal{O}^{r-s})=(r-s)^{2}-1$, so the dimension of the image of
$\phi$ is

$\displaystyle\dim\phi(\mathbb{P})$
$\displaystyle=s^{2}(2\Delta(E^{\prime})-1)+1-(r-s)\cdot(s+a-2)-(r-s)^{2}+1$
$\displaystyle=-s^{2}+(1-2a)s+2-(r-s)\cdot(s+a-2)-1-(r-s)^{2}+1$
$\displaystyle=-r^{2}+(s-a+2)r-s^{2}-(a+1)s+2.$

This is the actual dimension of the Brill-Noether locus; we now check that the
expected dimension of the Brill-Noether loci equals the actual dimension.

Set $m=r-s-\chi(\mathbf{v})$; the expected codimension of the Brill-Noether
locus $B_{m}(\mathbf{v})$ is

$\displaystyle\text{expcodim}_{m}(\mathbf{v})$ $\displaystyle=(r-s)\cdot m$
$\displaystyle=-(r-s)\cdot(r-s-\chi(\mathbf{v}))$
$\displaystyle=(s-r)\cdot(s-a-1)$ $\displaystyle=s^{2}+(1-r+a)s-ar-r.$

The expected dimension is then

$\displaystyle\text{expdim}_{m}(\mathbf{v})$ $\displaystyle=\dim
M(\mathbf{v})-(s^{2}+(1-r+a)s-ar-r)$
$\displaystyle=-r^{2}+(1-2a)r+2-s^{2}-(1-r+a)s+ar+r$
$\displaystyle=-r^{2}+(s-a+2)r-s^{2}-(a+1)s+2$

as required.

For the latter statement, we consider the map $\mathbb{P}_{k}\rightarrow
B^{k}(\mathbf{v})$. By Lemma 6.3 in [CH18], all fibers of this map are
$\mathbb{P}GL_{k}$. ∎

######  Remark 5.2.

When $h^{0}(E)=r$, the associated extension is of the form

$0\rightarrow\mathcal{O}^{r}\rightarrow
E\rightarrow\mathcal{O}_{L}(a)\rightarrow 0$

for $L\subseteq\mathbb{P}^{2}$ a line and $a\leq 0$. In this case
$\text{ch}_{2}(E)=a-1/2$. We regard $a$ as an invariant of $E$ for other
values of $h^{0}(E)$ via this formula.

###  5.1. Reducible Brill-Noether loci

When $\text{ch}_{1}(\mathbf{v})>1$, the Brill-Noether loci can be reducible.

When $\text{ch}_{0}(\mathbf{v})=1$, we consider twists of ideal sheaves
$I_{Z}(d)$, where $d>0$ and $Z\subseteq\mathbb{P}^{2}$ is a subscheme of
finite length. These moduli spaces are isomorphic to the Hilbert scheme
$\mathbb{P}^{2[n]}$, but their cohomological properties depend on the twist.
We often write $\mathbb{P}^{2[n]}(d)$ for these moduli spaces.

######  Proposition 5.3.

Let $\mathbf{v}=(1,d,d^{2}/2-n)$ be the Chern character of a twist $I_{Z}(d)$
of an ideal sheaf of a scheme of length $n$. If $n>d^{2}$, the Brill-Noether
loci $B^{2}(\mathbf{v})$ are reducible, and have at least $d-1$ components.

###### Proof.

Choose $Z$ such that $h^{0}(I_{Z}(d))\geq 2$, and let
$C_{1},C_{2}\subseteq\mathbb{P}^{2}$ be curves of degree $d$ containing $Z$.
If $C_{1}\cap C_{2}$ were zero-dimensional it would have length
$d^{2}<\ell(Z)$, so in particular it could not contain $Z$. We conclude that
$C_{1}\cap C_{2}$ has one-dimensional components. The general such $Z$ lies in
$B^{2}\smallsetminus B^{3}$ so the pencil spanned by $C_{1}$ and $C_{2}$ is
determined. In this case we call $C_{Z}$ the union of the one-dimensional
components of $C_{1}\cap C_{2}$; the general such $Z$ will have $C_{Z}$
irreducible.

For any $0<e<d$, we examine the following loci in $B^{2}$:

$B^{2}_{e}:=\overline{\\{Z:C_{Z}\text{ is irreducible with
}\deg(C_{Z})=e\\}}.$

Clearly $B^{2}$ is the union of the $B^{2}_{e}$ as $e$ ranges between 1 and
$d-1$; we claim that each lies in a distinct irreducible component of $B^{2}$.

Indeed, the association of a general element $I_{Z}(d)\in B^{2}$ to the pencil
of reducible degree $d$ curves containing it induces a rational map
$\phi:B^{2}\dashrightarrow G(2,p(d))$, and the pencils themselves are lines on
a Segre variety
$\Sigma_{d-k,k}\subseteq\mathbb{P}H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(d))$
lying in a fiber of one of its projections
$\Sigma_{d-k,k}\rightarrow\mathbb{P}H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(d-k)),\mathbb{P}H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(k))$.
One can check that each such line (given by fixing a smooth curve of degree
$d-k$ and a pencil of curves of degree $k$) lies in the image of $\phi$.
Denote by $P_{d-k,k}$ the loci of such lines in $G(2,p(d))$.

As $k$ varies between 1 and $d-1$, there are no containments among the Segre
varieties $\Sigma_{d-k,k}$ in
$\mathbb{P}H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(d))$, and the line classes in
the fibers of the projections form irreducible components of the image of
$\phi$. Each of the loci $B^{2}_{e}$ dominates $P_{e,d-e}$, so the claim
follows. ∎

$L_{1}$$L_{2}$$R$$Z$$C_{1}$$C_{2}$$L$$Z$

Figure 3. General members of the loci $B_{2}^{1}$ and $B_{2}^{2}$ on
$\mathbb{P}^{2[10]}(3)$.

We will use reducibility in rank 1 to form reducible Brill-Noether loci in
higher rank via the Serre construction.

######  Theorem 5.4 ([HL10], 5.1.1).

Let $X$ be a smooth surface, $L,M\in\text{Pic}(X)$ be line bundles on $X$, and
$Z\subseteq X$ a local complete intersection subscheme of codimension 2. Then
there is a locally free extension

$0\rightarrow L\rightarrow E\rightarrow M\otimes I_{Z}\rightarrow 0$

(and hence the general such extension is locally free), if and only if $Z$
satisfies the Cayley-Bacharach property with respect to $L^{\vee}MK_{X}$: for
any $Z^{\prime}\subseteq Z$ of length $\ell(Z)-1$, the map

$H^{0}(L^{\vee}MK_{X}\otimes I_{Z})\rightarrow H^{0}(L^{\vee}MK_{X}\otimes
I_{Z^{\prime}})$

is surjective.

Assume that $Z\subseteq\mathbb{P}^{2}$ satisfies the Cayley-Bacharach property
for $\mathcal{O}_{\mathbb{P}^{2}}(b-a-3)$. Then the general extension

$0\rightarrow\mathcal{O}(a)\rightarrow E\rightarrow I_{Z}(b)\rightarrow 0$

is locally free. To check that the general such $E$ is slope stable, it is
enough to check that there are no maps
$\mathcal{O}_{\mathbb{P}^{2}}(d)\rightarrow E$ with $d>\mu(E)=(a+b)/2$.
Indeed, any destabilizing subbundle $S\subseteq E$ has rank 1, and since $E$
is locally free, this inclusion induces $S^{\vee\vee}\subseteq
E^{\vee\vee}\simeq E$, and $S^{\vee\vee}$ is locally free of rank 1, i.e., a
line bundle.

######  Example 5.5.

The extensions we consider are in the above form, with $(a,b)=(0,3)$ and
$\ell(Z)=10$:

$0\rightarrow\mathcal{O}\rightarrow E\rightarrow I_{Z}(3)\rightarrow 0.$

Here $\chi(E)=h^{0}(E)=1$ for generic choice of $Z$. Set
$\mathbf{v}=\text{ch}(E)$. Note that the Cayley-Bacharach condition is
trivially satisfied, as the map in question is

$H^{0}(I_{Z})=0\rightarrow 0=H^{0}(I_{Z^{\prime}}).$

Stability is also easy to check. We need to rule out the existence of maps
$\mathcal{O}(d)\rightarrow E$ where $d\geq 2$. Any such map induces a nonzero
map to from $\mathcal{O}(d)$ to either $\mathcal{O}$ or $I_{Z}(3)$, but either
such map vanishes by stability when we choose $Z$ noncollinear. Thus the
general extension $E$ is both locally free and slope-stable. We construct
multiple components in $B^{3}(\mathbf{v})$ by considering the components of
$B^{2}(I_{Z}(d))$.

The Brill-Noether locus $B_{2}$ on the twisted Hilbert scheme
$\mathbb{P}^{2[10]}(3)$ has (at least) two components: $B^{2}_{1}$ and
$B^{2}_{2}$. The general member $Z\in B^{2}_{1}$ is contained in two cubics
$X$ and $Y$ each of which is the union of a fixed line $L$ with a conic
$C_{1}$ or $C_{2}$. The general $Z\in B^{2}_{2}$ is contained in two cubics
$U$ and $V$ each of which is the union of a fixed conic $R$ with a line
$L_{1}$ or $L_{2}$. See Figure 3.

We consider the following families of stable sheaves $E$: construct a
projective bundle $P_{1}$ over $B^{2}_{1}(I_{Z}(d))$ with fiber
$\mathbb{P}\text{Ext}^{1}(I_{Z}(3),\mathcal{O})$ for $Z\in B^{2}_{1}$. It is
generically a family of stable sheaves, so admits a map $P_{1}\dashrightarrow
M(\text{ch}(E))$; construct $P_{2}\dashrightarrow M(\mathbf{v})$ similarly.
One can check that the image closures $B_{1}$ and $B_{2}$ of $P_{1}$ and
$P_{2}$ have dimensions 23, above the expected dimension 22 for a component of
$B^{2}(\mathbf{v})$.

To conclude that $B^{3}(\mathbf{v})$ is reducible, we need to verify that
$B_{1}$ and $B_{2}$ are not nested, and there is no component of
$B_{3}(\mathbf{v})$ containing both. The first statement is straightforward,
following from the definitions of the loci. For the second, suppose
$V\subseteq B^{3}(\mathbf{v})$ is irreducible and contains both $B_{1}$ and
$B_{2}$. Then the general $E\in V$ has three global sections, and since the
general points of $B_{1}$ and $B_{2}$ have global sections
$\mathcal{O}\rightarrow F$ with torsion-free cokernel, so too does the general
point of $V$. Thus $E$ appears in an extension

$0\rightarrow\mathcal{O}\rightarrow E\rightarrow I_{Z}(3)\rightarrow 0$

with $\ell(Z)=10$. Since $E\in B^{3}(\mathbf{v})$, $h^{0}(I_{Z}(3))\geq 2$ so
$Z$ lies in $B^{2}_{1}$ or $B^{2}_{2}$. However a general point of $V$
specializing to a general point of $B_{1}$ or $B_{2}$ determines a
specialization of $B^{2}_{1}$ to $B_{2}^{2}$, or vice versa. To see this, let
$\mathcal{V}/V$ be a universal family, perhaps after shrinking $V$. Then we
claim there is a distinguished section $\mathcal{O}_{\mathbb{P}^{2}\times
V}\rightarrow\mathcal{V}$ whose cokernel $\mathcal{I}$ is generically a family
of ideal sheaves $I_{Z}(3)$, fitting in an exact sequence

$0\rightarrow\mathcal{O}_{\mathbb{P}^{2}\times
V}\rightarrow\mathcal{V}\rightarrow\mathcal{I}\rightarrow 0$

which on fibers recovers the sequences defining the loci $B_{1}$ and $B_{2}$.
In particular, it will follow that a specialization of a general point of $V$
to a general point of $B_{1}$ and to a general point of $B_{2}$ determines a
specialization of $B^{2}_{1}$ to $B^{2}_{2}$, contradicting Proposition 5.3.
To see this, let $E\in B_{1}$ be given, and choose a section $t\in H^{0}(E)$
fitting in an exact sequence

$0\rightarrow\mathcal{O}\stackrel{{\scriptstyle t}}{{\rightarrow}}E\rightarrow
I_{Z_{t}}(3)\rightarrow 0$

for $Z_{t}\subseteq\mathbb{P}^{2}$ a zero-dimensional subscheme of length 10
with $I_{Z_{t}}(3)\in B^{2}_{1}\subseteq\mathbb{P}^{2[10]}(3)$. Then choose
general $s_{1},s_{2}\in H^{0}(E)$ independent from $t$. The section $s_{1}$
determines another sequence

$0\rightarrow\mathcal{O}\stackrel{{\scriptstyle
s_{1}}}{{\rightarrow}}E\rightarrow I_{Z_{s_{1}}}(3)\rightarrow 0$

where $Z_{s_{1}}\subseteq\mathbb{P}^{2}$ is a zero-dimensional subscheme,
necessarily of length 10. We claim that $Z_{s_{1}}\in B^{2}_{1}$ as well. To
see this, consider the dependency loci $X(s_{1},t)$ and $X(s_{1},s_{2})$
where, respectively, $s_{1}$ and $t$ are dependent and where $s_{1}$ and
$s_{2}$ are dependent. Each is a degree 3 curve in $\mathbb{P}^{2}$, and
$Z_{s_{1}}\subseteq X(s_{1},t)\cap X(s_{1},s_{2})$. However since
$\ell(Z_{s_{1}})=10$, the intersection cannot be transverse, and the
dependency loci share a component. Since $Z_{t}\in B^{2}_{1}$, there is a line
in the shared loci containing $Z_{s_{1}}$. The claim is now proved, which
completes the example.

In fact in higher rank many Brill-Noether loci are reducible, as we now show.
The context for the following result is established in Section 4. Recall that
the parabola $\xi_{r}$ is the locus of characters $\mathbf{v}\in
K(\mathbb{P}^{2})$ satisfying $\chi(\mathbf{v})=\text{ch}_{0}(\mathbf{v})$,
thought of in the $(\mu,\Delta)$-plane, and we set $\xi_{r}(a)$ to be the
intersection of $\xi_{r}$ with the vertical line $\mu=a$.

######  Remark 5.6.

In [CH18, Definition 3.1] a somewhat different definition of the extremal
character to $\mathbf{v}$ is given. We will need to use this other definition
alongside ours in what follows; we set $\mathbf{v}^{\prime}_{CH}$ to be the
extremal character in the sense of Coskun-Huizenga in [CH18]. The character
$\mathbf{v}^{\prime}_{CH}$ is defined as in Definition 3.2 but replacing
axioms (D1) and (D2) as follows:

  1. (D1’)

$\text{ch}_{0}(\mathbf{v}^{\prime}_{CH})\leq\text{ch}_{0}(\mathbf{v})$ and if
$\text{ch}_{0}(\mathbf{v}^{\prime}_{CH})=\text{ch}_{0}(\mathbf{v})$ then
$\text{ch}_{1}(\mathbf{v})-\text{ch}_{1}(\mathbf{v}^{\prime}_{CH})>0$;

  2. (D2’)

$\mu(\mathbf{v}^{\prime}_{CH})<\mu(\mathbf{v})$.

It will follow from Lemma 5.10 that
$\mu(\mathbf{v}^{\prime}_{CH})\leq\mu(\mathbf{v}^{\prime})$. The variant
$\mathbf{v}_{CH}^{\prime}$ has the property that when
$\text{ch}_{0}(\mathbf{v})$ and $\text{ch}_{1}(\mathbf{v})$ are not coprime,
we still have $\mu(\mathbf{v}_{CH}^{\prime})<\mu(\mathbf{v})$.

######  Theorem 5.7.

Let $\mathbf{v}\in K(\mathbb{P}^{2})$ be a stable Chern character with
$\mu(\mathbf{v})>0$. Let $\mu_{CH}^{\prime}$ denote the slope of the extremal
character $\mathbf{v}_{CH}^{\prime}$ for $\mathbf{v}$. If
$\mu_{CH}^{\prime}>1/3$, $\Delta(\mathbf{v})\gg 0$ and
$\mu(\mathbf{v})\notin\mathbb{Z}$, then $B^{r}(\mathbf{v})$ is reducible.

  1. (1)

If $\Delta(\mathbf{v})\geq\xi_{r}(\mu(\mathbf{v}))$ and $\mu_{CH}^{\prime}\geq
1/3$, then $B^{r}(\mathbf{v})$ contains a component of the expected dimension.

  2. (2)

If $\Delta(\mathbf{v})\gg 0$ and $\mu_{CH}^{\prime}>1/3$, then
$B^{r}(\mathbf{v})$ contains a component of dimension larger than the expected
dimension.

Note that the assumptions in the theorem are implied by the assumptions in
Theorem 1.1 (4).

We first construct the families of sheaves necessary for the proof and compile
some statements about them. Our first family will consist of sheaves $E$
appearing in extensions determined by $r(E)$ global sections of $E$, and our
second family will consist of sheaves $F$ appearing in extension classes
determined by the extremal subsheaves $E^{\prime}$.

First, let $\text{Pic}_{\mathcal{C}/U}^{d}$ be the relative Picard scheme over
the universal curve $\mathcal{C}\rightarrow U$, where
$U\subseteq\mathbb{P}H^{0}\mathcal{O}_{\mathbb{P}^{2}}(c_{1})$ is the space of
smooth plane curves of degree $c_{1}$. We set
$g=\frac{1}{2}(c_{1}-1)(c_{1}-2)$. Consider the projective bundle $Y$ whose
fiber over $\mathcal{O}_{C}(D)$ is
$\mathbb{P}\text{Ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}^{r})$ defined over
the open subset of $\text{Pic}_{\mathcal{C}/U}^{d}$ where the dimension of
this group is constant. We form a family of sheaves $\mathcal{E}$ on
$\mathbb{P}^{2}$ parametrized by $Y$ by setting $E_{y}$, $y\in Y$, to be an
extension sheaf

$0\rightarrow\mathcal{O}^{r}\rightarrow
E_{y}\rightarrow\mathcal{O}_{C}(D)\rightarrow 0$

determined by $y=(\mathcal{O}_{C}(D),e)$ for
$\mathcal{O}_{C}(D)\in\text{Pic}^{d}_{\mathcal{C}/U}$ and
$e\in\mathbb{P}\text{Ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}^{r})$.

Recall that a sheaf $E$ on $\mathbb{P}^{2}$ is called prioritary if
$\text{Ext}^{2}(E,E(-1))=0$.

######  Lemma 5.8.

Let $\mathcal{E}$ be the family of sheaves just constructed. Then for $d=g-1$,

  1. (1)

$\mathcal{E}$ is complete;

  2. (2)

the general member of $\mathcal{E}$ is prioritary.

When $c_{1}/r\geq 1/3$, the general member is stable.

###### Proof.

(1) First, we have a natural identification

$T_{\mathcal{O}_{C}(D)}\text{Pic}^{d}_{\mathcal{C}/U}\simeq\text{Ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}_{C}(D)).$

Let $T_{y}$ be the tangent space to $Y$ at $y\in Y$, and let $E=E_{y}$ be the
associated sheaf. The Kodaira-Spencer map
$\kappa:T_{y}\rightarrow\text{Ext}^{1}(E,E)$ fits into the following diagram:

${0}$${\text{Ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}^{r})}$${T_{y}}$${\text{Ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}_{C}(D))}$${0}$${\text{Ext}^{1}(E,\mathcal{O}^{r})}$${\text{Ext}^{1}(E,E)}$${\text{Ext}^{1}(E,\mathcal{O}_{C}(D))}$$\scriptstyle{\kappa}$

The left-hand vertical map is obtained by applying
$\text{Hom}(-,\mathcal{O}^{r})$ to the sequence

$0\rightarrow\mathcal{O}^{r}\rightarrow
E\rightarrow\mathcal{O}_{C}(D)\rightarrow 0;$

since $\text{Ext}^{1}(\mathcal{O}^{r},\mathcal{O}^{r})=0$, it is surjective.
The right-hand vertical map is obtained by applying
$\text{Hom}(-,\mathcal{O}_{C}(D))$ to the same sequence. The next term in the
sequence is

$\text{Ext}^{1}(\mathcal{O}^{r},\mathcal{O}_{C}(D))\simeq
H^{1}(C,\mathcal{O}_{C}(D))^{r}$

whose dimension is

$\text{ext}^{1}(\mathcal{O}^{r},\mathcal{O}_{C}(D))=r\cdot
h^{1}(C,\mathcal{O}_{C}(D))=r\cdot h^{0}(C,\mathcal{O}_{C}(K_{C}-D).$

We have

$\deg(K_{C}-D)=2g-2-d=g-1.$

For general $D$, Riemann-Roch on $C$ then gives
$h^{0}(\mathcal{O}_{C}(K_{C}-D))=0$, so the right-hand vertical map in the
diagram is also surjective. The snake lemma implies that $\kappa$ is
surjective as well, i.e., $\mathcal{E}$ is complete.

(2) Apply $\text{Hom}(E,-)$ to the sequence

$0\rightarrow\mathcal{O}(-1)^{r}\rightarrow
E(-1)\rightarrow\mathcal{O}_{C}(D-H)\rightarrow 0$

to get a sequence

(4)
$\text{Ext}^{2}(E,\mathcal{O}(-1)^{r})\rightarrow\text{Ext}^{2}(E,E(-1))\rightarrow\text{Ext}^{2}(E,\mathcal{O}_{C}(D-H))\rightarrow
0.$

To show general $E$ is prioritary, it is enough to show that the outer two
terms generically vanish.

The first term is Serre dual to $\text{Hom}(\mathcal{O}(-1),E(-3))^{r}\simeq
H^{0}(E(-2))^{r}$. This space sits in an exact sequence

$0\rightarrow H^{0}(\mathcal{O}(-2)^{r})\rightarrow H^{0}(E(-2))\rightarrow
H^{0}(\mathcal{O}_{C}(D-2H))\rightarrow 0.$

The first term vanishes, and as above for general $D$,
$h^{0}(\mathcal{O}_{C}(D-2H))\leq h^{0}(\mathcal{O}_{C}(D))=0$. We conclude
the first term in the sequence (4) vanishes.

The last term in sequence (4) is Serre dual to
$\text{Hom}(\mathcal{O}_{C}(D-H),E(-3))$. To show this vanishes, it is enough
to show that generic $E$ appearing in $\mathcal{E}$ is torsion-free. If
$T\subseteq E$ is the torsion part of $E$, write $E^{\prime}=E/T$. The induced
map $T\rightarrow\mathcal{O}_{C}(D)$ is nonzero, so we obtain a diagram

${0}$${0}$${K}$${T}$${\mathcal{O}_{C}(D)}$${Q}$${\mathcal{O}^{r}}$${E}$${\mathcal{O}_{C}(D)}$${0}$${E^{\prime}}$${E^{\prime}}$${0}$${0}$${0}$${0}$

The snake lemma produces an exact sequence

$0\rightarrow K\rightarrow\mathcal{O}^{r}\rightarrow E^{\prime}\rightarrow
Q\rightarrow 0.$

Since $\mathcal{O}^{r}$ is torsion-free, $K=0$. Arguing in similar fashion to
the proof of Lemma 4.5 (1), we see that the inclusion
$T\hookrightarrow\mathcal{O}_{C}(D)$ cannot have a nonzero cokernel, which
would be supported in codimension two. Thus $T\simeq\mathcal{O}_{C}(D)$, but
then the map $T\rightarrow E$ splits the sequence defining $E$. This
contradiction implies that $E$ is torsion-free, as required. We conclude that
the general member of the family $\mathcal{E}$ is prioritary.

Now $\mathbf{v}(E)$ lies on the curve $\xi_{r}$. When $\mu(E)\geq 1/3$,
$\mathbf{v}(E)$ lies on or above the Drézet-Le Potier curve. To conclude that
the general member of $\mathcal{E}$ is also stable, one proceeds in entirely
the same way as in [LP97, §16.2]. Specifically, the stack of prioritary
sheaves is irreducible and its general member is stable. The family
$\mathcal{E}$ induces a map to the stack of prioritary sheaves, and
completeness implies the general member of $\mathcal{E}$ is a general member
of the stack of prioritary sheaves, so it is stable. ∎

######  Corollary 5.9.

A general member of $\mathcal{E}$ is slope-stable for $d\leq g-1$.

###### Proof.

It suffices to exhibit one slope-stable member. By the lemma, for $d=g-1$ a
general member of $\mathcal{E}$ is a general member of
$M(r,\frac{g-1}{r},\xi_{r}(\frac{g-1}{r}))$. Because the general member is
slope-stable ([DLP85, Corollaire 4.12]), any elementary modification of a
general member of $\mathcal{E}$ is again slope-stable. ∎

It follows that the family $\mathcal{E}$ determines a map $Y\dashrightarrow
M(\mathbf{v})$, and we set $Z_{1}\subseteq M(\mathbf{v})$ to be the closure of
its image.

For the second family, we first assume $\gcd(r,c_{1})=1$. Let
$\mathbf{v}^{\prime}$ be the extremal character associated to $\mathbf{v}$ in
the sense of Definition 3.2. Because
$\mu(\mathbf{v}^{\prime}_{CH})\leq\mu(\mathbf{v}^{\prime})$, we have
$\mu(\mathbf{v}^{\prime})>1/3$. Setting
$\mathbf{v}^{\prime\prime}=\mathbf{v}-\mathbf{v}^{\prime}$ and letting
$\Delta(\mathbf{v})\gg 0$, we may choose stable $E^{\prime\prime}\in
M(\mathbf{v}^{\prime\prime})$ and semistable $E^{\prime}\in
M(\mathbf{v}^{\prime})$ and form extensions

$0\rightarrow E^{\prime}\rightarrow E\rightarrow E^{\prime\prime}\rightarrow
0.$

Then $\text{ch}(E)=\mathbf{v}$ and $E$ is generically stable (see Proposition
3.3). When $r(E)$ and $c_{1}(E)$ are coprime,
$\mu(\mathbf{v}^{\prime})<\mu(\mathbf{v})$, and $\mathbf{v}^{\prime}$ has
minimal discriminant. If $r(E)$ and $c_{1}(E)$ are not coprime, our
construction will differ slightly. Because $\mu(\mathbf{v}^{\prime})>1/3$, we
have $\Delta(\mathbf{v}^{\prime})<\xi_{r}(\mu(\mathbf{v}^{\prime}))$, i.e.,
$\chi(E^{\prime})>r^{\prime}$ (see the proof of Theorem 4.7). Write
$\chi(E^{\prime})=r^{\prime}+\epsilon^{\prime}$, and let
$\epsilon:=\min\\{\epsilon^{\prime},r^{\prime\prime}\\}$. By Definition 3.2,
$r^{\prime\prime}>0$, hence $\epsilon>0$. We want to examine the Brill-Noether
locus $B^{r^{\prime\prime}-\epsilon}(\mathbf{v}^{\prime\prime})$, i.e., the
locus of sheaves $E^{\prime\prime}\in M(\mathbf{v}^{\prime\prime})$ with
$h^{0}(E^{\prime\prime})\geq r^{\prime\prime}-\epsilon$. Our second family
will consist of extension sheaves $E$ as above, where $E^{\prime}$ has general
cohomology $h^{0}(E^{\prime})=r^{\prime}+\epsilon^{\prime}$ and
$E^{\prime\prime}$ has potentially special cohomology
$h^{0}(E^{\prime\prime})=r^{\prime\prime}-\epsilon$. If we show that the
appropriate Brill-Noether locus is nonempty, we may construct a projective
bundle $\mathbb{P}$ over $M(\mathbf{v}^{\prime})\times
B^{r^{\prime\prime}-\epsilon}(\mathbf{v}^{\prime\prime})$ supporting the
family of sheaves $E$ as above. Then the family determines a map
$\mathbb{P}\dashrightarrow M(\mathbf{v})$; let $Z_{2}$ be the closure of the
image.

###### Proof of Theorem 5.7.

We have as above rational maps $Y\dashrightarrow Z_{1}$ and
$\mathbb{P}\dashrightarrow Z_{2}$ into $M(\mathbf{v})$. We will show that
$Z_{2}$ is well-defined (i.e., the Brill-Noether locus
$B^{r^{\prime\prime}-\epsilon}(\mathbf{v}^{\prime\prime})$ is nonempty), that
$Z_{1}$ is an irreducible component of $B^{r}(\mathbf{v})$, and
$\dim(Z_{2})>\dim(Z_{1})$.

We now consider the first family of sheaves. By construction, the fiber of the
map $Y\dashrightarrow Z_{1}$ has fiber $\text{Aut}(\mathcal{O}^{r})$. Recall
that we set $\Delta=\Delta_{0}+k/r$, where $\Delta_{0}$ is minimal and $k\geq
0$. We have

$\displaystyle\dim Z_{1}$ $\displaystyle=\dim
Y-\dim\text{Aut}(\mathcal{O}^{r})$
$\displaystyle=\dim\text{Pic}^{d}_{\mathcal{C}/U}+r\cdot\text{ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O})-r^{2}$
$\displaystyle=c_{1}^{2}+1+r(3c_{1}+k)-r^{2}$
$\displaystyle=rk+(\text{const}).$

Let $V\subseteq B^{r}(\mathbf{v})$ be an irreducible subvariety containing
$Z_{1}$, and let $E\in Z_{1}$ be general. Because $h^{0}(E)=r$ and the
evalutation map $\text{ev}_{E}$ for $E$ has full rank, we can shrink $V$ to
assume that each member of $V$ does as well. In particular, the map $\psi$
constructed in the proof of Theorem 4.6 carries $E$ to a smooth curve. It
follows that the general point of $V$ is as well, and the inverse image
$\psi|_{V}^{-1}(U)\subseteq V$ is an open dense subset; in particular,
$V\subseteq Z_{1}$. We conclude finally that $Z_{1}$ is an irreducible
component of $B^{r}(\mathbf{v})$.

We now consider the second family of sheaves. We first assume that
$\gcd(r(E),c_{1}(E))=1$. By Definition 3.2, we have $r^{\prime\prime}>0$. We
conclude from Theorem 4.7 that the Brill-Noether locus
$B^{r^{\prime\prime}}(\mathbf{v}^{\prime\prime})$ is nonempty, so in
particular the larger Brill-Noether locus
$B^{r^{\prime\prime}-\epsilon}(\mathbf{v}^{\prime\prime})$ is as well. If
$r^{\prime}=1$, then $E^{\prime}$ is a line bundle since it has minimal
discriminant, and $H^{1}(E^{\prime})=0$. If $r^{\prime}\geq 2$, by Theorem
2.2, $H^{1}(E)=0$ since by construction $E^{\prime}$ is chosen to be general.
So for generic $E$ as in the construction of $Z_{2}$ we have

$h^{0}(E)\geq(r^{\prime}+\epsilon^{\prime})+(r^{\prime\prime}-\epsilon)\geq
r(E),$

as desired.

We compute the dimension of $Z_{2}$. As usual, write
$\Delta(\mathbf{v}^{\prime\prime})=\Delta_{0}+k/r^{\prime\prime}$, where
$\Delta_{0}$ is the minimal discriminant of a stable bundle of slope
$\mu(\mathbf{v}^{\prime\prime})$ and rank $r^{\prime\prime}$. The expected
dimension of $B^{r^{\prime\prime}-\epsilon}(\mathbf{v}^{\prime\prime})$ is

$\displaystyle\text{expdim}B^{r^{\prime\prime}-\epsilon}(\mathbf{v}^{\prime\prime})$
$\displaystyle=\dim
M(\mathbf{v}^{\prime\prime})-\text{expcodim}B^{r^{\prime\prime}-\epsilon}(\mathbf{v}^{\prime\prime})$
$\displaystyle=r^{\prime\prime
2}(2\Delta(\mathbf{v}^{\prime\prime})-1)+1-(r^{\prime\prime}-\epsilon)(r^{\prime\prime}-\epsilon-\chi(\mathbf{v}^{\prime\prime}))$
$\displaystyle=2r^{\prime\prime}k-(r^{\prime\prime}-\epsilon)k+(\text{const}_{1})$

where (const1) is independent of $k$. The dimension of $\mathbb{P}$ is given
by

(5) $\begin{split}\dim\mathbb{P}&=\dim
B^{r^{\prime\prime}-\epsilon}(\mathbf{v}^{\prime\prime})+\dim
M(\mathbf{v}^{\prime})+\text{ext}^{1}(E^{\prime\prime},E^{\prime})-1\\\ &\geq
2r^{\prime\prime}k-(r^{\prime\prime}-\epsilon)k+(\text{const}_{1})+r^{\prime
2}(2\Delta(\mathbf{v}^{\prime})-1)\\\
&=(r+\epsilon)k+(\text{const}_{2}).\end{split}$

where again (const2) is independent of $k$.

We claim that the fiber dimension of the map $\mathbb{P}\rightarrow Z_{2}$ is
bounded independent of $k$; for $E\in Z_{2}$ let $\mathbb{P}_{E}$ denote the
fiber over $E$. To show this we consider the forgetful maps
$\mathbb{P}_{E}\rightarrow\text{Quot}(E,\mathbf{v}^{\prime\prime})$ to the
Quot scheme that assigns to an extension class $0\rightarrow
E^{\prime}\rightarrow E\rightarrow E^{\prime\prime}\rightarrow 0$ the quotient
$E\rightarrow E^{\prime\prime}$. The fiber of the forgetful map has dimension
bounded by

$\dim\text{Aut}(E^{\prime})\leq\hom(E^{\prime},E^{\prime})\leq s^{2}$

where the Jordan-Hölder filtration of $E^{\prime}$ has length $s$. In
particular it is independent of $k$, so it is enough to bound the dimension of
the Quot scheme. There is a canonical identification $T_{[E\rightarrow
E^{\prime\prime}]}\text{Quot}(E,\mathbf{v}^{\prime\prime})\simeq\text{Hom}(E^{\prime},E^{\prime\prime})$
and its dimension bounds the dimension of
$\text{Quot}(E,\mathbf{v}^{\prime\prime})$. Now consider a general line
$L\subseteq\mathbb{P}^{2}$ in the locally free loci for $E^{\prime}$ and
$E^{\prime\prime}$ and the restriction

$0\rightarrow E^{\prime\prime}(-1)\rightarrow E^{\prime\prime}\rightarrow
E^{\prime\prime}|_{L}\rightarrow 0$

of $E^{\prime\prime}$ to $L$. Then
$\text{Hom}(E^{\prime},E^{\prime\prime}(-1))=0$, and by the Grauert-Mülich
Theorem ([HL10, Theorem 3.2.1]),
$\hom(E^{\prime},E^{\prime\prime}|_{L})=h^{0}(E^{\prime\vee}\otimes
E^{\prime\prime}|_{L})$ is bounded independent of $\Delta(E^{\prime\prime})$,
hence $k$, and the claim follows.

From the dimension count above, we see that

(6) $\dim
Z_{2}\geq\dim\mathbb{P}-\dim\mathbb{P}_{E}=(r+\epsilon)k+(\text{const}).$

When the rank and degree fail to be coprime, let $\mathbf{v}_{CH}^{\prime}$ be
the extremal character of $\mathbf{v}$. Since $\Delta(\mathbf{v})\gg 0$, an
extremal decomposition of $\mathbf{v}$ exists. Denote the quotient character
by $\mathbf{v}_{CH}^{\prime\prime}$. Now by Lemma 5.10 below,
$r(\mathbf{v}_{CH}^{\prime\prime})>0$. Since
$\mu(\mathbf{v}_{CH}^{\prime})>1/3$,
$\mu(\mathbf{v}_{CH}^{\prime\prime})>1/3$. Since $\mathbf{v}_{CH}^{\prime}$
has minimal discriminant, $\Delta(\mathbf{v}_{CH}^{\prime})\gg 0$, so
Corollary 5.9 implies that the general member of
$Z_{1}(\mathbf{v}_{CH}^{\prime\prime})$ is slope-stable. The character
$\mathbf{v}_{CH}^{\prime}$ is below $\xi_{r}$ (see the proof of Theorem 4.7),
so we may write $\chi(\mathbf{v}_{CH}^{\prime})=r^{\prime}+\epsilon^{\prime}$.
Set $\epsilon=\min\\{\epsilon^{\prime},r^{\prime\prime}\\}>0$. We have
$Z_{1}(\mathbf{v}_{CH}^{\prime\prime})\subseteq
B^{r^{\prime\prime}}(\mathbf{v}_{CH}^{\prime\prime})\subseteq
B^{r^{\prime\prime}-\epsilon}(\mathbf{v}_{CH}^{\prime\prime})$. Let
$B^{\prime\prime}$ be the component of
$B^{r^{\prime\prime}-\epsilon}(\mathbf{v}_{CH}^{\prime\prime})$ that contains
$Z_{1}(\mathbf{v}_{CH}^{\prime\prime})$. Then a general member of
$B^{\prime\prime}$ is slope-stable, and since $B^{r^{\prime\prime}-\epsilon}$
is determinantal variety, $\dim
B^{\prime\prime}\geq(r^{\prime\prime}+\epsilon)k+\text{const}^{\prime\prime}$,
where const′′ is independent of $k$ and
$\Delta(\mathbf{v})=\Delta_{\min}+k/r$. By precisely the same argument as in
the proof of [CH18, Theorem 6.4], a general extension of the form

$0\rightarrow E^{\prime}\rightarrow E\rightarrow E^{\prime\prime}\rightarrow
0$

is stable, where $E^{\prime\prime}\in B^{\prime\prime}$ and $E^{\prime}\in
M(\mathbf{v}^{\prime})$ are general members. In particular, $h^{0}([E])$ is
well-defined for such $[E]\in M(\mathbf{v})$ (since there is no other sheaf
that is $S$-equivalent to $E$). If $r(\mathbf{v}_{CH}^{\prime})=1$, since
$\mathbf{v}_{CH}^{\prime}$ has minimal discriminant, $E^{\prime}$ is a line
bundle, $h^{1}(E^{\prime})=0$. If $r(\mathbf{v}_{CH}^{\prime})>1$, since
$E^{\prime}$ is general in $M(\mathbf{v}^{\prime})$, by Theorem 2.2,
$h^{0}(E^{\prime})=r^{\prime}+\epsilon^{\prime}$ and $h^{1}(E^{\prime})=0$. In
either case,

$h^{0}(E)=(r^{\prime}+\epsilon^{\prime})+(r^{\prime\prime}-\epsilon)\geq(r^{\prime}+\epsilon)+(r^{\prime\prime}-\epsilon)=r^{\prime}+r^{\prime\prime}=r.$

We obtain a rational map $\mathbb{P}\dasharrow B^{r}(\mathbf{v})\subseteq
M(\mathbf{v})$ where $\mathbb{P}$ is the projective bundle over
$M(\mathbf{v}^{\prime})\times B^{\prime\prime}$ whose fiber over
$(E^{\prime},E^{\prime\prime})$ is
$\mathbb{P}\text{Ext}^{1}(E^{\prime\prime},E^{\prime})$. Denote the image of
this rational map by $Z_{2}$. Then the dimension counts (5) and (6) apply, and
we see that

$\dim Z_{2}=(r+\epsilon)k+(\text{const})>rk+(\text{const})=\dim Z_{1}$

when $k\gg 0$. This completes the proof. ∎

######  Lemma 5.10.

With the same assumptions as in Theorem 5.7, if $\gcd(r,c_{1})>1$, then the
extremal character $\mathbf{v}_{CH}^{\prime}$ has
$r(\mathbf{v}_{CH}^{\prime})<r(\mathbf{v})$.

###### Proof.

If not, then $\mu(\mathbf{v}^{\prime})=\frac{c_{1}-1}{r}$. We claim that there
exists an integer $n$ such that
$\frac{c_{1}-1}{r}<\frac{n}{r-1}<\frac{c_{1}}{r}$. This is equivalent to the
inequalities $\frac{(c_{1}-1)(r-1)}{r}<n<\frac{(c_{1})(r-1)}{r}$. The sequence
of $r$ consecutive integers

$(c_{1}-1)(r-1),\quad(c_{1}-1)(r-1)+1,\quad...,\quad c_{1}(r-1)$

contains exactly one entry divisible by $r$. By assumption,
$\frac{c_{1}}{r}\notin\mathbb{Z}$, so $r$ does not divide $c_{1}(r-1)$. It
suffices to prove that $r$ does not divide $(c_{1}-1)(r-1)$. If
$r|(c_{1}-1)(r-1)$, then $r|(c_{1}-1)$. Set $m=\gcd(c_{1},r)$. Then $m$
divides $r,c_{1}-1$, and $c_{1}$, which contradicts the assumption that $m>1$.
∎

######  Example 5.11.

Let $\mathbf{v}_{k}\in K(\mathbb{P}^{2})$ be the Chern character

$\text{ch}(\mathbf{v})=(\text{ch}_{0}(\mathbf{v}),\text{ch}_{1}(\mathbf{v}),\text{ch}_{2}(\mathbf{v}))=(3,2,-1-k)$

so that

$\mu(\mathbf{v})=\frac{2}{3},\quad\Delta(\mathbf{v})=\frac{5}{9}+\frac{k}{3}.$

Then $\mathbf{v}_{k}$ is stable for all $k\geq 0$. As in Theorem 5.7, there
are two components in the Brill-Noether loci $B^{3}(\mathbf{v}_{k})$, whose
general members are given as extensions

$0\rightarrow\mathcal{O}^{3}\rightarrow
E\rightarrow\mathcal{O}_{C}(D)\rightarrow 0$

with $C\subseteq\mathbb{P}^{2}$ a smooth conic and $D$ of degree $1-k$, and

$0\rightarrow T_{\mathbb{P}^{2}}(-1)\rightarrow E\rightarrow
I_{Z}(1)\rightarrow 0$

where $Z\subseteq\mathbb{P}^{2}$ is a codimension two subscheme of length
$k+1$. For both extension types the general extension bundle $E$ is stable, as
in Theorem 5.7, and the locus $Z_{1}$ whose points correspond to the first
extension type form an irreducible component of $B^{3}(\mathbf{v}_{k})$. Its
dimension is

$\dim
Z_{1}=\dim\text{Pic}^{1-k}_{\mathcal{C}/U}+\text{ext}^{1}(\mathcal{O}_{C}(D),\mathcal{O}^{3})-1+\dim
GL_{3}=3k+8.$

The locus $Z_{2}$ corresponding to the second extension type has dimension

$\dim
Z_{2}=\dim\mathbb{P}^{2[k+1]}+\text{ext}^{1}(I_{Z}(1),T_{\mathbb{P}^{2}}(-1))-1+(\text{const})=4k+(\text{const})$

Thus $Z_{2}$ lies in an irreducible component distinct from $Z_{1}$, and we
conclude the Brill-Noether locus $B^{3}(\mathbf{v}_{k})$ is reducible for
$k\gg 0$.

##  6\. References

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