# Instanton sheaves on Fano threefolds

Gaia Comaschi  and  Marcos Jardim  IMECC - UNICAMP
Departamento de Matemática
Rua Sérgio Buarque de Holanda, 651
13083-970 Campinas-SP, Brazil<EMAIL_ADDRESS>
###### Abstract.

Generalizing the definitions originally presented by Kuznetsov and Faenzi, we
study (possibly non locally free) instanton sheaves of arbitrary rank on Fano
threefolds. We classify rank 1 instanton sheaves and describe all curves whose
structure sheaves are rank 0 instanton sheaves. In addition, we show that
every rank 2 instanton sheaf is an elementary transformation of a locally free
instanton sheaf along a rank 0 instanton sheaf. To complete the paper, we
describe the moduli space of rank 2 instanton sheaves of charge 2 on a quadric
threefold $X$, and show that the full moduli space of rank 2 semistable
sheaves on $X$ with Chern classes $(c_{1},c_{2},c_{3})=(-1,2,0)$ is connected
and contains, besides the instanton component, just one other irreducible
component which is also fully described.

###### Contents

  1. 1 Introduction
  2. 2 Background and notation
    1. 2.1 Classification of Fano threefolds
    2. 2.2 Stability of sheaves
  3. 3 Serre correspondence for torsion free sheaves
  4. 4 Instanton sheaves on Fano threefolds
    1. 4.1 Classification of rank 1 instanton sheaves
    2. 4.2 Classification of rank 2 instanton sheaves
    3. 4.3 Stability of rank 2 instanton sheaves
    4. 4.4 Instantons via Serre
  5. 5 Instanton sheaves on quadric threefolds
    1. 5.1 Instanton sheaves of charge 1
    2. 5.2 Instantons of charge 2
    3. 5.3 A description of ${\mathcal{L}}(2)$ via Serre correspondence.
  6. 6 The moduli space ${\mathcal{M}}_{X}(2;-1,2,0)$
    1. 6.1 The moduli space ${\mathcal{M}}_{X}(2;-1,2,2)$
    2. 6.2 Description of ${\mathcal{C}}$
    3. 6.3 Intersection of ${\mathcal{C}}$ and $\overline{{\mathcal{L}}(2)}$

##  1\. Introduction

In their seminal work [2], Atiyah, Drinfeld, Hitchin and Manin presented the
notion of mathematical instantons, rank 2 holomorphic vector bundles on
${\mathbb{P}^{3}}$ that correspond to anti-self-dual connections, a.k.a.
instantons, on the sphere $S^{4}$. More precisely, a mathematical instanton of
charge $n$ is defined as a stable rank 2 vector bundle $E$ with Chern classes
$c_{1}(E)=0,\ c_{2}(E)=n$ and such that $H^{i}(E(-2))=0$ for $i=1,2$.

In the following years several authors presented different generalizations of
mathematical instanton first to odd dimensional projective spaces [26], then
to non locally free sheaves of any rank on arbitrary projective spaces [19],
and to other Fano threefolds besides ${\mathbb{P}^{3}}$ [11, 24, 7], and more
recently to arbitrary polarized projective varieties [3, 4].

In the present paper, we introduce a definition of instanton sheaves on a Fano
threefold $X$ of Picard rank one, compatible with all the aforementioned ones.
Namely, a torsion free sheaf $E$ on $X$ is said to be an instanton sheaf of
charge $n$ if it is $\mu$-semistable and it satisfies $c_{1}(E)=-r_{X},\
c_{2}(E)=n$ and $h^{i}(E(-q_{X}))=0$ for $i=1,2$, with $r_{X},\ q_{X}$
integers such that $K_{X}\sim-(2q_{X}+r_{X})H_{X}$ where $K_{X}$ and $H_{X}$
are, respectively the anticanonical class and the ample generator of
$\operatorname{{Pic}}(X)$. These requirements are nevertheless sufficient to
guarantee that several expected properties for instanton sheaves still hold in
this more general setting.

Once the notion of an instanton sheaf has been provided and their main
features have been illustrated, we focus our attention on how they behave in
families with a particular emphasis on the rank 2 case. Indeed, there exists a
vast literature on moduli spaces of rank 2 instanton sheaves.

On ${\mathbb{P}^{3}}$, the moduli space ${\mathcal{I}}(n)$ of rank 2 instanton
bundles on ${\mathbb{P}^{3}}$ has been proved to be an irreducible [33, 34]
smooth [20] affine [8] variety of dimension $8n-3$. A more comprehensive
picture of $\overline{{\mathcal{I}}(n)}$, the closure of ${\mathcal{I}}(n)$
within the Gieseker–Maruyama moduli scheme
${\mathcal{M}}_{{\mathbb{P}^{3}}}(0,n,0)$ of rank 2 semistable sheaves on
${\mathbb{P}^{3}}$ with Chern classes $c_{1}=0,\ c_{2}=n,\ c_{3}=0$, can then
be obtained taking into account also the non locally free instanton sheaves,
as shown in [22, 23]. Moreover, the moduli space ${\mathcal{L}}(n)$ of all
rank 2 instanton sheaves of charge $n$ was shown in [21] to be connected for
$n\leq 4$.

Moduli spaces of rank 2 instanton bundles on other Fano threefolds $X$ have
been widely inspected as well: among them we mention for example [9, 27, 32].
In these works, a frequently used technique to construct instantons and study
their moduli is the so called Serre correspondence. This latter establishes a
correspondence between global sections of rank 2 reflexive sheaves on $X$ and
locally Cohen Macaulay (l.c.m.) curves on $X$; it is then possible to deduce
geometrical properties of moduli spaces of sheaves from the geometry of the
Hilbert scheme of curves on $X$.

In order to carry on these pursuits, we present in this paper a more general
form of the Serre correspondence that applies to torsion free sheaves. This
will allow us to describe in greater detail the geometry of the curves
corresponding to the non locally free rank 2 instanton sheaves, and then apply
these results to study moduli spaces of rank 2 instanton sheaves on the
quadric threefold.

Here is the plan of the paper. In Section 2 we set up the notation for the
rest of the paper by recalling the classification of Fano threefolds of Picard
rank one, and some features of (semi)stable sheaves. Section 3 is dedicated to
formulating the Serre correspondence for torsion free sheaves, generalizing
the classical correspondence presented in [1, 13, 14].

Instanton sheaves on Fano threefolds, the main characters of our tale, are
then introduced in Section 4. After going over some basic properties and
examples of instanton sheaves, we also introduce the notion of rank 0
instanton sheaves, that is 1-dimensional sheaves $T$ on $X$ satisfying the
cohomological vanishing $H^{i}(T(-q_{X}))=0,\ i=0,1$; for
$X={\mathbb{P}^{3}}$, rank 0 instantons were originally introduced in [15] and
further studied in [12], and play a key role in the study of non-reflexive
instanton sheaves via a procedure known as elementary transformation. We
present a classification of the rank 0 instantons $T$ of the form
$T\simeq{\mathcal{O}}_{C}$ where $C\subset X$ a l.c.m. curve (that will be
therefore referred to as an instanton curve) and as a direct consequence of
this result we obtain a classification of rank 1 instanton sheaves, see
Proposition 21.

The first main result of the paper is a classification of rank 2 instanton
sheaves, see Theorem 24. To be precise, we prove that in this case an
instanton sheaf $E$ is either locally free or its singular locus has pure
dimension one. This implies in particular that every non locally free rank 2
instanton sheaf $E$ is obtained via an elementary transformation of a locally
free instanton sheaf $F$ along a rank 0 instanton sheaf $T$; if this occur we
moreover have $E^{\vee\vee}\simeq F$ and
$\operatorname{Sing}(E)=\operatorname{Supp}(T)$.

We complete Section 4 with a detailed description of the Serre correspondence
for non locally free rank 2 instanton sheaves. In particular, we describe how
curves corresponding to locally free instanton sheaves behave under elementary
transformation; in this way we can also relate the geometry of the curves
corresponding to non locally free instanton sheaves to the singularities of
these sheaves.

This formulation of the Serre correspondence is the main tool used in Section
5 to study the moduli space of instantons of charge $2$ on the quadric
threefold $X\subset{\mathbb{P}^{4}}$. The Serre correspondence was used in
[32] to describe the moduli space ${\mathcal{I}}(2)$ of instanton bundles of
charge 2 on $X$ and to prove that the latter is an irreducible smooth variety
of dimension 6. Our study of the families of curves corresponding to non
locally free instanton sheaves allows us to prove that these always deform to
locally free instanton sheaves, and that they are parameterized by an
irreducible divisor in $\overline{{\mathcal{I}}(2)}$.

Still relying on the Serre correspondence, we give a complete description of
the moduli space ${\mathcal{M}}_{X}(2;-1,2,0)$ of rank 2 semistable sheaves
with Chern classes $c_{1}=-1,\ c_{2}=2,\ c_{3}=0$ in Section 6. We prove that
this moduli space consists of two irreducible components, namely
$\overline{{\mathcal{I}}(2)}$, and a 10-dimensional irreducible component
${\mathcal{C}}$ whose general point is the kernel of an epimorphism
$F\twoheadrightarrow{\mathcal{O}}_{p}$, with $F$ a stable reflexive sheaf with
$c_{1}(F)=-1,\ c_{2}(F)=2,\ c_{3}(F)=2$ and $p\in X$ a point. We will finally
show that $\overline{{\mathcal{I}}(2)}\cap{\mathcal{C}}\neq\emptyset$ proving
the connectedness of ${\mathcal{M}}_{X}(2;-1,2,0)$.

### Acknowledgments

MJ is supported by the CNPQ grant number 302889/2018-3 and the FAPESP Thematic
Project 2018/21391-1. GC is supported by the FAPESP grant number 2019/21140-1.

##  2\. Background and notation

###  2.1. Classification of Fano threefolds

Let $X$ be a smooth 3 dimensional projective variety whose Picard group has
rank one. Letting $H_{X}$ denote the ample generator of
$\operatorname{{Pic}}(X)$, the canonical class $K_{X}$ can be written in the
form

$K_{X}=-i_{X}H_{X}\hskip 28.45274pti_{X}\in{\mathbb{Z}};$

$X$ is said to be Fano whenever $i_{X}>0$. For each Fano threefold $X$, we
define the following numerical invariants:

  * •

the index, defined as the integer $i_{X}$;

  * •

the degree $d_{X}:=H_{X}^{3}$;

In addition, we let $q_{X}$ and $r_{X}$ denote the quotient and the remainder
of the division of $i_{X}$ by 2, so that we can write $i_{X}=2q_{X}+r_{X},$
with $q_{X}\geq 0$ and $r_{X}\in\\{0,1\\}$.

The cohomology groups of a Fano threefold $X$ satisfies the following
properties. All the groups $H^{i,i}(X)$ have dimension one, and for this
reason towards the entire article we will write the Chern classes of any sheaf
$F\in Coh(X)$ as integers. By Kodaira vanishing we then compute:

$\displaystyle h^{i}({\mathcal{O}}_{X}(k))$ $\displaystyle=0,\ i=1,2,\
k\in{\mathbb{Z}}$ $\displaystyle h^{0,p}(X)$ $\displaystyle=h^{p,0}(X)=0.$

Fano threefolds with Picard rank one were classified by Iskovskikh and Mukai
[18, 29]. Recall that $i_{X}\leq 4$, and

  * •

if $i_{X}=4$, then $X\simeq{\mathbb{P}}^{3}$;

  * •

if $i_{X}=3$, then $X$ is a smooth quadric hypersurface in ${\mathbb{P}}^{4}$.

  * •

there are five families of Fano threefolds with $i_{X}=2$, up to deformation;
these families are classified according to its degree
$d_{X}\in\\{1,2,3,4,5\\}$.

  * •

there are ten deformation families of Fano threefolds with $i_{X}=1$, which
are also classified according to its degree $d_{X}$ taking all even values
between 2 and 22, except 18.

Note that even if in some cases we have different isomorphism classes of Fano
threefolds, these belongs to the same deformation family.

###  2.2. Stability of sheaves

Let $E$ be a coherent sheaf on a non-singular projective variety $X$ with
$\operatorname{{Pic}}(X)={\mathbb{Z}}$; let $E(t):=E\otimes H_{X}^{\otimes
t}$, where $H_{X}$ denotes the ample generator of $\operatorname{{Pic}}(X)$.

Recall that $E$ is _(semi)stable_ if for every non-trivial subsheaf $F\subset
E$ we have

$p_{F}(t)<\leavevmode\nobreak\ (\leq)\leavevmode\nobreak\ p_{E}(t),$

where $p_{E}(t)$ denotes the reduced Hilbert polynomial of the sheaf $E$.
Furthermore, when $E$ is a torsion free sheaf, $E$ is _$\mu$ -(semi)stable_ if
for every non-trivial subsheaf $F\subset E$ such that $E/F$ is also torsion
free we have

$\dfrac{c_{1}(F)\cdot
H_{X}^{d-1}}{\operatorname{{rk}}(F)}<\leavevmode\nobreak\
(\leq)\leavevmode\nobreak\ \dfrac{c_{1}(E)\cdot
H_{X}^{d-1}}{\operatorname{{rk}}(E)}.$

Remark that, for torsion free sheaves:

$\mu-{\rm stability}\leavevmode\nobreak\ \leavevmode\nobreak\
\Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
stability}\leavevmode\nobreak\ \leavevmode\nobreak\
\Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
semistability}\leavevmode\nobreak\ \leavevmode\nobreak\
\Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \mu-{\rm
semistability},$

see [16, Lemma 1.2.13]. In addition, $E$ is $\mu$-(semi)stable if and only if
$E^{\vee}$ is $\mu$-(semi)stable.

Here is a simple characterization of $\mu$-(semi)stable rank 2 sheaves, which
generalizes well known results for reflexive sheaves, cf. [30, Lemma
II.1.2.5]. Recall that a torsion free sheaf $E$ is said to be _normalized_ if
$c_{1}(E)\in\\{0,-1,\dots,-\operatorname{{rk}}(E)+1\\}$; every torsion free
sheaf can be normalized after a twist by ${\mathcal{O}}_{X}(k)$ for some
suitable integer $k$. Recall also that a normalized torsion free sheaf with
$c_{1}(E)<0$ is $\mu$-stable if and only if it is $\mu$-semistable.

######  Lemma 1.

Let $E$ be a normalized torsion free sheaf.

  * (1)

Assuming $c_{1}(E)=0$, we have that

    * (1.1)

if $E$ is $\mu$-stable then $h^{3}(E\otimes\omega_{X})=0$;

    * (1.2)

if $E$ is $\mu$-semistable then $h^{3}(E\otimes\omega_{X}(1))=0$;

    * (1.3)

the converse statements hold when $\operatorname{{rk}}(E)=2$.

  * (2)

Assuming $c_{1}(E)<0$, we have that

    * (2.1)

if $E$ is $\mu$-semistable then $h^{3}(E\otimes\omega_{X})=0$;

    * (2.2)

the converse statement holds when $\operatorname{{rk}}(E)=2$.

###### Proof.

By Serre duality, we know that
$H^{3}(E\otimes\omega_{X})\simeq\operatorname{Hom}(E,{\mathcal{O}}_{X})^{*}$.
If $h^{3}(E\otimes\omega_{X})>0$, then there is a non-trivial morphism
$\varphi:E\to{\mathcal{O}}_{X}$; let $F:=\ker(\varphi)$. Since
$c_{1}(F)=-c_{1}(\operatorname{im}(\varphi))\geq 0$, we conclude that $E$
cannot be $\mu$-stable.

Conversely, assume that $\operatorname{{rk}}(E)=2$; if $E$ is not
$\mu$-stable, let $F$ be a destabilizing subsheaf; since $E$ has rank 2, we
must have that both $F$ and $G:=E/F$ are rank 1 torsion free sheaves. It
follows that $G={\mathcal{I}}_{\Gamma}(k)$ for some subscheme $\Gamma\subset
X$ and $k\leq 0$, thus there exists a monomorphism
$G\hookrightarrow{\mathcal{O}}_{X}$; composing the epimorphism
$E\twoheadrightarrow G$ with the latter, we obtain a non trivial morphism
$E\to{\mathcal{O}}_{X}$, showing that $h^{3}(E\otimes\omega_{X})>0$.

The proofs for item (1.2) and (2) are completely analogous. ∎

We will need the following result regarding 1-dimensional sheaves.

######  Lemma 2.

Let $T$ be a pure 1-dimensional sheaf with $\chi(T(t))=d\cdot(t+e)$. If
$h^{0}(T(-e))=0$, then $T$ is semistable.

###### Proof.

If $S\hookrightarrow T$ is a subsheaf, then $h^{0}(S(-e))=0$; if we set
$\chi(S(t))=s\cdot t+x$, then $\chi(S(-e))=-se+x=-h^{1}(L(-1))\leq 0$, thus
$x\leq se$; note that $s>0$ because $T$ has pure dimension 1. It follows that

$\dfrac{\chi(T(t))}{d}-\dfrac{\chi(S(t))}{s}=e-\dfrac{x}{s}\geq 0,$

proving that $T$ is semistable. ∎

##  3\. Serre correspondence for torsion free sheaves

The so-called _Serre correspondence_ is one of the most efficient tools to
construct and study rank 2 sheaves on a threefold $X$.

Recall from [14, Theorem 4.1] (which generalizes [13, Theorem 1.1]) that this
is a correspondence relating pairs $(C,\xi)$ consisting of a curve $C$ in $X$
and a global section $\xi$ of a twist of the dualizing sheaf
$\omega_{C}:={\mathcal{E}}{\it xt}^{2}({\mathcal{O}}_{C},\omega_{X})$, with
pairs $(E,s)$ consisting of a rank 2 reflexive sheaf $E$ and a global section
$s\in H^{0}(E)$ whose cokernel is torsion-free. Another version of the Serre
correspondence was given by Arrondo in [1, Theorem 1.1], including locally
free sheaves of higher rank.

The main goal of this section is to consider a generalization of the Serre
correspondence for torsion free sheaves on projective varieties generalizing
all of the results mentioned above.

######  Theorem 3.

Let $X$ be a non-singular, projective variety and let $L$ be a line bundle on
$X$ such that $H^{1}(L^{\vee})=H^{2}(L^{\vee})=0$. There is a correspondence
between

  * •

sets $(E,s_{1},\cdots,s_{r-1})$ consisting of a rank r torsion free sheaf $E$
with $\det(E)=L$, and global sections $s_{1},\dots,s_{r-1}\in H^{0}(E)$ whose
dependency locus has codimension at least 2;

  * •

sets $(C,\xi_{1},\dots,\xi_{r-1})$ consisting of a codimension 2 subscheme
$C\subset X$ and sections $\xi_{1},\dots,\xi_{r-1}\in
H^{0}(\omega_{C}\otimes\omega_{X}^{-1}\otimes L^{-1})$.

###### Proof.

Starting with a set $(E,s_{1},\cdots,s_{r-1})$ as described in the first item,
we form a morphism

$\sigma:=(s_{1},\cdots,s_{r-1}):{\mathcal{O}}_{X}^{\oplus(r-1)}\longrightarrow
E;$

the hypothesis on $(s_{1},\cdots,s_{r-1})$, which means that the common zeros
of $s_{i}$ have codimension at least 2, imply that $\sigma$ is surjective and
$\operatorname{coker}(\sigma)$ is a torsion free sheaf. It follows that
$\operatorname{coker}(s)\simeq{\mathcal{I}}_{C}\otimes L$ where $C$ is a
subscheme of codimension 2, which is precisely the dependency locus of
$(s_{1},\cdots,s_{r-1})$, and $L$ is a line bundle. Therefore, we obtain a
short exact sequence of the form

(1)
$0\longrightarrow{\mathcal{O}}_{X}^{\oplus(r-1)}\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}E\longrightarrow{\mathcal{I}}_{C}\otimes
L\longrightarrow 0;$

in addition, this exact sequence defines an extension class

$\xi\in\operatorname{Ext}^{1}({\mathcal{I}}_{C}\otimes
L,{\mathcal{O}}_{X}^{\oplus(r-1)}).$

Using the spectral sequence for local-to-global Ext’s

(2) $H^{p}({\mathcal{E}}{\it
xt}^{q}({\mathcal{I}}_{C},{\mathcal{O}}_{X}))\Rightarrow\operatorname{Ext}^{p+q}({\mathcal{I}}_{C},{\mathcal{O}}_{X})$

one checks that the hypothesis $H^{1}(L)=H^{2}(L)=0$ yields the first of the
following isomorphisms

(3) $\operatorname{Ext}^{1}({\mathcal{I}}_{C}\otimes
L,{\mathcal{O}}_{X})\simeq H^{0}({\mathcal{E}}{\it
xt}^{1}({\mathcal{I}}_{C}\otimes L,{\mathcal{O}}_{X}))\simeq
H^{0}({\mathcal{E}}{\it
xt}^{2}({\mathcal{O}}_{C},\omega_{X})\otimes\omega_{X}^{-1}\otimes L^{-1}).$

This means that the extension class $\xi$ can be regarded as $r-1$ section
$\xi_{1},\dots,\xi_{r-1}\in H^{0}(\omega_{C}\otimes\omega_{X}^{-1}\otimes
L^{-1})$. We have thus obtained a set $(C,\xi_{1},\dots,\xi_{r-1})$ as
described in the second item.

Conversely, given a set $(C,\xi_{1},\dots,\xi_{r-1})$ we can use the
isomorphisms in display (3) to re-interpret $\xi_{1},\dots,\xi_{r-1}$ as an
extension class in $\operatorname{Ext}^{1}({\mathcal{I}}_{C}\otimes
L,{\mathcal{O}}_{X}^{\oplus(r-1)})$ leading to an exact sequence as in display
(1), which yields a set $(E,s_{1},\cdots,s_{r-1})$. ∎

In general, the abelian groups $\operatorname{Ext}^{1}({\mathcal{I}}_{C},L)$
and $H^{0}({\mathcal{E}}{\it xt}^{1}({\mathcal{I}}_{C},L))$ are related via
the following exact sequence

$0\longrightarrow
H^{1}(L^{\vee})\longrightarrow\operatorname{Ext}^{1}({\mathcal{I}}_{C}\otimes
L,{\mathcal{O}}_{X})\longrightarrow H^{0}({\mathcal{E}}{\it
xt}^{1}({\mathcal{I}}_{C}\otimes L,{\mathcal{O}}_{X}))\longrightarrow
H^{2}(L^{\vee});$

here, we used the isomorphism ${\mathcal{H}}{\it om}({\mathcal{I}}_{C}\otimes
L,{\mathcal{O}}_{X})\simeq L^{\vee}$. Therefore, if one only assumed that
$H^{2}(L^{\vee})=0$, then every set $(C,\xi_{1},\dots,\xi_{r-1})$ defines an
extension class in $\operatorname{Ext}^{1}({\mathcal{I}}_{C}\otimes
L,{\mathcal{O}}_{X}^{\oplus(r-1)})$ and thus a torsion free sheaf of rank $r$.

In this paper, we will only be concerned with threefolds, so we fix
$\dim(X)=3$ once and for all. Moreover, we will mostly consider only rank 2
sheaves.

######  Remark 4.

Fix $r=2$.

  1. (1)

$E$ is reflexive if and only if the scheme $C$ is locally Cohen–Macaulay
(l.c.m.) and $\xi:{\mathcal{O}}_{X}\to{\mathcal{E}}{\it
xt}^{2}({\mathcal{O}}_{C},{\mathcal{O}}_{X})\otimes L^{-1}$ only vanishes on a
0-dimensional subscheme $Z\subset C$. In addition, $Z$ coincides with the
singular locus of $E$.

  2. (2)

$E$ is locally free if and only if the scheme $C$ is locally complete
intersection (l.c.i.) and
$\xi:{\mathcal{O}}_{X}\to\omega_{C}\otimes\omega_{X}^{-1}\otimes L^{-1}$ is
non vanishing.

Detailed explanation for these claims can be found in the classical references
[14, Theorem 4.1] and [13, Theorem 1.1], respectively.

When $E$ is not reflexive, set $T_{E}:=E^{\vee\vee}/E$ and consider the
following commutative diagram

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where $G:=\operatorname{coker}(\iota(s))$ and $L=\det(E)$; we argue that $G$
is torsion free, so that $G\simeq{\mathcal{I}}_{C^{\prime}}\otimes L$ for some
l.c.m. curve $C^{\prime}\subset C$.

Indeed, assume that $G$ is not torsion free and assume that $P\hookrightarrow
G$ is the maximal torsion subsheaf, so that $G/P$ is torsion free; the exact
sequence in the middle column implies that ${\mathcal{E}}{\it
xt}^{p}(G,{\mathcal{O}}_{X})=0$ for $p>1$, thus ${\mathcal{E}}{\it
xt}^{p}(P,{\mathcal{O}}_{X})=0$ for $p>1$ as well (since ${\mathcal{E}}{\it
xt}^{3}(G/P,{\mathcal{O}}_{X})=0$); it follows that $P$ and ${\mathcal{E}}{\it
xt}^{1}(P,{\mathcal{O}}_{X})$ must be a 2-dimensional sheaf. On the other
hand, since $\dim T_{E}\leq 1$, one can dualize the exact sequence in the
bottom line and conclude that $G^{\vee}\simeq L^{\vee}$ and

(5) $0\longrightarrow{\mathcal{E}}{\it
xt}^{1}(G,{\mathcal{O}}_{X})\longrightarrow{\mathcal{E}}{\it
xt}^{1}({\mathcal{I}}_{C}\otimes
L,{\mathcal{O}}_{X})\longrightarrow{\mathcal{E}}{\it
xt}^{2}(T_{E},{\mathcal{O}}_{X})\longrightarrow 0,$

since ${\mathcal{E}}{\it xt}^{p}(G,{\mathcal{O}}_{X})=0$ for $p>1$. This means
that $\dim{\mathcal{E}}{\it xt}^{1}(G,{\mathcal{O}}_{X})\leq 1$, providing a
contradiction. Therefore, $G$ does not admit a torsion subsheaf.

In general, the quotient sheaf $T_{E}$ is not pure dimensional. Our next
result characterizes those torsion free sheaves $E$ for which $T_{E}$ has pure
dimension 1.

######  Lemma 5.

Assume that the pairs $(E,s)$ and $(C,\xi)$ correspond via the Serre
correspondence outlined in Theorem 3. The scheme $C$ is l.c.m. if and only if
$T_{E}$ is a pure 1-dimensional sheaf.

###### Proof.

Dualizing the bottom line of the diagram in display (4) yields the first of
the following isomorphisms

${\mathcal{E}}{\it xt}^{3}(T_{E},{\mathcal{O}}_{X})\simeq{\mathcal{E}}{\it
xt}^{2}({\mathcal{I}}_{C}\otimes L,{\mathcal{O}}_{X})\simeq{\mathcal{E}}{\it
xt}^{3}({\mathcal{O}}_{C},{\mathcal{O}}_{X});$

the left most sheaf is 0-dimensional, so one can disregard the twist by the
line bundle $L$. Therefore, $T_{E}$ contains a 0-dimensional subsheaf if and
only if ${\mathcal{O}}_{C}$ also does, meaning that $C$ is not l.c.m. ∎

Let $U\hookrightarrow{\mathcal{O}}_{C}$ be the maximal 0-dimensional subsheaf,
so that ${\mathcal{E}}{\it
xt}^{3}({\mathcal{O}}_{C},{\mathcal{O}}_{X})\simeq{\mathcal{E}}{\it
xt}^{3}(U,{\mathcal{O}}_{X})$. As a by-product of the previous proof, we also
conclude that ${\mathcal{E}}{\it
xt}^{3}(T_{E},{\mathcal{O}}_{X})\simeq{\mathcal{E}}{\it
xt}^{3}(U,{\mathcal{O}}_{X})$. In other words, the 0-dimensional components of
the support of $T_{E}$ are always contained in the 0-dimensional components of
$C$, regardless of the choice of section $s$.

##  4\. Instanton sheaves on Fano threefolds

Let $X$ be a Fano threefold of Picard rank one and of index $i_{X}$, following
all the notation and definitions posed in Section 2.

The key point of the present paper is the introduction of the following
definition, which generalizes Faenzi’s and Kuznetsov’s definitions of
instanton bundles on a Fano threefold, compare with [11, Definition 1] and
[24, Definition 1.1].

######  Definition 6.

An _instanton sheaf_ $E$ on $X$ is a $\mu$-semistable sheaf with first Chern
class $c_{1}(E)=-r_{X}$ and such that:

(6) $H^{1}(E(-q_{X}))=H^{2}(E(-q_{X}))=0.$

The _charge_ of $E$ is defined to be $c_{2}(E)$.

The $\mu$-semistability condition rules out
${\mathcal{O}}_{X}(-r_{X})\oplus{\mathcal{O}}_{X}^{\oplus(r-1)}$ as instanton
sheaf when $r_{X}=1$ (ie. when $i_{X}$ is odd); however,
${\mathcal{O}}_{X}^{\oplus r}$ is considered an instanton sheaf when $r_{X}=0$
(ie. when $i_{X}$ is even).

######  Remark 7.

When $X={\mathbb{P}^{3}}$ this definition is, in general, a bit more
restrictive that the definition of instanton sheaves adopted in [19, 12, 21,
23]; in these references, an instanton sheaf on ${\mathbb{P}^{3}}$ was defined
as a torsion free sheaf $E$ with $c_{1}(E)=0$ and

$h^{0}(E(-1))=h^{1}(E(-2))=h^{2}(E(-2))=h^{3}(E(-3))=0.$

Using this definition, one can find examples of instanton sheaves of rank 4
and larger that are not $\mu$-semistable, see [19, Example 3]. However, both
definitions are equivalent when $\operatorname{{rk}}(E)=2$, since every rank 2
sheaf on ${\mathbb{P}^{3}}$ satisfying the conditions above is automatically
$\mu$-semistable.

The following technical results will be useful later on.

######  Lemma 8.

Let $E$ be an instanton sheaf of rank $r$.

  1. (1)

$H^{0}(E(-n))=0\ \forall\>n\geq 1-r_{X}$ and $H^{3}(E(n))=0,\ \forall\>n\geq-
i_{X}+1$.

  2. (2)

$H^{i}(E(-q_{X}))=0,\
\operatorname{Ext}^{i}(E,{\mathcal{O}}_{X}(-q_{X}-r_{X}))=0,\ \forall\>i.$

In particular, we conclude that $\chi(E(-q_{X}))=0$.

###### Proof.

Since $E$ is $\mu$-semistable, $\operatorname{Hom}({\mathcal{O}}_{X}(n),E)=0$
whenever $n>\frac{-r_{X}}{r}$ and since $r_{X}=0$ or $1$, this happens if and
only if $n\geq 1-r_{X}$. An equivalent argument leads to
$\operatorname{Hom}(E,{\mathcal{O}}_{X}(n))=0$ whenever $n<\frac{-r_{X}}{r}$
that is to say, whenever $n\geq-i_{X}+1$. Therefore $H^{0}(E(n))=0$ for $n\geq
1-r_{X}$ and, by duality,
$\operatorname{Ext}^{3}({\mathcal{O}}_{X}(-n),E)\simeq H^{3}(E(n))=0$ for
$n\geq-i_{X}+1$. Since $1-r_{X}\leq q_{X}\leq i_{X}-1$ we get
$H^{i}(E(-q_{X}))=0$, for $i=0,3$ and this together with (6) leads to
$H^{i}(E(-q_{X}))=0,\ \forall i$. By Serre duality, we have isomorphisms

$H^{i}(E(-q_{X}))\simeq\mathrm{Ext}^{3-i}(E(-q_{X}),\omega_{X}))^{*}\simeq\mathrm{Ext}^{3-i}(E,{\mathcal{O}}_{X}(-q_{X}-r_{X}))^{*},$

therefore $\operatorname{Ext}^{i}(E,{\mathcal{O}}_{X}(-q_{X}-r_{X}))=0,\
\forall\>i$. ∎

From these computations we determine the value of the third Chern class of
instanton sheaves.

######  Corollary 9.

Let $E$ be an instanton sheaf of rank $r$.

  1. (1)

If $i_{X}>1,\ \chi(E(-q_{X}))=\frac{c_{3}(E)}{2}=0,$ hence $c_{3}(E)=0$.

  2. (2)

If $i_{X}=1,\ \chi(E(-q_{X}))=\chi(E)=(r-2)+\frac{c_{3}(E)}{2}=0$ hence
$c_{3}(E)=2(2-r)$.

###### Proof.

$\chi(E(-q_{X}))=0$ by Lemma 8; by Grothendieck–Riemann–Roch theorem we
compute $\chi(E(-q_{X}))=\frac{c_{3}(E)}{2}$ whenever $i_{X}>1$ so that
$c_{3}(E)=0$ and $\chi(E)=(r-2)+\frac{c_{3}(E)}{2}$ for $i_{X}=1$ so that
$c_{3}(E)=2(2-r)$. ∎

The main motivation behind Definition 6 is that non locally free instanton
sheaves naturally arise as degenerations of locally free ones. When
$X={\mathbb{P}^{3}}$, this phenomenon has been studied in [22, 23]. To see it
in greater generality, let us consider some explicit examples of rank 2
instanton sheaves.

Let $C:=L_{1}\sqcup\cdots\sqcup L_{n}$ be the disjoint union of lines in $X$,
and set $G:={\mathcal{I}}_{C}(q_{X}-1)$; note that for $p=1,2$, we have

$H^{p}(G(-q_{X}))=H^{p}({\mathcal{I}}_{C}(-1))\simeq\bigoplus_{k=1}^{n}H^{p-1}({\mathcal{O}}_{L_{k}}(-1))=0.$

We can then consider extensions of the form

(7) $0\longrightarrow{\mathcal{O}}_{X}(-q_{X}-r_{X}+1)\longrightarrow
E\longrightarrow{\mathcal{I}}_{C}(q_{X}-1)\longrightarrow 0;$

clearly, $c_{1}(E)=-r_{X}$ and one easily checks that $H^{p}(E(-q_{X}))=0$ for
$p=1,2$. Applying Lemma 1, we verify that $E$ is $\mu$-semistable when
$i_{X}\geq 2$; however, such sheaves are always properly $\mu$-semistable when
$i_{X}=2$ and are not $\mu$-semistable when $i_{X}=1$. Therefore, E is a rank
2 instanton sheaf provided $i_{X}\geq 2$. Inspired by the traditional
nomenclature for $X={\mathbb{P}^{3}}$, instanton sheaves given by an extension
as in display (7) are called _’t Hooft instantons_ ; the charge of a ’t Hooft
instanton sheaf corresponding to $n$ lines is $n-1$

######  Example 10.

Here is an example of a family of rank 2 locally free instanton sheaves
degenerating into a non locally free one. Assume that $i_{X}\geq 2$, and let
$C$ be a disjoint union of $n\geq 2$ lines in $X$, as above.

Since

$\operatorname{Ext}^{1}({\mathcal{I}}_{C}(q_{X}-1),{\mathcal{O}}_{X}(-q_{X}-r_{X}+1))=\operatorname{Ext}^{1}({\mathcal{I}}_{C},\omega_{X}(2))\simeq
H^{0}(\omega_{C}(2))=\bigoplus_{k=1}^{n}H^{0}({\mathcal{O}}_{L_{k}}),$

we can consider extension classes $\xi_{t}=(1,\dots,1,t)$ with
$t\in\mathbb{C}$, inducing a family of instanton sheaves $E_{t}$, parametrized
by $\mathbb{C}$.

Note that $E_{t}$ is locally free when $t\neq 0$, since $\xi_{t}$ is non
vanishing in this case. On the other hand $\xi_{0}$ vanishes along $L_{n}$, so
the corresponding ’t Hooft instanton sheaf $E_{0}$ is not locally free. Note
that $E_{0}$ satisfies the following short exact sequence

$0\longrightarrow E_{0}\longrightarrow
F\longrightarrow{\mathcal{O}}_{L}(q_{X}-1)\longrightarrow 0,$

where $F$ is a locally free ’t Hooft instanton sheaf of charge $n-1$.

When $i_{X}$ is even, instanton sheaves of rank larger than 2 can easily be
produced using rank 2 instantons and ideal sheaves, via the following claim.

######  Lemma 11.

Assume that $i_{X}=2,4$, so that $r_{X}=0$. If $E_{1}$ and $E_{2}$ are
instanton sheaves, then any extension of $E_{1}$ by $E_{2}$ is also an
instanton sheaf.

###### Proof.

If $E$ is an extension of $E_{1}$ by $E_{2}$, then it is easy to check that
$E$ satisfies the cohomological conditions in Definition 6. Since $E_{1}$ and
$E_{2}$ are $\mu$-semistable sheaves with vanishing slope, then so is $E$. ∎

Next, we consider the generalization of a definition first introduced in [15,
Definition 6.1] for $X={\mathbb{P}^{3}}$, and further studied in [12, 21].

######  Definition 12.

A _rank 0 instanton sheaf_ on a Fano threefold $X$ is a pure 1-dimensional
sheaf $T$ satisfying $h^{0}(T(-q_{X}))=h^{1}(T(-q_{X}))=0$.

If $T$ is a rank 0 instanton sheaf on $X$, then $\chi(T(t))=d\cdot(t+q_{X})$,
and the coefficient $d$ is called the _degree_ of $T$. Moreover, Lemma 2
implies that $T$ is always semistable.

######  Proposition 13.

If $T$ is a rank 0 instanton sheaf, then so is $T^{\rm
D}\otimes\omega_{X}^{-1}(-r_{X})$, where $T^{\rm D}:={\mathcal{E}}{\it
xt}^{2}(T,\omega_{X})$.

###### Proof.

Note first that $T^{\rm D}$ is pure 1-dimensional sheaf. Using the spectral
sequence (2) for local to global Ext’s, we check that, for $p=0,1$

$H^{p}(T^{\rm D}\otimes\omega_{X}^{-1}(-r_{X}-q_{X}))=H^{p}({\mathcal{E}}{\it
xt}^{2}(T,{\mathcal{O}}_{X}(-q_{X}-r_{X})))\simeq\operatorname{Ext}^{p+2}(T,{\mathcal{O}}_{X}(-q_{X}-r_{X})).$

Serre duality yields the isomorphism

$\operatorname{Ext}^{p+2}(T,{\mathcal{O}}_{X}(-q_{X}-r_{X}))\simeq
H^{1-p}(T(-q_{X}))^{*}$

and the latter vanishes by the instantonic condition on $T$. ∎

######  Lemma 14.

If $T$ is a rank 0 instanton sheaf, then $h^{0}(T(-q_{X}-n))=0$ and
$h^{1}(T(-q_{X}+n))=0$ for every $n\geq 0$.

###### Proof.

Given a rank 0 instanton sheaf $T$, let $S\subset X$ be a hyperplane section
transversal to the support of $T$ (ie. $\dim(\operatorname{Supp}(T)\cap
S)=0$), so that ${\mathcal{T}\\!}{\it or}^{1}(T,{\mathcal{O}}_{S})=0$. This
implies that we can twist the exact sequence
$0\to{\mathcal{O}}_{X}(-1)\to{\mathcal{O}}_{X}\to{\mathcal{O}}_{S}\to 0$ by
$T(k)$ to obtain the short exact sequence

$0\longrightarrow T(k-1)\longrightarrow T(k)\longrightarrow
T\otimes{\mathcal{O}}_{S}(k)\longrightarrow 0.$

Taking cohomology, we conclude that $h^{0}(T(k-1))=0$ whenever
$h^{0}(T(k))=0$, while $h^{1}(T(k))=0$ whenever $h^{1}(T(k-1))=0$, since
$\dim(T\otimes{\mathcal{O}}_{S})=0$. The desired claims follow by induction. ∎

We are now interested in detecting when a l.c.m. curve $C$ is such that the
structure sheaf ${\mathcal{O}}_{C}$ is a rank 0 instanton. We refer to a curve
$C$ of such a kind as an instanton curve.

######  Lemma 15.

Let $X$ be a Fano threefold of Picard rank one.

  1. (1)

There are no instanton curves when $i_{X}=1,4$.

  2. (2)

When $i_{X}=2,3$, every instanton curve $C$ of degree $d$ fits in a short
exact sequence of the form

(8)
$0\longrightarrow{\mathcal{O}}_{l}\longrightarrow{\mathcal{O}}_{C}\longrightarrow{\mathcal{O}}_{C^{\prime}}\longrightarrow
0$

where $l\subset X$ is a line and $C^{\prime}$ is an instanton curve of degree
$d-1$.

###### Proof.

The fact that there are no instanton curves on a Fano threefold $X$ of index
$i_{X}=1$ is simply due to the fact that we can not have a projective
algebraic curve $C\subset X$ such that $H^{0}({\mathcal{O}}_{C})=0$ (the
restriction map $H^{0}({\mathcal{O}}_{X})\to H^{0}({\mathcal{O}}_{C})$ is
necessarily an injection).

Let us now treat the cases $q_{X}>0$. Consider an instanton curve $C\subset X$
of degree $d$; if $C$ is reduced, then

$h^{0}({\mathcal{O}}_{C})=\chi({\mathcal{O}}_{C})+h^{1}({\mathcal{O}}_{C})\geq
d\cdot q_{X},$

thus $C$ has at least $d\cdot q_{X}$ connected components.

If $q_{X}=2$ (i.e. $X={\mathbb{P}^{3}}$), then this is impossible since $C$
can have at most $d$ connected components.

If $q_{X}=1$, then $C$ is a reduced curve of degree $d$ with with at least $d$
connected components; the only possibility is for $C$ to be a disjoint union
of lines which ensures that ${\mathcal{O}}_{C}$ fits in a short exact sequence
of the form (8).

If $C$ is not reduced, let $C_{\rm red}$ be its reduction. This latter is a
l.c.m. curve satisfying $h^{1}({\mathcal{O}}_{C_{\rm red}}(-q_{X}))=0$ and
$h^{0}({\mathcal{O}}_{C_{\rm red}}(-q_{X}))=0$ (since $q_{X}>0$ and $C_{\rm
red}$ is reduced and l.c.m.): in other words $C_{\rm red}$ is an instanton
curve.

In particular, there are no instanton curves $C$ for $i_{X}=1,4$ and for
$i_{X}=2,3,$ $C_{\rm red}$ is a disjoint union of $d^{\prime}<d$ lines. It
follows that

${\mathcal{O}}_{C}\simeq\bigoplus_{j=1}^{d^{\prime}}{\mathcal{O}}_{C_{j}}$

where $\ell_{j}:=(C_{j})_{\rm red}$ are disjoint lines, and
$H^{i}({\mathcal{O}}_{C_{j}}(-1)),\ i=0,1$. To prove our claim it is therefore
enough to show that each ${\mathcal{O}}_{C_{j}}$ fits in a short exact
sequence of the form (8); this is obtained applying Lemma 16 below. ∎

######  Lemma 16.

Let $C$ be a multiple structure of degree $d$ on a line $\ell\subset X$.
Suppose that $H^{i}({\mathcal{O}}_{C}(-1))=0$ for $i=0,1$. Then $C$ fits in a
short exact sequence of the form

$0\rightarrow{\mathcal{O}}_{\ell}\rightarrow{\mathcal{O}}_{C}\rightarrow{\mathcal{O}}_{C^{\prime}}\rightarrow
0$

where $C^{\prime}$ is a multiple structure of degree $d-1$ on ${\ell}$ such
that $H^{i}({\mathcal{O}}_{C^{\prime}}(-1))=0$ for $i=0,1$.

###### Proof.

According to [5], a curve $C$ satisfying the hypotheses above admits a
filtration:

(9) $l=C_{1}\subset C_{2}\subset\ldots\subset C_{m}=C$

where each $C_{j}$ is a multiple structure on $\ell$ whose sheaf of ideals
${\mathcal{I}}_{C_{j}}$ is the kernel of a surjection
${\mathcal{I}}_{C_{j-1}}\twoheadrightarrow L_{j-1}$ with $L_{j-1}$ a vector
bundle on $\ell$. Furthermore there exist induced generically surjective
morphisms $L_{i}\otimes L_{j}\to L_{i+j}$ for each $i,j\in\\{1,\ldots m\\}$
and, in particular, generically surjective maps $L_{1}^{\otimes j}\to L_{j}$.
Note that since in our case $C_{1}=C_{\rm red}$ is the line $\ell$, each
vector bundle $L_{j}$ splits as
$L_{j}=\bigoplus_{i=1}^{k_{j}}{\mathcal{O}}_{\ell}(a_{j}^{i})$,
$k_{j}\in\\{1,2\\}$ (as $l$ has codimension 2). Therefore, $C=C_{m}$ fits into
a sort exact sequence of the form

$0\rightarrow
L_{m-1}\rightarrow{\mathcal{O}}_{C}\rightarrow{\mathcal{O}}_{C_{m-1}}\rightarrow
0;$

in order to prove the lemma it is therefore sufficient to show that each
summand of $L_{m-1}$ has degree 0.

Consider the second step $C_{2}$ of the filtration (9). This must satisfy
$H^{1}({\mathcal{O}}_{C_{2}}(-1))=0$ so that $a_{1}^{i}\geq 0$ for $1\leq
i\leq k_{1}$. Since we have a generically surjective morphism $L_{1}^{\otimes
m-1}\to L_{m-1}$ we deduce that $a_{m-1}^{i}\geq 0,\ 1\leq i\leq k_{m-1}$; but
as $H^{0}({\mathcal{O}}_{C}(-1))=0,\ H^{0}(L_{m-1}(-1))=0$ we have
$a_{m-1}^{i}\leq 0,\ 1\leq i\leq k_{m-1}$. The only possibility is thus
$a_{m-1}^{i}=0$ for $1\leq i\leq k_{m-1}$. ∎

From now on the l.c.m. curve of degree $d$ constructed “inductively” via the
short exact sequences of the form (8), will be referred to as degree $d$ line
arrangements.

######  Remark 17.

From the proof of Lemma 15 we learn that the degree $d$ lines arrangements are
the only degree $d$ l.c.m. curves $C$ such that $\chi({\mathcal{O}}_{C})=d$.
In particular we notice that for a degree $d$ l.c.m. curve $C$, since
$\chi({\mathcal{O}}_{C}(-1))=-h^{1}({\mathcal{O}}_{C}(-1))=-d+\chi({\mathcal{O}}_{C})\leq
0$, we always have $\chi({\mathcal{O}}_{C})\leq d$ and equality holds if and
only if $C$ is a degree $d$ line arrangement.

One of the main reason that justifies our interest in rank 0 instanton sheaves
is that these sheaves play a primary role in the study of non-reflexive
instanton sheaves. It is indeed possible to construct instanton sheaves out of
non-reflexive ones performing an _elementary transformation_ along a rank 0
instanton sheaf.

We recall that the elementary transformation consists of the following
procedure. Let $F$ be a reflexive instanton sheaf, let $T$ be a rank 0
instanton sheaf, and consider an epimorphism $q:F\twoheadrightarrow T$. It is
easy to check that $E:=\ker q$ is also an instanton sheaf. Indeed, consider
the exact sequence

(10) $0\longrightarrow E\longrightarrow F\stackrel{{\scriptstyle
q}}{{\longrightarrow}}T\longrightarrow 0;$

$E$ is $\mu$-semistable because $F$ is $\mu$-semistable and $\mu(F)=\mu(E)$;
the exact sequence in cohomology (here, $p=1,2$)

$H^{p-1}(T(-q_{X}))\longrightarrow H^{p}(F(-q_{X}))\longrightarrow
H^{p}(E(-q_{X}))$

implies that $h^{p}(E(-q_{X}))=0$ ($p=1,2$) since $T$ and $F$ are instanton
sheaves.

It is almost immediate to prove that whenever $E$ is the elementary
transformation of $F$ along $T$ then the following holds

######  Lemma 18.

Let $E$ be an instanton sheaf obtained by elementary transformation of a
reflexive instanton $F$ along a rank 0 instanton $T$. Then $F\simeq
E^{\vee\vee}$.

###### Proof.

Applying the functor ${\mathcal{H}}{\it om}(\>\cdot\>,{\mathcal{O}}_{X})$ to
(10), we get $E^{\vee}\simeq F^{\vee}$ (since $T$ is one-dimensional) hence
$E^{\vee\vee}\simeq F^{\vee\vee}\simeq F$. ∎

As it turns out not all the non-reflexive instanton sheaves are necessarily
obtained in this way. We can indeed prove that for a non reflexive instanton
sheaf $E$, $E^{\vee\vee}/E$ is always purely one-dimensional but not
necessarily a rank 0 instanton shef; accordingly $E^{\vee\vee}$ is not
necessarily an instanton sheaf either.

######  Proposition 19.

Let $E$ be a non reflexive instanton sheaf of rank $r>0$. Then the following
hold:

  1. (1)

$T_{E}:=E^{\vee\vee}/E$ has pure dimension one;

  2. (2)

$E$ has homological dimension 1;

  3. (3)

$H^{p}(E^{\vee\vee}(-q_{X}))=0,\ p=0,2,3$

  4. (4)

$E^{\vee\vee}$ is an instanton if and only if $T$ is a rank 0 instanton sheaf.
This condition is equivalent to $c_{3}(E^{\vee\vee})=0$ for $i_{X}>1$ and
$c_{3}(E^{\vee\vee})=2(2-r)$ for $i_{X}=1$.

###### Proof.

Since $E$ is torsion free, it injects in its double dual, leading to a short
exact sequence:

(11) $0\rightarrow E\rightarrow E^{\vee\vee}\rightarrow T_{E}\rightarrow 0$

where $T_{E}:=E^{\vee\vee}/E$ is a torsion sheaf supported on the locus of
points where $E$ fails to be reflexive. Applying the functor
${\mathcal{H}}{\it om}(\>\cdot\>,{\mathcal{O}}_{X})$ to (11) we obtain an
exact sequence $0\rightarrow E^{\vee}\rightarrow
E^{\vee\vee}\rightarrow{\mathcal{E}}{\it
xt}^{1}(T_{E},{\mathcal{O}}_{X})\rightarrow{\mathcal{E}}{\it
xt}^{1}(E^{\vee\vee},{\mathcal{O}}_{X})$ from which we deduce that
$\dim(T_{E})\leq 1$. Indeed, since $E^{\vee}$ is reflexive, $E^{\vee}\simeq
E^{\vee\vee}$ which implies that ${\mathcal{E}}{\it
xt}^{1}(T_{E},{\mathcal{O}}_{X})$ injects in ${\mathcal{E}}{\it
xt}^{1}(E^{\vee\vee}{\mathcal{O}}_{X})$; but this can clearly not happen if
ever $T_{E}$ had dimension $>1$. If ever this was the case, ${\mathcal{E}}{\it
xt}^{1}(T_{E},{\mathcal{O}}_{X})$ would have dimension bigger then one as well
leading to a contradiction since ${\mathcal{E}}{\it
xt}^{1}(E^{\vee\vee},{\mathcal{O}}_{X})$ is zero-dimensional due to the
reflexivity of $E^{\vee\vee}$. Twisting now (11) by
${\mathcal{O}}_{X}(-q_{X})$ and taking cohomology we get an exact sequence:

$H^{0}(E^{\vee\vee}(-q_{X}))\rightarrow H^{0}(T_{E}(-q_{X}))\rightarrow
H^{1}(E(-q_{X})).$

The l.s.t. vanishes since $E^{\vee\vee}$ is $\mu$-semistable, the r.s.t.
vanishes since $E$ is an instanton; accordingly $H^{0}(T_{E}(-q_{X}))=0$ thus
$H^{0}(T_{E}(-n))=0,\ \forall n\geq q_{X}$ (apply 14) which allows us to
conclude that $T_{E}$ has pure dimension one ending the proof of $(1)$. As a
consequence of $(1)$ we get that ${\mathcal{E}}{\it
xt}^{2}(E,{\mathcal{O}}_{X})\simeq{\mathcal{E}}{\it
xt}^{3}(T_{E},{\mathcal{O}}_{X})=0$ and since moreover $0={\mathcal{E}}{\it
xt}^{3}(E^{\vee\vee},{\mathcal{O}}_{X})$ surjects onto ${\mathcal{E}}{\it
xt}^{3}(E,{\mathcal{O}}_{X})$, we can conclude that $E$ has homological
dimension 1, proving (2). The long exact sequence in cohomology from (11)
twisted by ${\mathcal{O}}_{X}(-q_{X})$ now leads to

$H^{i}(E(-q_{X}))\simeq H^{i}(E^{\vee\vee}(-q_{X}))=0,\ i=2,3,\hskip
5.69054ptH^{1}(E^{\vee\vee}(-q_{X}))=H^{1}(T_{E}(-q_{X})).$

These equalities lead to $(3)$ (as we have already pointed out that
$H^{0}(E^{\vee\vee}(-q_{X}))$ vanishes by $\mu$-semistability) and ensure that
$E^{\vee\vee}$ is an instanton if and only if $T_{E}$ is a rank 0 instanton.
Finally, we compute that $\chi(T_{E}(-q_{X}))=\frac{c_{3}(E^{\vee\vee})}{2}$
for $i_{X}>1$ (resp. $\chi(T_{E})=\frac{c_{3}(E^{\vee\vee})}{2}+r-2$ for
$i_{X}=1$) which holds if and only if $c_{3}(E^{\vee\vee})=0$ (resp. if and
only if $c_{3}(E^{\vee\vee})=2(2-r))$. ∎

###  4.1. Classification of rank 1 instanton sheaves

Since $\operatorname{{Pic}}(X)={\mathbb{Z}}$, a locally free (or equivalently
reflexive) instanton of rank one is uniquely determined by its first Chern
class, thus the only instanton line bundle is ${\mathcal{O}}_{X}(-r_{X})$. Let
us now consider the non locally free case.

######  Lemma 20.

Let $L$ be a non locally free instanton sheaf of rank one. Then
$L^{\vee\vee}\simeq{\mathcal{O}}_{X}(-r_{X})$ and $L^{\vee\vee}/L$ is a rank 0
instanton.

###### Proof.

Applying proposition (19) (recall that in rank one reflexivity is equivalent
to local freeness) we have that $L$ always fits in a short exact sequence of
the form:

(12) $0\rightarrow L\rightarrow L^{\vee\vee}\rightarrow T\rightarrow 0$

with $T$ being a torsion sheaf of pure dimension one. Accordingly,
$L^{\vee\vee}$ is a rank one reflexive sheaf with
$c_{1}(L^{\vee\vee})=c_{1}(L)=-r_{X}$, that is to say
$L^{\vee\vee}\simeq{\mathcal{O}}_{X}(-r_{X})$. Since
${\mathcal{O}}_{X}(-r_{X})$ is an instanton, one can easily check that is a
rank 0 instanton sheaf. ∎

We therefore understand that the classification of rank 1 instanton sheaves
reduces to the classification of rank 0 instanton sheaves $T$ admitting an
epimorphism ${\mathcal{O}}_{X}(-r_{X})\twoheadrightarrow T$. This latter can
be attained as a consequence of Lemma 15.

######  Proposition 21.

Let $L$ be a rank one instanton sheaf of charge $d$ on a Fano threefold $X$
with Picard rank one. The following hold:

  1. (1)

if $i_{X}=3,4$, then $d=0$ and $L\simeq{\mathcal{O}}_{X}(-r_{X})$;

  2. (2)

if $i_{X}=1,2$, we have $L\simeq{\mathcal{O}}_{X}(-r_{X})$ whenever $d=0$
whilst for $d>0$ $L$ always fits in a short exact sequence of the form:

$0\rightarrow L\rightarrow
L^{\prime}\rightarrow{\mathcal{O}}_{\ell}\rightarrow 0$

for a line ${\ell}\subset X$ and $L^{\prime}$ a rank one instanton sheaf of
charge $d-1$.

###### Proof.

By Lemma 20, the classification of rank one instanton sheaves $L$ of charge
$d$, reduces to the classification of degree $d$ rank 0 instanton sheaves $T$
admitting a surjection ${\mathcal{O}}_{X}(-r_{X})\twoheadrightarrow T$, that
is to say, to the classification of degree $d$ l.c.m. curves $C\subset X$ such
that ${\mathcal{O}}_{C}(-r_{X})$ is a rank 0 instanton. But this means that
$H^{i}({\mathcal{O}}_{C}(-2))=0,\ i=0,1$ for $i_{X}=3,4$ and
$H^{0}({\mathcal{O}}_{C}(-1))$ for $i_{X}=1,2$. The arguments used to prove
Lemma 15 (i) show then that there are no rank 1 non locally free instantons of
rank one on Fano threefolds of index $3$ or $4$. Similarly, from Lemma 15 (2)
and Remark 17, we know that the only degree $d$ l.c.m. curves such that
$H^{i}({\mathcal{O}}_{C}(-1))=0$ are the degree $d$ lines arrangements. This
proves point (2). ∎

######  Remark 22.

Proposition 21 can be rephrased by saying that each rank one instanton $L$ of
charge $d>0$, on a Fano threefold $X$ of index $i_{X}\in\\{1,2\\}$, is always
isomorphic to ${\mathcal{I}}_{C}(-r_{X})$ for $C$ a degree $d$ line
arrangement. Recall that a curve $C$ of such a kind is supported of
$d^{\prime}\leq d$ disjoint lines and can be constructed “inductively” from an
extension:

$0\rightarrow{\mathcal{O}}_{\ell}\rightarrow{\mathcal{O}}_{C}\rightarrow{\mathcal{O}}_{C}^{\prime}\rightarrow
0$

with $C^{\prime}$ a degree $d-1$ line arrangement. As a consequence, for
$i_{X}=2$, rank one instantons of strictly positive charge coincide with ideal
sheaves of instanton curves.

###  4.2. Classification of rank 2 instanton sheaves

In general, the double dual $E^{\vee\vee}$ of a torsion free sheaf $E$ is a
reflexive (possibly non locally free) sheaf; if $E$ is an instanton sheaf, we
have that $c_{1}(E^{\vee\vee})=-r_{X}$ and that $E$ is $\mu$-semistable, but
$E^{\vee\vee}$ may not satisfy the instantonic vanishing conditions.

The main result of this section guarantees that $E^{\vee\vee}$ is a locally
free instanton sheaf when $\operatorname{{rk}}(E)=2$. Recall that, since $X$
has Picard rank one, we have in this case an isomorphism:

(13) $E^{\vee\vee}\simeq E^{\vee}(-r_{X}).$

In particular this implies that the Serre’s duality establishes isomorphisms:

(14) $H^{i}(E^{\vee\vee}(n))\simeq H^{3-i}(E^{\vee}(-n-i_{X}))^{*}\simeq
H^{3-i}(E^{\vee\vee}(-2q_{X}-n))^{*},\ i=0,3.$

When $X$ is a Fano threefold, Lemma 1 says that a normalized rank 2 torsion
free sheaf $E$ on X is $\mu$-semistable if and only if $h^{3}(E(-i_{X}+1))=0$.
In addition, note that $\mu$-semistability implies that $h^{0}(E(r_{X}-1))=0$,
when we assume that $c_{1}(E)=r_{X}$.

Let us now focus on rank 2 instantons. To begin with, we show that, in the
rank 2 case, the reflexivity of an instanton implies its local freeness.

######  Lemma 23.

Let $E$ be a rank 2 reflexive instanton. Then $E$ is locally free.

###### Proof.

Applying Lemma 9, $\chi(E(-q_{X}))=0$ leading to $c_{3}(E)=0$. The proof of
[14, Proposition 2.6] applies verbatim to arbitrary Fano threefolds of Picard
rank one which allows to conclude that $c_{3}(E)=0$ if and only if $E$ is
locally free. ∎

We are finally ready to prove our classification of non locally free rank 2
instanton sheaves.

######  Theorem 24.

Let $E$ be a rank 2 instanton sheaf. Then $E^{\vee\vee}$ is an instanton
bundle and $T_{E}:=E^{\vee\vee}/E$ is a rank 0 instanton sheaf whenever
$T_{E}\neq 0$.

###### Proof.

By Propositions 19 and 23 it is enough to prove that for a non locally free
instanton $E$, $h^{1}(E^{\vee\vee}(-q_{X}))=0$. Consider the local to global
spectral sequence

$E^{p,q}_{2}=H^{p}({\mathcal{E}}{\it
xt}^{q}(E^{\vee\vee},{\mathcal{O}}_{X}(-q_{X}-r_{X})))\Rightarrow\operatorname{Ext}^{p+q}(E^{\vee\vee},{\mathcal{O}}_{X}(-q_{X}-r_{X})).$

${\mathcal{H}}{\it om}(E^{\vee\vee},{\mathcal{O}}_{X}(-q_{X}-r_{X}))\simeq
E^{\vee\vee}(-q_{X})$ hence $E^{p,0}=0$ for $p\neq 1$ which implies that the
spectral sequence already degenerates at the $r=2$ sheet. Therefore

$\operatorname{Ext}^{1}(E^{\vee\vee},{\mathcal{O}}_{X}(-q_{X}-r_{X}))\simeq
H^{0}({\mathcal{E}}{\it
xt}^{1}(E^{\vee\vee},{\mathcal{O}}_{X}(-q_{X}-r_{X})))\oplus
H^{1}(E^{\vee\vee}(-q_{X}));$

since
$\operatorname{Ext}^{1}(E^{\vee\vee},{\mathcal{O}}_{X}(-r_{X}-q_{X}))\simeq
H^{2}(E^{\vee\vee}(-q_{X}))^{*}=0$, we then deduce that
$H^{0}({\mathcal{E}}{\it
xt}^{1}(E^{\vee\vee},{\mathcal{O}}_{X}(-q_{X}-r_{X})))=0$ and
$H^{1}((E^{\vee\vee}(-q_{X}))=0.$ This ensure that $E^{\vee\vee}$ is an
instanton bundle. ∎

######  Remark 25.

Theorem 24 generalizes [9, Proposition 3,1, Theorem 3.5] and [35, Theorem
1,2], [36, Theorem 1.3].

######  Remark 26.

It is worth pointing out that if in the definition of instanton, we replace
$\mu$-semistability with a weaker cohomological condition, Theorem 24 no
longer holds. We can prove this with the following counterexample. Let us
consider a Fano threefold $X$ of index $i_{X}=1$ and a sheaf $E\in Coh(X)$
defined by:

$0\rightarrow
E\rightarrow{\mathcal{O}}_{X}\oplus{\mathcal{O}}_{X}(-1)\rightarrow{\mathcal{O}}_{p}\rightarrow
0$

with $p\in X$ a point. $E$ is a $\mu$-unstable rank 2 torsion free (it is
destabilized by ${\mathcal{I}}_{p}$) such that $H^{i}(E)=0$ for $0\leq i\leq
2$, but $E^{\vee\vee}/E\simeq{\mathcal{O}}_{p}$.

######  Corollary 27.

Let $E$ be a rank 2 non-locally free instanton. Then the sheaf
$S_{E}:={\mathcal{E}}{\it xt}^{1}(E,{\mathcal{O}}_{X}(-r_{X}))$ is a rank 0
instanton.

###### Proof.

Whenever $E$ is not locally free, it fits in an exact sequence of the form
(11) and since $E^{\vee\vee}$ is locally free, we get an isomorphism:

${\mathcal{E}}{\it xt}^{1}(E,{\mathcal{O}}_{X}(-r_{X}))\simeq{\mathcal{E}}{\it
xt}^{2}(T_{E},{\mathcal{O}}_{X}(-r_{X})).$

Since $T_{E}$ is a rank 0 instanton, by Lemma 13, $T_{E}^{\rm
D}\otimes\omega_{X}^{-1}(-r_{X})$ is a rank 0 instanton as well. The statement
of the corollary then follows from the fact that we have an isomorphism
${\mathcal{E}}{\it xt}^{2}(T_{E},{\mathcal{O}}_{X}(-r_{X}))\simeq T_{E}^{\rm
D}\otimes\omega_{X}^{-1}(-r_{X})$. ∎

######  Remark 28.

If $E$ is a rank 2 non locally free instanton, we then have an equality
$\operatorname{Supp}(T_{E})=\operatorname{Sing}(E)$ for $\
T_{E}:=E^{\vee\vee}/E$. Note that this might not hold for arbitrary rank since
a priori $\operatorname{Supp}(T_{E})\subset\operatorname{Sing}(E)$ and
equality holds if and only if $E^{\vee\vee}$ is locally free.

Summing up these last results, we can affirm that in rank 2 an instanton sheaf
$E$ is either locally free or $\operatorname{Sing}(E)$ has pure dimension one
and $E$ is obtained by elementary transformation of an instanton bundle along
a rank 0 instanton supported on $\operatorname{Sing}(E)$. Elementary
transformation of rank 2 instanton bundles has been widely used in [23] to
construct and study families of non-locally free instantons on
$X={\mathbb{P}^{3}}$. Most importantly for the present paper, Faenzi made a
very interesting use of this construction in [11]: in loc. cit. elementary
trasnformation is indeed used to prove the existence of rank 2 instanton
bundle of charge $k,\forall\>k\geq 2$ on Fano threefolds of index 2. Mimicking
this approach, we can state the following theorem:

######  Theorem 29.

Let $X$ be a Fano threefold of index $i_{X}$. Assume that $F$ is a rank 2
locally free instanton sheaf, and $\ell\subset X$ is a line such that the
following hypotheses hold:

  * •

$F$ is unobstructed, i.e. $\operatorname{Ext}^{2}(F,F)=0$;

  * •

${\mathcal{N}}_{\ell/X}\simeq{\mathcal{O}}_{\ell}(q_{X}-1)\oplus{\mathcal{O}}_{\ell}(i_{X}-q_{X}-1)$
and $F|_{\ell}\simeq{\mathcal{O}}_{\ell}\oplus{\mathcal{O}}_{\ell}(-r_{X})$.

Then $X$ admits rank 2 locally free instanton sheaves of charge $k$ for every
$k\geq c_{2}(F)$.

###### Proof.

The induction argument presented in [11, Theoreom D] applies to any Fano
threefold $X$ carrying an instanton bundle $F$ and a line $\ell$ satisfying
the hypotheses of the theorem. We summarize here the main steps of the proof.
For any pair $(F,\ell)$ as above, we get the existence of an epimorphism
$\phi:F\twoheadrightarrow{\mathcal{O}}_{\ell}(q_{X}-1)$ whose kernel $E$ is a
non locally free instanton (as ${\mathcal{O}}_{\ell}(q_{X}-1)$ is a rank 0
instanton) of charge $c_{2}(F)+1$. One then proves that the assumptions made
on $(F,\ell)$ ensure then that $\operatorname{Ext}^{2}(E,E)=0$ and that a
general deformation of $E$ is an instanton bundle. This is done showing at
first that a general non-locally free deformation of $E$ is still an instanton
singular along a line and obtained from a deformation of $(\phi,F,\ell)$. We
then compute that the family of instantons whose singular locus is a line has
dimensions strictly less then $\operatorname{ext}^{1}(E,E)$. As a consequence
$E$ deforms to a locally free sheaf and by semicontinuity, a general
deformation is a non-obstructed instanton bundle. By induction we therefore
get existence of unobstructed instanton bundle of rank 2 for each charge $\geq
c_{2}(F)$. ∎

The theorem always applies on Fano threefold of index $i_{X}\geq 2$.

  * •

On $X\simeq{\mathbb{P}}^{3}$ a t’Hooft instanton of charge $1$ and a general
line lead to the existence of rank 2 instantons of charge $k$ for each $k\geq
1$. The same result is proved, with different techniques in [23].

  * •

For $i_{X}=3$ the spinor bundle and any line $\ell\subset X$ ensure the
existence of instanton bundles of charge $k\geq 1$.

  * •

The case $i_{X}=2$ was treated in [11] where the existence of instanton
bundles of charge $k\geq 2$ is proved.

Fano threefolds of index one for which the theorem applies are treated in [6,
Theorem 3.7].

We end the section characterizing the dimensions of the intermediate
cohomology groups $H^{i}(E(n)),\ i=1,2$ of rank 2 instantons (the groups
$H^{i}(E(n))$ for $i=0,3$ where studied in Lemma 8).

######  Lemma 30.

Let $E$ be a rank 2 instanton and let $m_{0}$ denote the smallest integer such
that $m_{0}H$ is very ample. Then the following hold:

  1. (1)

$H^{1}(E(-q_{X}-n))=0$ for $n=m_{0}$ and $\forall n\geq 2m_{0}$

  2. (2)

$H^{2}(E(-q_{X}+n))=0$ for $n=m_{0}$ and $\forall n\geq 2m_{0}$

###### Proof.

Let $D$ be a general element in the linear system $|m_{0}H|$. By generality
assumption $E|_{D}$ is $\mu$-semistable (see [28]), therefore
$H^{0}(E|_{D}(-q_{X}))=0$, and $H^{2}(E|_{D}(-q_{X}+m_{0}))\simeq
Hom(E|_{D},{\mathcal{O}}_{D}(-q_{X}-r_{X}))^{*}=0$. Taking cohomology in the
short exact sequence:

(15) $0\rightarrow E(-q_{X}-m_{0})\rightarrow E(-q_{X})\rightarrow
E|_{D}(-q_{X})\rightarrow 0$

we therefore get that $H^{1}(E(-q_{X}-m_{0}))=0$; twisting then (15) by
${\mathcal{O}}_{X}(m_{0})$ and taking cohomology, we obtain
$H^{2}(E(-q_{X}+m_{0}))$. These arguments apply to the letter to any general
divisor $D\in|nH|,n\geq 2m_{0}$ since under these assumption $D$ is very ample
and $E|_{D}$ is $\mu$-semistable. ∎

###  4.3. Stability of rank 2 instanton sheaves

Clearly, when $i_{X}$ is odd, every rank 2 instanton sheaf $E$ on $X$ is
$\mu$-stable, simply because $\mu$-stability coincides with
$\mu$-semistability when $c_{1}(E)=-1$.

In [21], the authors showed that every non trivial rank 2 instanton sheaf on
$X={\mathbb{P}^{3}}$ is stable, so that
${\mathcal{O}}_{\mathbb{P}^{3}}^{\oplus 2}$ is the only properly semistable
(meaning semistable but not stable) rank 2 instanton sheaf. In addition, a
rank 2 instanton sheaf is properly $\mu$-semistable only when
$E^{\vee\vee}={\mathcal{O}}_{\mathbb{P}^{3}}^{\oplus 2}$.

The situation is quite different for Fano threefolds of index 2. Indeed, let
$E$ be a properly $\mu$-semistable rank 2 instanton sheaf on a Fano threefold
$X$ with $i_{X}=2$. When $E$ is locally free, this is equivalent to say that
$h^{0}(E)>0$; choosing a non trivial section $s\in H^{0}(E)$, we obtain the
exact sequence

(16) $0\longrightarrow{\mathcal{O}}_{X}\longrightarrow
E\longrightarrow{\mathcal{I}}_{C}\longrightarrow 0,$

and it is easy to check that ${\mathcal{I}}_{C}$ is a rank one instanton,
meaning that $C$ is an instanton curve; recall that the latter have been
classified in Lemma 15. Summing up, we proved the following claim.

######  Lemma 31.

Let $E$ be a rank 2 locally free instanton sheaf on a Fano threefold $X$ of
index $i_{X}=2$. If $E$ is properly $\mu$-semistable, then $E$ fits into an
exact sequence as in display (16) where $C$ is an instanton curve. In
particular, such sheaves are not semistable.

When $E$ is not locally free, we have that $E^{\vee\vee}$ is a properly
$\mu$-semistable locally free instanton sheaf of rank 2; taking the unique (up
to scalar multiple) non trivial section $s\in H^{0}(E^{\vee\vee})$, we get the
following commutative diagram

(17)
${0}$${0}$${{\mathcal{I}}_{C_{1}}}$${{\mathcal{O}}_{X}}$${0}$${E}$${E^{\vee\vee}}$${T}$${0}$${0}$${{\mathcal{I}}_{C_{2}}}$${{\mathcal{I}}_{C}}$${\operatorname{coker}(\phi)}$${0}$${0}$${0}$$\scriptstyle{s}$$\scriptstyle{\phi}$$\scriptstyle{q}$$\scriptstyle{Id}$

Here $\phi=q\circ s$; clearly, the kernel of $\phi$ is the ideal sheaf of a
pure 1-dimensional scheme, which we denote by $C_{1}$; remark that if $q=0$,
then $C_{1}$ is empty. Since the cokernel of the inclusion
${\mathcal{I}}_{C_{1}}\hookrightarrow E$ must also be torsion free, we
complete the leftmost column. Note that

(18)
$H^{1}({\mathcal{I}}_{C_{1}}(-1))=H^{2}({\mathcal{I}}_{C_{2}}(-1))=0\leavevmode\nobreak\
\leavevmode\nobreak\ \Longleftrightarrow\leavevmode\nobreak\
\leavevmode\nobreak\
H^{0}({\mathcal{O}}_{C_{1}}(-1))=H^{1}({\mathcal{O}}_{C_{2}}(-1))=0.$

######  Lemma 32.

Let $E$ be a rank 2 instanton sheaf on a Fano threefold of index 2. If $E$ is
properly semistable, then $E$ is S-equivalent to
${\mathcal{I}}_{C_{1}}\oplus{\mathcal{I}}_{C_{2}}$ where $C_{1}$ and $C_{2}$
are instanton curves of the same degree.

###### Proof.

As we have seen above, the hypothesis imply that $E$ must be an extension of
ideal sheaves ${\mathcal{I}}_{C_{1}}$ and ${\mathcal{I}}_{C_{2}}$ satisfying

$\chi({\mathcal{I}}_{C_{j}}(t))=\dfrac{1}{2}\chi(E(t))=\chi({\mathcal{O}}_{X}(t))-\dfrac{c_{2}(E)}{2}(t+1).$

In particular, $\chi({\mathcal{I}}_{C_{j}}(-1))=0$; the vanishings in display
(18) imply that $C_{1}$ and $C_{2}$ must be instanton curves. ∎

From Lemma 32 we deduce the following corollaries.

######  Corollary 33.

There are no properly semistable rank 2 instanton of odd charge.

###### Proof.

Each semistable instanton $E$ is S-equivalent to a sheaf of the form
${\mathcal{I}}_{C_{1}}\oplus{\mathcal{I}}_{C_{i}}$ where $C_{1}$ and $C_{2}$
are instanton curves of the same degree $d$; thus $c_{2}(E)=2d$. ∎

######  Corollary 34.

There exists no properly semistable locally free instanton sheaves of charge
$>0$.

###### Proof.

If $E$ is semistable and $c_{2}>0$ then $E$ fits in

$0\rightarrow{\mathcal{I}}_{C_{1}}\rightarrow
E\rightarrow{\mathcal{I}}_{C_{2}}\rightarrow 0$

with $C_{i}$ being instanton curves of degree
$\deg(C_{1})=\deg(C_{2})=\frac{c_{2}(E)}{2}.$ From this short exact sequence
we compute that $E$ has depth 2 along all points $x\in C_{1}$ hence $E$ can
not be locally free. ∎

At the beginning of the section we observed that there are no properly
$\mu$-semistable non locally-free instanton sheaves on Fano threefolds of odd
index (since on these varieties $\mu$-semistable rank 2 bundles are
$\mu$-stable) and that each properly $\mu$-semistable non locally instanton
$E$ on ${\mathbb{P}}^{3}$ satisfies
$E^{\vee\vee}\simeq{\mathcal{O}}_{{\mathbb{P}}^{3}}^{\oplus 2}$. The Fano
threefolds $X$ of index $i_{X}=2$ are the only ones carrying families of
properly $\mu$-semistable non locally free instanton sheaves $E$ such that
$c_{2}(E^{\vee\vee})>0$. Moreover, even if every properly $\mu$-semistable
instanton bundle $F$ with $c_{2}(F)>0$ is Gieseker unstable, the instanton
sheaves $E$ obtained as elementary transformation of $F$ along rank 0
instantons might be semistable or even stable.

######  Lemma 35.

Let $(F,T,q)$ be, respectively, a properly $\mu$-semistable rank 2 instanton
bundle $F$ of charge $n>0$ a rank 0 instanton $T$ of multiplicity $d$ and an
epimorphism $q:F\twoheadrightarrow T$. Let $E$ be the sheaf defined as
$E:=\ker(q)$. Then $E$ is stable, resp. properly semi-stable, if and only if
$\forall s\in H^{0}(F),\ s\neq 0$, $ch_{2}(\operatorname{im}(q\circ
s))>\frac{n+d}{2}$, resp. if and only if $T$ is an instanton curve and
$\forall s\in H^{0}(F),\>s\neq 0$ $\operatorname{im}(q\circ s)$ is an
instanton curve of degree $\frac{n+d}{2}$.

###### Proof.

As usual we start considering the short exact sequence

$0\rightarrow E\rightarrow F\xrightarrow{q}T\rightarrow 0;$

note that the charge of $E$ is $n+d$.

Considering a diagram analogous to the one in display (17), we see that every
subsheaf of $E$ with trivial determinant is the ideal sheaf
${\mathcal{I}}_{B}$ of a scheme $B$ such that
${\mathcal{O}}_{B}=\operatorname{im}(q\circ s)$ for a (hence for each) non
zero section $s\in H^{0}(F)\simeq{\mathbb{C}}$. Denoting by $d^{\prime}$ and
$x$ the degree and the Euler characteristic of the curve $B$, respectively, we
have:

$\dfrac{1}{2}P_{E}(t)-P_{{\mathcal{I}}_{B}}(t)=-\dfrac{n+d}{2}(t+1)+d^{\prime}t+x=t(d^{\prime}-\frac{(n+d)}{2})+x^{\prime}-\frac{(n+d)}{2}$

It is therefore clear that if ever $d^{\prime}>\frac{n+d}{2}$, then $E$ is
stable whilst $d^{\prime}<\frac{n+d}{2}$ leads to the unstability of $E$. In
the case $d^{\prime}=\frac{(n+d)}{2}$, we can never have stability since
$x^{\prime}\leq\frac{(n+d)}{2}$ and equality occurs if and only if $B$ is an
instanton curve (or, equivalently, a degree $d^{\prime}$ line arrangement).
Indeed for a l.c.m. curve $B$ of degree $d^{\prime}$, since
$h^{0}({\mathcal{O}}_{B}(-1))=0$,
$\chi({\mathcal{O}}_{B}(-1))=-d^{\prime}+x^{\prime}\leq 0$ and equality holds
if and only if $B$ is an instanton curve.

This shows that $E$ is stable, resp. properly semistable, if and only if for
each non-zero global section $s\in H^{0}(F),\ ch_{2}(\operatorname{im}(q\circ
s))>\frac{n+d}{2}$ (resp. $ch_{2}(\operatorname{im}(q\circ s))=\frac{n+d}{2}$
and $\operatorname{im}(q\circ s)$ is an instanton curve). To conclude the
proof of the proposition we still need to show that if $E$ is properly
semistable then $T$ itself is an instanton curve. Once again we consider the
diagram (17) induced by $s\in H^{0}(F)$ (recall that $F\simeq E^{\vee\vee}$);
since $\operatorname{im}(q\circ s)$ is an instanton curve,
$\operatorname{coker}(q\circ s)$ is an instanton as well; moreover by the fact
that ${\mathcal{I}}_{C}$ surjects onto $\operatorname{coker}(q\circ s)$, we
deduce that $\operatorname{coker}(q\circ s)$ must as well be isomorphic to
${\mathcal{O}}_{B^{\prime}}$ with $B^{\prime}$ an instanton curve of degree
$\frac{(d-n)}{2}$. This last assertion is due to the fact that a rank 0
instanton $T^{\prime}$ on a l.c.m. curve $C$ has always degree (as a
${\mathcal{O}}_{C}$-module) $\deg(C)-\chi({\mathcal{O}}_{C})\geq 0$ with
equality holding if and only if $C$ is an instanton curve (indeed whenever
$P_{C}(t)=\deg(C)t+\deg(C)$, $h^{0}({\mathcal{O}}_{C}(-1))=0$ implies
$h^{1}({\mathcal{O}}_{C}(-1))=0$) and $T^{\prime}\simeq{\mathcal{O}}_{C}$.
Therefore $T$ is an instanton curve extension of ${\mathcal{O}}_{B^{\prime}}$
by ${\mathcal{O}}_{B}$. ∎

###  4.4. Instantons via Serre

Let $X$ be a Fano threefold of Picard rank 1, index $i_{X}$ and take a rank 2
instanton sheaf $E$ of charge $c_{2}$. Following the Serre correspondence
outlined in Section 3, we choose a section $s\in H^{0}(E(n))$ with torsion-
free cokernel and obtain a short exact sequence

$0\longrightarrow{\mathcal{O}}_{X}(-n)\longrightarrow
E\longrightarrow{\mathcal{I}}_{C}(n-r_{X})\longrightarrow 0$

which yields to a l.c.m. curve $C\subset X$; its arithmetic genus $p_{a}(C)$
and the degree $d$ are given by

(19) $\displaystyle p_{a}(C)$
$\displaystyle=1-[(d_{X}n^{2}-d_{X}nr_{X}+c_{2})(\frac{i_{X}}{2}+r_{X}-n)+\frac{1}{2}r_{X}(nr_{X}-n^{2}-c_{2})]$
(20) $\displaystyle d$ $\displaystyle=d_{X}n^{2}-d_{X}nr_{X}+c_{2};$

moreover, its sheaf of ideals will satisfy the cohomological conditions:

(21) $H^{0}({\mathcal{I}}_{C}(n-r_{X}-1))=0,\
H^{i}({\mathcal{I}}_{C}(n-r_{X}-q_{X}))=0,\ i=1,2.$

Our aim now is to characterize in detail those curves that ”Serre correspond”
to non locally free instanton sheaves of rank 2.

We consider therefore a non locally free instanton sheaf $E$ and we let $n$ be
a non-negative integer such that $h^{0}(E(n)>0$; take $s\in H^{0}(E(n))$ such
that $\operatorname{coker}(s)$ is a torsion free sheaf, and let
$s^{\vee\vee}\in H^{0}(E^{\vee\vee}(n))$ be the image of $s$ via the injective
map $H^{0}(E(n))\hookrightarrow H^{0}(E^{\vee\vee}(n)).$ According to the
argument just below the diagram in display (4), we obtain the following
commutative diagram

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where $C$ and $C^{\prime}$ are the curves corresponding to the pairs $(E,s)$
and $(E^{\vee\vee},s^{\vee\vee})$, respectively; Lemma 5 guarantees that
$C^{\prime}$ is l.c.m. To figure out the associated extension classes, we note
that the short exact sequence in display (5) can be rewritten as follows

(23)
$0\rightarrow\omega_{C^{\prime}}(r_{X}+i_{X}-2n)\rightarrow\omega_{C}(r_{X}+i_{X}-2n)\rightarrow{\mathcal{E}}{\it
xt}^{2}(T_{E}(n),{\mathcal{O}}_{X})\rightarrow 0.$

######  Lemma 36.

If $(C,\xi)$ corresponds to a pair $(E(n),s)$ where $E$ is a non locally free
instanton sheaf of rank 2, and $n$ is a non negative integer, then the pair
$(C^{\prime},\xi^{\prime})$ corresponding to $(E^{\vee\vee}(n),s^{\vee\vee})$
satisfies the following conditions:

  1. (1)

$0\to{\mathcal{I}}_{C}\to{\mathcal{I}}_{C^{\prime}}\to T_{E}(r_{X}-n)\to 0$;

  2. (2)

$\xi$ is the image of $\xi^{\prime}\in
H^{0}(\omega_{C^{\prime}}(i_{X}+r_{X}-2n))$ under the inclusion:

$0\rightarrow
H^{0}(\omega_{C^{\prime}}(i_{X}+r_{X}-2n))\xrightarrow{\iota}H^{0}(\omega_{C}(i_{X}+r_{X}-2n))$

###### Proof.

The first item is just the bottom line of diagram in display (22). As for the
second item, applying ${\mathcal{H}}{\it om}(\ \cdot\
,{\mathcal{O}}_{X}(r_{X}-2n))$ to the short exact sequence of point $(1)$ and
taking global section, we obtain:

(24) $0\rightarrow
H^{0}(\omega_{C^{\prime}}(i_{X}+r_{X}-2n))\xrightarrow{\iota}H^{0}(\omega_{C}(i_{X}+r_{X}-2n))\to
H^{0}({\mathcal{E}}{\it xt}^{2}(T_{E}(n),{\mathcal{O}}_{X}))$

where the r.s.t and the middle term are isomorphic to
$\operatorname{Ext}^{1}({\mathcal{I}}_{C^{\prime}}(2n-r_{X}),{\mathcal{O}}_{X})$
and $\operatorname{Ext}^{1}({\mathcal{I}}_{C}(2n-r_{X}),{\mathcal{O}}_{X})$,
respectively. Note now that, more generally, given $\zeta^{\prime}\in
H^{0}(\omega_{C^{\prime}}(i_{X}+r_{X}-2n))$ corresponding to
$0\rightarrow{\mathcal{O}}_{X}\xrightarrow{s^{\prime}}F^{\prime}\rightarrow{\mathcal{I}}_{C}^{\prime}(2n-r_{X})\rightarrow
0$, applying the snake Lemma we obtain a commutative diagram:

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In particular, $F^{\prime}\simeq E^{\prime\vee\vee},$
$s^{\prime}=r^{\vee\vee}$ and $\iota(\zeta^{\prime})\in
H^{0}(\omega_{C}(i_{X}+r_{X}-2n))$ corresponds to the short exact sequence
defined by the first column of the diagram. Accordingly
$\xi=\iota(\xi^{\prime})$ which also implies that $\xi$ lies in the kernel of
the morphism $H^{0}(\omega_{C}(i_{X}+r_{X}-2n))\to H^{0}({\mathcal{E}}{\it
xt}^{2}(T_{E}(n),{\mathcal{O}}_{X}))$. ∎

######  Remark 37.

Note that whenever we are given a pair of l.c.m curve $C,\>C^{\prime}$ whose
sheaves of ideals fit $0\to{\mathcal{I}}_{C}\to{\mathcal{I}}_{C^{\prime}}\to
T_{E}(r_{X}-n)\to 0$, we obtain a short exact sequence like the one in display
(23) and, taking global sections, a short exact sequence like the one in
display (24). In particular, if $n\geq r_{X}+q_{X}$, then
$H^{0}({\mathcal{E}}{\it xt}^{2}(T_{E}(n),{\mathcal{O}}_{X}))=0$ (recall that
${\mathcal{E}}{\it xt}^{2}(T,{\mathcal{O}}_{X}(-r_{X}))$ is a rank 0
instanton) which means that there are no instanton bundles corresponding to
the curve $C$ and every $\xi\in H^{0}(\omega_{C}(r_{X}+i_{X}-2n))$ corresponds
to a non locally free instanton sheaf that is singular along
$\operatorname{Supp}(T_{E})$. The only cases in which a curve can correspond
both to a locally free and and to a non locally free instanton sheaf occur
therefore for $i_{X}=3,4$ and $n=1$. Since a curve $C$ os such a kind
satisfies (cf. 21)
$H^{i}({\mathcal{I}}_{C}(-1))=H^{i-1}({\mathcal{O}}_{C}(-1))=0,\ i=1,2$, from
remark 17, we see that the only curves corresponding both locally free and
non-locally free rank 2 instantons of charge $n$ are the lines arrangement of
degree $n+1-r_{X}$ (degree of $C$ is computed applying 19). An example of a
family of instanton bundles corresponding to a degree $n$ line arrangement $C$
and degenerating to a non locally free instanton, still corresponding to $C$
was exhibited in example 10.

Next, we consider the reverse construction: let $(C^{\prime},\xi^{\prime})$ be
a pair consisting of a l.c.i curve $C$ satisfying

$H^{i}({\mathcal{I}}_{C^{\prime}}(n-r_{X}-q_{X}))=0,\ i=1,2,$

and a nowhere vanishing section $\xi^{\prime}\in
H^{0}(\omega_{C^{\prime}}(i_{X}+r_{X}-2n))$. Considering the short exact
sequence

(26) $0\longrightarrow{\mathcal{O}}_{X}(-n)\stackrel{{\scriptstyle
r}}{{\longrightarrow}}F\longrightarrow{\mathcal{I}}_{C^{\prime}}(n-r_{X})\longrightarrow
0$

given by regarding $\xi^{\prime}$ as a class in
$\operatorname{Ext}^{1}({\mathcal{I}}_{C^{\prime}}(n-r_{X}),{\mathcal{O}}_{X}(-n))$
and the second part of Remark 4, it follows that the rank 2 sheaf $F$ in the
corresponding pair $(F(n),r)$ is a locally free instanton sheaf.

######  Lemma 38.

Any pair $(C,\xi)$ consisting of

  1. (1)

a l.c.m. curve $C$ containing $C^{\prime}$ such that
$T:={\mathcal{I}}_{C^{\prime}}(n-r_{X})/{\mathcal{I}}_{C}(n-r_{X})$ is a rank
0 instanton sheaf;

  2. (2)

a section $\xi\in H^{0}(\omega_{C}(i_{X}+r_{X}-2n))$ lying in the image of the
induced map

$H^{0}(\omega_{C^{\prime}}(i_{X}+r_{X}-2n))\hookrightarrow
H^{0}(\omega_{C}(i_{X}+r_{X}-2n))$

corresponds to a pair $(E(n),s)$ where $E$ is a non locally free rank 2
instanton sheaf which is singular along $\operatorname{Supp}(T)$.

###### Proof.

The sequence in display 26 and the way the curve $C$ is chosen provides us
with the diagram

(27) $\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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The sheaf $E:=\ker(q)$ satisfies two short exact sequences; the first one

$0\longrightarrow E\longrightarrow F\longrightarrow T\longrightarrow 0$

implies that $E$ is an instanton sheaf, while the second

$0\longrightarrow{\mathcal{O}}_{X}(-n)\longrightarrow
E\longrightarrow{\mathcal{I}}_{C}(n-r_{X})\longrightarrow 0$

induces the section $\xi\in H^{0}(\omega_{C}(i_{X}+r_{X}-2n))$ which vanishes
on $\operatorname{Supp}(T)$, and therefore lies in the image of the map given
in the statement of the lemma. ∎

The reason why we chose to portray in detail how the Serre correspondence
works for rank 2 instantons is simply due to the fact that we are mainly
concerned with moduli spaces of rank 2 sheaves. Of course these arguments can
be generalized to instantons of arbitrary rank. Doing so we can in particular
provide examples of non-locally free reflexive instantons of rank $>2$ (we
recall indeed that in rank 2 the reflexivity of instantons is equivalent to
their local freeness).

######  Example 39.

We can construct a non locally free reflexive instanton on $X$ being either
${\mathbb{P}^{3}}$ or a quadric threefold as follows. Let $C\subset X$ be a
smooth rational curve of degree $d\geq 4$ and we consider two linearly
independent sections $\xi_{i}\in H^{0}({\mathcal{O}}_{C}((d-2){\rm pt})),\
i=1,2$, whose zero loci intersect along a 0-dimensional scheme $Z\subset C$ of
length $d^{\prime}<d-2$. Since ${\mathcal{O}}_{C}((d-2){\rm
pt})\simeq\omega_{C}(1)$, these two sections correspond to an extension class
in
$\operatorname{Ext}^{1}({\mathcal{I}}_{C}(2-r_{X}),{\mathcal{O}}_{X}(-1)^{\oplus
2})$, thus giving rise to a short exact sequence

(28) $0\rightarrow{\mathcal{O}}_{X}(-1)^{\oplus 2}\rightarrow
E\rightarrow{\mathcal{I}}_{C}(2-r_{X})\rightarrow 0;$

we argue that the middle term $E$ is the sheaf we are looking for.

Indeed, the exact sequence in display (28) yield

$H^{i}(E(-q_{X}))\simeq H^{i}({\mathcal{I}}_{C})=0,\ \forall\>i.$

Dualizing the same exact sequence and recalling that ${\mathcal{E}}{\it
xt}^{1}({\mathcal{I}}_{C}(2-r_{X}),{\mathcal{O}}_{X})\simeq\omega_{C}(2)$, we
obtain:

$0\rightarrow{\mathcal{O}}_{X}(r_{X}-2)\rightarrow
E^{\vee}\rightarrow{\mathcal{O}}_{X}(1)^{\oplus
2}\xrightarrow{\xi}\omega_{C}(2)\rightarrow{\mathcal{E}}{\it
xt}^{1}(E,{\mathcal{O}}_{X})\rightarrow 0$

where the morphism $\xi$ is defined by the two sections $(\xi_{1},\xi_{2})\in
H^{0}(\omega_{C}(1)^{\oplus 2})$ we started with. By construction, $\xi$ fails
to be surjective along $Z$, so that ${\mathcal{E}}{\it
xt}^{1}(E,{\mathcal{O}}_{X})$ is supported on $Z$. This, together with the
vanishing of ${\mathcal{E}}{\it xt}^{i}(E,{\mathcal{O}}_{X})$ for $i>1$
implies that $E$ is reflexive with $\operatorname{Sing}(E)=Z$, thus non
locally free.

Finally, to see that $E$ is $\mu$-semistable it is enough to check that
$H^{0}(E(r_{X}-1))=H^{0}(E^{\vee}(r_{X}-1))=0$, cf. [30]. $H^{0}(E(r_{X}-1))$
vanishes since it is isomorphic to $H^{0}({\mathcal{I}}_{C}(1-2r_{X}))=0$
(this is clearly zero for $r_{X}=1$ whilst for $r_{X}=0$ it is ensured by the
fact that $C$ is not a planar curve). Setting $F:=\ker(\xi)$, note that
$H^{0}(E^{\vee}(r_{X}-1))=H^{0}(F(r_{X}-1))$, and that the latter coincides
with the kernel of the induced map

$H^{0}({\mathcal{O}}_{X}(r_{X})^{\oplus
2})\stackrel{{\scriptstyle(\xi_{1},\xi_{2})}}{{\longrightarrow}}H^{0}(\omega_{C}(r_{X}+1)),$

given by multiplication by the sections $\xi_{i},\ i=1,2$.

When $r_{X}=0$ (ie. $X={\mathbb{P}^{3}}$) the fact that $\xi_{i}$ are linearly
independent is enough to guarantee that this map is injective, thus
$H^{0}(F(r_{X}-1))=0$, as desired.

If $r_{X}=1$, one must argue that $(\xi_{1},\xi_{2})$ does have not a syzygy
$(\sigma_{1},\sigma_{2})\in H^{0}({\mathcal{O}}_{C}(d\leavevmode\nobreak\ {\rm
pt}))$ of degree $d$ that lies in the image of the restriction map

$H^{0}({\mathcal{O}}_{X}(1)^{\oplus 2})\to H^{0}({\mathcal{O}}_{C}(1)^{\oplus
2})\simeq H^{0}({\mathcal{O}}_{C}(d\leavevmode\nobreak\ {\rm pt})).$

This seems to be a generic condition when $d-d^{\prime}$ is sufficiently
large, but we have not been able to prove it.

##  5\. Instanton sheaves on quadric threefolds

Let $V$ be a 5-dimensional vector space and consider a smooth quadric
hypersurface $X\subset{\mathbb{P}}(V)\simeq{\mathbb{P}^{4}}$. $X$ is the only
Fano 3-fold of Picard rank one and index 3, therefore an instanton sheaf $E$
on $X$ is defined as a torsion free $\mu$-semistable sheaf with $c_{1}(E)=-1$
and such that:

(29) $H^{i}(E(-1))=0,\ i=1,2.$

Recall that since $c_{1}(E)$ is odd, every instanton sheaf on $X$ is actually
$\mu$-stable; this ensures the vanishing of $H^{i}(E(-1))$ for $i=0,3$ as well
(cf. Lemma 8). From now on we will only be concerned with instanton sheaves of
rank 2 (therefore when referring to an instanton sheaf we will always imply
that its rank is 2).

The Chern character of a rank 2 instanton $E$ of charge $n$ is:

(30) ${\rm ch}(E)=\left(2,-[H],\leavevmode\nobreak\
(1-n)[l],\leavevmode\nobreak\ \frac{-1}{3}+\frac{n}{2}\right)$

(by Corollary 9, $c_{3}(E)=0$); applying Riemann–Roch, we compute the Hilbert
polynomial of $E$:

(31) $P_{E}(t)=\frac{2}{3}t^{3}+2t^{2}+\left(\frac{7}{3}-n\right)t+(1-n)$

In this section we present some results on instanton sheaves on $X$. We will
focus our attention on instanton sheaves $E$ of charge 2, emphasising the
relation that these sheaves have with the curves corresponding to global
sections of $E(1)$ via Serre correspondence. The Serre correspondence allows
us not only to describe the instanton moduli space, but to obtain also a
complete picture of the entire Gieseker–Maruyama moduli space
$\mathcal{M}:=\mathcal{M}_{X}(2,-1,2,0)$ of semistable rank 2 sheaves with
Chern classes $(c_{1},c_{2},c_{3})=(-1,2,0)$, together with its relation with
the Hilbert scheme $\operatorname{Hilb}_{2t+2}(X)$.

###  5.1. Instanton sheaves of charge 1

Since every instanton sheaf $E$ is a $\mu$-semistable sheaf with
$c_{1}(E)=-1$, the Bogomolov inequality implies that $c_{2}(E)\geq 1$. In the
case $c_{2}=1$, a well known example of instanton bundle is provided by the so
called spinor bundle which will be henceforth denoted by ${\mathcal{S}}$.

Recall that it can be defined as follows, cf. [31, Definition 1.3] for which
we refer to for all details in this paragraph. There is an embedding
$s:X\to\mathbb{G}(1,3)$, the grassmannian of lines in ${\mathbb{P}^{3}}$, see
; then ${\mathcal{S}}:=s^{*}U$, where $U$ is the universal bundle on
$\mathbb{G}(1,3)$. This is a $\mu$-stable rank 2 bundle on $X$ with
$c_{1}({\mathcal{S}})=-1$ and $c_{2}({\mathcal{S}})=1$. In addition,
${\mathcal{S}}$ is rigid [31, Theorem 2.1], and every $\mu$-stable rank 2
bundle $E$ on $X$ with $c_{1}(E)=-1$ and $c_{2}(E)=1$ is isomorphic to the
spinor bundle ${\mathcal{S}}$.

Since $h^{0}({\mathcal{S}}(1))=4$, Serre construction provides the following
short exact sequence

(32)
$0\longrightarrow{\mathcal{O}}_{X}(-1)\longrightarrow{\mathcal{S}}\longrightarrow{\mathcal{I}}_{\ell}\longrightarrow
0$

where $\ell$ is a line in $X$. It is then easy to see that ${\mathcal{S}}$ is
a rank 2 instanton bundle of charge 1. This observation allows us to give the
following characterization of the family $F(X)$ of lines on $X$. Since
$h^{0}({\mathcal{S}})=0$ (by stability) we deduce then that $\forall s\in
H^{0}({\mathcal{S}}(1)),\ s\neq 0$, $\operatorname{coker}(s)$ is torsion free
and isomorphic to ${\mathcal{I}}_{l}(1)$ for a line $l\subset X$ (this last
assertion follows from a simple Chern character computation) . Conversely,
$\forall\>[l]\in F(X)$, every sheaf fitting in a non-split short exact
sequence of the form (32) is a $\mu$-stable vector bundle with $c_{1}=-1,\
c_{2}=1$ and is therefore isomorphic to ${\mathcal{S}}$. Accordingly
$F(X)\simeq{\mathbb{P}}^{3}\simeq{\mathbb{P}}(H^{0}({\mathcal{S}}(1))$.

######  Proposition 40.

Every rank 2 instanton sheaf of charge $1$ on $X$ is isomorphic to the spinor
bundle.

###### Proof.

Let $E$ be a rank 2 instanton sheaf of charge $1$. If $E$ is reflexive, then
it must be locally free and therefore it is isomorphic to the spinor bundle.

If $E$ is not reflexive, then Theorem 24 implies that $E^{\vee\vee}$ is a
locally free instanton sheaf of charge $c_{2}(E^{\vee\vee})\geq 1$. However,
$c_{2}(E^{\vee\vee})+\deg(T_{E})=c_{2}(E)=1$, thus in fact
$c_{2}(E^{\vee\vee})=1$ and $\deg(T_{E})=0$. It follows that $T_{E}=0$,
contradicting the hypothesis that $E$ was not reflexive. ∎

The following result will also be useful later on.

######  Lemma 41.

Every $\mu$-stable rank 2 reflexive sheaf with Chern classes $c_{1}=-1$ and
$c_{2}=1$ is isomorphic to the spinor bundle.

###### Proof.

Let $F$ be a $\mu$-stable rank 2 reflexive sheaf with $c_{1}(F)=-1$ and
$c_{2}(F)=1$, so that

$\chi(F)=\dfrac{c_{3}(F)}{2}=h^{2}(F)-h^{1}(F)$

since $h^{0}(F)=h^{3}(F)$ by $\mu$-stability. We claim that $h^{1}(F(n))=0,\
\forall n\leq 0$.

Indeed, take a general hyperplane section $Q\in|{\mathcal{O}}_{X}(1)|$ and
consider the restriction sequence

(33) $0\longrightarrow F(-1)\longrightarrow F\longrightarrow
F|_{Q}\longrightarrow 0,$

with $F|_{Q}$ being a $\mu$-semistable locally free sheaf on $Q$ with
$c_{1}(F|_{Q})=(-1,-1)$. The $\mu$-semistability of $F|_{Q}$ leads to the
vanishing of $H^{0}(F|_{Q}(n)),\ \forall n\leq 0$ and of $H^{2}(F|_{Q}(n)),\
\forall n\geq-1$. In addition, since $\chi(F|_{Q})=1-c_{2}(F)=0$, we conclude
that $h^{1}(F|_{Q})=0$; Serre duality then implies that $h^{1}(F|_{Q}(-1))=0$.
From the fact that $h^{1}(F|_{Q})=h^{2}(F|_{Q}(-1))=0$, we deduce that
$F|_{Q}$ is 1-regular which implies that $h^{1}(F|_{Q}(n))=0,\ \forall n\geq
0$. Since by Serre duality $h^{1}(F|_{Q}(n))=h^{1}(F|_{Q}(-n-1))$, we can
finally conclude that $h^{1}(F|_{Q}(n))=0,\ \forall n\in{\mathbb{Z}}$.
Twisting the sequence in display (33) and taking cohomology, we thus get
$h^{1}(F(n))=h^{1}(F(n+1)),\ \forall n\leq-1$; but from the reflexivity of
$F$, $H^{1}(F(n))=0$ for $n\ll 0$ hence $h^{1}(F(n))=0,\ \forall n\leq 0$, as
desired.

It follows that $h^{2}(F)=\dfrac{c_{3}(F)}{2}$; since $h^{i}(F|_{Q}(n))=0$ for
$i=1,2$ and $\forall n\geq-1$ we also get, inductively, that
$h^{2}(F(n))=h^{2}(F(n+1))$ $\forall n\geq-2$ so that
$h^{2}(F(n))=\dfrac{c_{3}(F)}{2},\ \forall n\geq-2$. By the Serre vanishing
theorem, we must have that $h^{2}(F(n))=0$ when $n\gg 0$, thus in fact
$h^{2}(F(n))=0$ for every $n\geq-2$, and hence $c_{3}(F)=0$, implying that $F$
must be locally free. But every rank 2 locally free sheaves with $c_{1}=-1$
and $c_{2}=1$ on $X$ is a spinor bundle. ∎

We end this preliminary section summoning some properties of $F(X)$, the
family of lines on $X$. We have already recalled that
$F(X)\simeq{\mathbb{P}}^{3}\simeq{\mathbb{P}}(H^{0}({\mathcal{S}}(1))$. One
”geometric” way to realize $F(X)$ as ${\mathbb{P}^{3}}$ is the following. We
start by constructing $X$ as a hyperplane section of the Grassmannian
$\mathbb{G}(1,3)\subset{\mathbb{P}^{5}}$ of lines in ${\mathbb{P}^{3}}$.
Recall now that we have 2 families of planes contained in $\mathbb{G}(1,3)$:
we have planes corresponding to families of lines passing trough a point (we
call them planes of type I), and planes parameterizing families of lines
contained in a plane ${\mathbb{P}}^{2}\subset{\mathbb{P}^{3}}$ (these will be
referred to as planes of type II). For each line $l\subset X$ there exists a
unique pair of planes $(\Delta_{I},\ \Delta_{II})$ with $\Delta_{I}$ of type
$I$, $\Delta_{II}$ of type $II$, containing $l$; these planes are both
parameterized by a 3-dimensional linear space ${\mathbb{P}^{3}}$.

Several of our next results will rely on the geometry of linear spaces of
lines; for this reason we recall here briefly some of their fundamental
property. We have two families of pencils of lines in $F(X)$. Consider indeed
a pencil ${\mathbb{P}^{1}}\subset F(X)$ and denote by $l_{0},\ l_{1}$ a pair
of generators. If ever $l_{0}\cap l_{1}=\emptyset$, then the entire
${\mathbb{P}^{1}}$ is a ruling in the quadric surface $Q:=\langle
l_{0},l_{1}\rangle\cap X$. In particular we must have that any pair of lines
in ${\mathbb{P}^{1}}$ are disjoint hence $Q$ must be smooth since we have no
disjoint lines in a singular hyperplane section of $X$. For the same reason we
deduce that if ever $l_{0}\cap l_{1}\neq\emptyset$, then any pair of lines in
${\mathbb{P}^{1}}$ must intersect so that this family must coincide we the
family of lines on a singular hyperplane section of $X$.

This implies in particular that these lines all have the form $\overline{qp}$
with $p$ fixed and $q$ varying along a conic. From these observations we
deduce that there exists a morphism

$\mathbb{G}(1,F(X))\xrightarrow{\gamma}{\mathbb{P}^{4}}^{*}$

and that moreover, denoting by
${\mathbb{P}^{4}}^{*}_{sm}:={\mathbb{P}^{4}}^{*}\setminus X^{*}$ the open of
smooth hyperplane sections and by
${\mathcal{U}}:=\gamma^{-1}({\mathbb{P}^{4}}^{*}_{sm})$,
$\gamma|_{{\mathcal{U}}}$ is a degree 2 covering over
${\mathbb{P}^{4}}^{*}_{sm}$.

Finally we recall that $\forall\ l\subset X$ we have a hyperplane
${\mathbb{P}^{2}}\subset F(X)$ of lines meeting $l$ (isomorphic to the family
of planes in ${\mathbb{P}^{4}}$ containing $l$) and that all the hyperplanes
in $F(X)$ are of this form.

###  5.2. Instantons of charge 2

Our study of the moduli space ${\mathcal{M}}$ starts with the study of
${\mathcal{L}}(2)$, the open subscheme parameterizing rank 2 instanton sheaves
of charge 2. We will prove the following:

######  Theorem 42.

${\mathcal{L}}(2)$ is a smooth, irreducible, 6-dimensional open subscheme of
${\mathcal{M}}$ whose general element is a locally free instanton sheaf. Its
closure $\overline{{\mathcal{L}}(2)}$ is an irreducible component of
${\mathcal{M}}$. The moduli space ${\mathcal{M}}$ is smooth along
${\mathcal{L}}(2)$.

The moduli space ${\mathcal{I}}(2)$ of locally free instanton sheaves of
charge 2 was studied in [32]. In loc. cit the authors proved the following.

######  Theorem 43.

[32, Theorem 4.1] The moduli space ${\mathcal{I}}(2)$ is locally a trivial
algebraic fibration over $({\mathbb{P}^{4}})^{*}_{sm}$ with fibre being two
disjoint copies of ${\mathbb{P}^{2}}\setminus C_{2}$, for a smooth conic
$C_{2}$. In particular it is a Stein manifold of dimension 6, rational
irreducible and smooth.

The key ingredient of this result is the description of the families of curves
arising as zero loci of global sections of $E(1)$ for
$[E]\in{\mathcal{I}}(2)$.

######  Proposition 44.

[32, Proposition 4.4] The zero set $V(s)$ of a global section $s$ of $E(1)$ is
a divisor of type $(2,0)$ on a smooth hyperplane section $Q\subset X$ (and
hence it is either the union of two disjoint lines or a double line of
arithmetic genus -1). The zero sets $V(s),\ V(t)$ of two sections $s,t$ of
$E(1)$ lie on the same smooth quadric $Q$ and cut a system $g_{2}^{1}$ without
base point.

This characterisation of the linear spaces ${\mathbb{P}}(H^{0}(E(1)))$ implies
indeed the existence of a morphism:

${\mathcal{I}}(2)\xrightarrow{\phi}{\mathbb{P}^{4}}^{*}_{sm},$

mapping a point $[E]\in{\mathcal{I}}(2)$ to the quadric surface containing all
the curves $V(s),\ s\in H^{0}(E(1))$. The fibre of $\phi$ over $Q$ consist of
the base point free pencils of divisors of type $(2,0)$ or $(0,2)$ on $Q$,
namely of two copies of ${\mathbb{P}^{2}}\setminus C_{2}$ where $C_{2}$ is a
smooth conic. The pencils of divisors of type $(2,0)$ are indeed parameterized
by the projective space
$\mathbb{G}(1,|{\mathcal{O}}_{Q}(2,0)|)\simeq|{\mathcal{O}}_{Q}(2,0)|^{*}\simeq{\mathbb{P}^{2}}$;
inside this projective space, the locus of pencils with a base point
identifies with $C_{2}$, the smooth conic of lines tangent to
$\Gamma_{2}\subset|{\mathcal{O}}_{Q}(2,0)|$, the conic parameterizing double
lines.

######  Remark 45.

Note that, by construction, the morphism $\phi$ factors through a morphism
$\phi_{{\mathcal{U}}}:{\mathcal{I}}(2)\rightarrow{\mathcal{U}}$ where we
recall that $\mathcal{U}\subset{\mathbb{G}(1,F(X))}$ is defined as the open
subset parameterizing rulings of smooth hyperplane sections of $X$.

We now pass to the study of non locally free instantons $[E]\in\mathcal{M}$.
By Theorem 24, if $E$ is a non locally free instanton, $E^{\vee\vee}$ is an
instanton bundle of charge $c_{2}(E^{\vee\vee})<c_{2}(E)$ and
$T_{E}:=E^{\vee\vee}/E$ is a rank 0 instanton of degree
$c_{2}(E)-c_{2}(E^{\vee\vee})$. Since $c_{2}(E)=2$ and the minimal charge of
an instanton sheaf on $X$ is 1, the only possibility is that
$E^{\vee\vee}\simeq{\mathcal{S}}$ so that $T_{E}$ is a rank 0 instanton of
degree 1. It is not difficult to prove that $T_{E}\simeq{\mathcal{O}}_{l}$ for
a line $l\subset X$. Since $P_{T_{E}}(t)=t+1$ and as $h^{1}(T_{E}(-1))$
implies $h^{1}(T_{E})=0$, we have $h^{0}(T_{E})=1$. For $s\in H^{0}(T_{E})$,
the image $\operatorname{im}(s)$ of the corresponding morphism
${\mathcal{O}}_{X}\xrightarrow{s}T_{E}$ must therefore be of the form
${\mathcal{O}}_{C}$ for $C$ a degree one l.c.m curve. But this means $C\simeq
l$ for a line $l\subset X$ and since $P_{{\mathcal{O}}_{l}}=P_{T_{E}}$ we
conclude that $T_{E}\simeq{\mathcal{O}}_{l}$. Summing up, each non locally
free instanton $E$ of charge 2 is defined by a short exact sequence of the
form:

(34) $0\rightarrow
E\rightarrow{\mathcal{S}}\xrightarrow{q}{\mathcal{O}}_{l}\rightarrow 0.$

Our next aim is to formulate results similar to Proposition 44 and Theorem 43
for non locally free instanton sheaf. We start describing the families of
curves corresponding to global sections of $E(1)$.

######  Proposition 46.

Let $E$ be a non locally free instanton of charge 2 singular along a line $l$.
Then $H^{0}(E(1))\simeq{\mathbb{C}}^{2}$ and $\forall s\in H^{0}(E(1)),\ s\neq
0$, $\operatorname{coker}(s)\simeq{\mathcal{I}}_{l^{\prime}\cup l}$ with
$l^{\prime}$ varying in a ruling of a smooth hyperplane section of $X$
containing $l$.

###### Proof.

Let us start with the computation of $H^{0}(E(1))$. Twisting (34) and taking
global sections, we obtain a linear map $H^{0}({\mathcal{S}}(1))\rightarrow
H^{0}({\mathcal{O}}_{l}(1))$ that can not be injective, (as
$h^{0}({\mathcal{S}}(1))=4$) hence $H^{0}(E(1))\neq 0$. Denote now by $\iota$
the inclusion $\iota:H^{0}(E(1))\hookrightarrow H^{0}({\mathcal{S}}(1))$.
Since every non-zero element in $H^{0}({\mathcal{S}}(1))$ has torsion free
cokernel, the same holds for any non-zero $s\in H^{0}(E(1))$; this implies
that $\operatorname{coker}(s)\simeq{\mathcal{I}}_{Y}(1)$ for a l.c.m subscheme
$Y\subset X$.

For any $s\in H^{0}(E(1)),\>s\neq 0$, we have
$\operatorname{coker}(\iota(s))\simeq{\mathcal{I}}_{l^{\prime}}(1)$ for a line
$l^{\prime}\subset X$ and we get a commutative diagram:

(35)
${0}$${{\mathcal{O}}_{X}}$${{\mathcal{O}}_{X}}$${0}$${0}$${0}$${E(1)}$${{\mathcal{S}}(1)}$${{\mathcal{O}}_{l}(1)}$${0}$${0}$${{\mathcal{I}}_{Y}(1)}$${{\mathcal{I}}_{l^{\prime}}(1)}$${{\mathcal{O}}_{l}(1)}$${0}$$\scriptstyle{s}$$\scriptstyle{\iota(s)}$$\scriptstyle{q}$$\scriptstyle{id}$

From it we compute that $P_{Y}(n)=2n+2$ and we deduce that
$\operatorname{Supp}(Y)=l^{\prime}\cup l$. Since $Y$ must be l.c.m. and as
${\mathcal{I}}_{l^{\prime}}$ surjects onto ${\mathcal{O}}_{l}$, the only
possibilities are either $l^{\prime}=l$, in which case
${\mathcal{I}}_{l}|_{l}\simeq{\mathcal{O}}_{l}\oplus{\mathcal{O}}_{l}(-1)$ and
$Y$ is a double structure on $l$ with arithmetic genus $-1$ and , or
$l^{\prime}\cap l=\emptyset$ in which case
${\mathcal{I}}_{l^{\prime}}|_{l}\simeq{\mathcal{O}}_{l}$ and $Y$ is simply the
union of $l$ and $l^{\prime}$. In each of these cases, the scheme $Y$ is
contained in the unique hyperplane $\langle Y\rangle\simeq{\mathbb{P}^{3}}$
thus, from the first column of (35) we compute that $h^{0}(E(1))=2$. To
complete the proof of the proposition we still need to describe the pencil
${\mathbb{P}}(H^{0}(E(1)))$. By construction the space of section $r_{l}\in
H^{0}({\mathcal{S}}(1))$ vanishing on $l$ locates a point in
${\mathbb{P}}(\iota(H^{0}(E(1)))\simeq{\mathbb{P}^{1}}$ (since
$r_{l}\otimes{\mathcal{O}}_{l}$=0); the arguments previously presented show
that a generic element in $\iota(H^{0}(E(1)))$ corresponds to a line disjoint
from $l$. From the discussion held at the end of section 5.1, the pencil
${\mathbb{P}}(\iota(H^{0}(E(1))))$ must therefore coincide with the ruling of
a smooth hyperplane section $Q$ of $X$ containing $l$ and the curves
corresponding to non zero sections of $E(1)$ are thus all of the form $l\cup
l^{\prime}$ with $l^{\prime}$ varying in ${\mathbb{P}}(\iota(H^{0}(E(1))))$. ∎

######  Remark 47.

Proposition 46 allows us to deduce that, given a l.c.m. curve $Y$ with Hilbert
polynomial $2t+2$ and $\operatorname{Supp}(Y)=l\cup l^{\prime}$, a pair
$(s,E(1))$ with $E$ a non locally free instanton singular along $l$,
corresponds to a pair $(Y,\xi)$ with $\xi\in H^{0}(\omega_{Y}(2))\simeq
H^{0}({\mathcal{O}}_{l})\oplus H^{0}({\mathcal{O}}_{l}^{\prime})$ of the form
$(0,e),\ e\neq 0.$