# Even and odd instanton bundles
on Fano threefolds

Vincenzo Antonelli, Gianfranco Casnati, Ozhan Genc

###### Abstract.

We define non–ordinary instanton bundles on Fano threefolds $X$ extending the
notion of (ordinary) instanton bundles introduced in [14]. We determine a
lower bound for the quantum number of a non–ordinary instanton bundle, i.e.
the degree of its second Chern class, showing the existence of such bundles
for each admissible value of the quantum number when $i_{X}\geq 2$ or
$i_{X}=1$, $\mathrm{rk}\operatorname{Pic}(X)=1$ and $X$ is ordinary. In these
cases we deal with the component inside the moduli spaces of simple bundles
containing the vector bundles we construct and we study their restriction to
lines. Finally we give a monadic description of non–ordinary instanton bundles
on ${\mathbb{P}^{3}}$ and the smooth quadric studying their loci of jumping
lines, when of the expected codimension.

###### Key words and phrases:

Fano threefold, vector bundle, $\mu$–(semi)stable bundle, simple bundle,
instanton bundle

###### 2020 Mathematics Subject Classification:

Primary: 14J60. Secondary: 14D21, 14F08, 14J30, 14J45.

The first and second authors are a member of GNSAGA group of INdAM and are
supported by the framework of the MIUR grant Dipartimenti di Eccellenza
2018-2022 (E11G18000350001). The third author is supported by the grant
MAESTRO NCN - UMO-2019/34/A/ST1/00263 - Research in Commutative Algebra and
Representation Theory.

##  1\. Introduction

Let $X$ be a threefold over the complex field $\mathbb{C}$. We say that $X$ is
a Fano threefold if its anticanonical line bundle $\omega_{X}^{-1}$ is ample.
The greatest integer $i_{X}$ such that
$\omega_{X}\cong{\mathcal{O}}_{X}(-i_{X}h)$ for some ample
${\mathcal{O}}_{X}(h)\in\operatorname{Pic}(X)$ is called the index of $X$: it
is well–known that such an ${\mathcal{O}}_{X}(h)$ is uniquely determined and
it is called the fundamental line bundle of $X$.

The very first examples of Fano threefold are ${\mathbb{P}^{3}}$ and the
smooth quadric $Q\subseteq{\mathbb{P}^{4}}$. In this cases
$i_{{\mathbb{P}^{3}}}=4$ and $i_{Q}=3$ respectively and the fundamental line
bundle is the one cut out by the hyperplanes. Thus the origin of the study of
vector bundles supported on Fano threefolds may be traced back to the seminal
papers [7, 29, 24], inspecting the case of the projective $3$–space, and to
[49, 48] for $Q$.

In the aforementioned papers, the authors focus their attention on
$\mu$–stable bundles. Recall that the number
$\mu({\mathcal{F}})=c_{1}({\mathcal{F}})h^{2}/\mathrm{rk}({\mathcal{F}})$ is
defined for each torsion–free sheaf ${\mathcal{F}}$. The torsion–free sheaf
${\mathcal{F}}$ is called $\mu$–semistable (resp. $\mu$–stable) if for all
proper subsheaves ${\mathcal{G}}$ with
$0<\mathrm{rk}({\mathcal{G}})<\mathrm{rk}({\mathcal{F}})$ we have
$\mu({\mathcal{G}})\leq\mu({\mathcal{F}})$ (resp.
$\mu({\mathcal{G}})<\mu({\mathcal{F}})$).

Among the $\mu$–stable rank two vector bundles on ${\mathbb{P}^{3}}$, a
relevant role is played by instanton bundles, i.e. $\mu$–stable rank two
vector bundles ${\mathcal{E}}$ such that $c_{1}({\mathcal{E}})=0$ and
$h^{1}\big{(}{\mathcal{E}}(-2)\big{)}=0$. Instanton bundles on
${\mathbb{P}^{3}}$ carry many interesting properties and they have been
thoroughly studied in the past and recent years.

Starting from ${\mathbb{P}^{3}}$, the notion of instanton bundle has been
widely generalized, e.g. to projective space and smooth quadrics of any
dimension. A first generalization to the Fano threefolds with Picard number
$\varrho_{X}:=\mathrm{rk}\operatorname{Pic}(X)=1$ can be found in [25] (see
also [38]). More precisely, an instanton bundle is defined in [25] as a
$\mu$–stable rank $2$ bundle with $c_{1}({\mathcal{E}})=(2q_{X}-i_{X})h$ and
$h^{1}\big{(}{\mathcal{E}}(-q_{X}h)\big{)}=0$ where
$q_{X}=\left[\frac{i_{X}}{2}\right]$. The existence of instanton bundles
according to this definition is proved for $i_{X}=2,3$ in [25] and for
$i_{X}=1$ in [10].

In [41], the authors extended the notion of instanton bundle to the flag
manifold $F(0,1,2)$ which is a Fano threefold with $\varrho_{X}=2$. More
recently, the definition has been further generalized to each Fano threefold,
regardless of $i_{X}$ or $\varrho_{X}$: see [14, Definition 1.2]. Moreover,
the existence of instanton bundles according to the latter definition has been
settled on some Fano threefolds: see [14, 17, 4, 3].

In any case, each instanton bundle ${\mathcal{E}}$ on $X$ satisfies
$c_{1}({\mathcal{E}})\in\\{0,-h\\}$ according to the parity of $i_{X}$. E.g.
$c_{1}({\mathcal{E}})=0$ when $X\cong{\mathbb{P}^{3}}$. Nonetheless, rank $2$
vector bundles ${\mathcal{E}}$ with $c_{1}({\mathcal{E}})=-1$ on
${\mathbb{P}^{3}}$ are certainly interesting and they have been classical
object of study (e.g. see [31, 11, 23]) for low values of
$c_{2}({\mathcal{E}})$.

Motivated by the above considerations, in this paper we give the following
definition, where for $\varepsilon\in\left\\{0,1\right\\}$ we set

$q_{X}^{\varepsilon}=\left[\frac{i_{X}+1-\varepsilon}{2}\right]$

and we denote the integral cohomology of $X$ by $H^{i}(X)$.

######  Definition 1.1.

Let $X$ be a Fano threefold.

A vector bundle ${\mathcal{E}}$ of rank 2 on $X$ is called an instanton bundle
if the following properties hold:

  * •

$c_{1}({\mathcal{E}})=-\varepsilon h$, where
$\varepsilon\in\left\\{0,1\right\\}$;

  * •

${\mathcal{E}}$ is $\mu$–semistable with respect to ${\mathcal{O}}_{X}(h)$ and
$(1-\varepsilon)h^{0}\big{(}{\mathcal{E}}\big{)}=0$;

  * •

$h^{1}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h)\big{)}=0$.

The class $c_{2}({\mathcal{E}})\in H^{4}(X)$ and its degree are respectively
called (topological) charge and quantum number of ${\mathcal{E}}$.

We spend now a few words in order to better explain Definition 1.1 above. When
$\varepsilon=1$ we do not require the vanishing
$h^{0}\big{(}{\mathcal{E}}\big{)}=0$ in the above definition. On the one hand,
in spite of this apparent asymmetry, the vanishing of
$h^{0}\big{(}{\mathcal{E}}\big{)}$ when $\varepsilon=1$ is actually a direct
consequence of the $\mu$–semistability of ${\mathcal{E}}$, because
$\mu({\mathcal{E}})<0$. On the other hand, when $\varepsilon=0$ the vector
bundle ${\mathcal{E}}:={\mathcal{O}}_{X}^{\oplus 2}$ certainly satisfies all
the properties of Definition 1.1 but $h^{0}\big{(}{\mathcal{E}}\big{)}=0$.
Moreover, when $i_{X}\leq 2$ it is not difficult to give further examples of
non–trivial rank $2$ bundles with an analogous behaviour and with arbitrarily
large quantum number: e.g. every rank $2$ bundle associated via the Serre
correspondence to a disjoint set of integral curves of degree $3-i_{X}$ in
$X$.

In order to have a rough idea of the meaning of the instantonic condition,
i.e. the vanishing condition for the degree $1$ cohomology in Definition 1.1
above, we will show in Lemma 2.4 that $h^{1}\big{(}{\mathcal{E}}(th)\big{)}=0$
for $t\leq-q_{X}^{\varepsilon}$ for each $\mu$–semistable rank $2$ vector
bundle ${\mathcal{E}}$ on $X$ with $c_{1}({\mathcal{E}})=-\varepsilon h$,
$\varepsilon\in\left\\{0,1\right\\}$. If $X\cong{\mathbb{P}^{3}}$, thanks to
[40, Theorem 7] the vanishing

$h^{1}\big{(}{\mathcal{E}}((1-q_{{\mathbb{P}^{3}}}^{\varepsilon})h)\big{)}=h^{1}\big{(}{\mathcal{E}}(-h)\big{)}=0,$

implies that ${\mathcal{E}}$ is a direct sum of line bundles: the same is true
for $X\cong Q$ and $\varepsilon=0$. In particular, in these cases,
$-q_{X}^{\varepsilon}=-2$ is the greatest integer such that there could exist
an indecomposable ${\mathcal{E}}$ as above with
$h^{1}\big{(}{\mathcal{E}}(th)\big{)}=0$ for $t\leq-q_{X}^{\varepsilon}$.

In Section 2 we list some helpful facts about vector bundles and Fano
threefolds that we will use throughout the whole paper.

When $\varrho_{X}=1$, each instanton bundle ${\mathcal{E}}$ is $\mu$–stable,
hence simple, i.e.

$\mathrm{Hom}_{X}\big{(}{\mathcal{E}},{\mathcal{E}}\big{)}\cong
H^{0}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}\cong\mathbb{C}.$

When $\varrho_{X}\geq 2$ this is no longer true. Nevertheless in Section 3 we
prove the following result characterizing the group of endomorphisms of an
instanton bundle.

######  Theorem 1.2.

Let $X$ be a Fano threefold.

If ${\mathcal{E}}$ is an instanton bundle on $X$, the following assertions
hold.

  1. (1)

If ${\mathcal{E}}$ is indecomposable, then it is simple.

  2. (2)

If ${\mathcal{E}}$ is decomposable, then
${\mathcal{E}}\cong{\mathcal{O}}_{X}(D)\oplus{\mathcal{O}}_{X}(-D-\varepsilon
h)$, where $Dh^{2}=-\varepsilon h^{3}/2$ and

$\begin{gathered}h^{0}\big{(}{\mathcal{O}}_{X}(D)\big{)}=h^{0}\big{(}{\mathcal{O}}_{X}(-D-\varepsilon
h)\big{)}=0,\\\
h^{1}\big{(}{\mathcal{O}}_{X}(D-q_{X}^{\varepsilon}h)\big{)}=h^{1}\big{(}{\mathcal{O}}_{X}(-D-(\varepsilon+q_{X}^{\varepsilon})h)\big{)}=0.\end{gathered}$
(1.1)

In this case,
$\mathrm{Hom}_{X}\big{(}{\mathcal{E}},{\mathcal{E}}\big{)}\cong\mathbb{C}^{\oplus
2}$ acting diagonally.

When $i_{X}-\varepsilon$ is even, instanton bundles in the sense of Definition
1.1 coincide with instanton bundles as defined in [25] and [14], because
$q_{X}=q_{X}^{\varepsilon}$. For this reason, we give the following
definition.

######  Definition 1.3.

Let $X$ be a Fano threefold.

An instanton bundle ${\mathcal{E}}$ on $X$ with
$c_{1}({\mathcal{E}})=-\varepsilon h$ is called even or odd (resp. ordinary,
non–ordinary), if $\varepsilon$ (resp. $i_{X}-\varepsilon$) is even or odd
respectively.

As we already pointed out above, ordinary instanton bundles have been widely
studied. The present paper is essentially focused on the non–ordinary
instanton bundle, studying their elementary properties, constructing examples
and studying the corresponding points in the moduli spaces where they sit.

More precisely, in Section 4 we deal with non–ordinary instanton bundles and
we prove the following lower bound for the quantum numbers of such bundles
(see [14, Lemma 4.2] for a similar bound in the ordinary case).

######  Theorem 1.4.

Let $X$ be a Fano threefold.

If ${\mathcal{E}}$ is an indecomposable non–ordinary instanton bundle on $X$,
then its quantum number is even and satisfies

$k\geq\left\\{\begin{array}[]{ll}2&\text{if either $i_{X}\geq 3$ or $i_{X}=2$
and $h^{3}\leq 5$,}\\\ 4&\text{if either $i_{X}=2$ and $h^{3}\geq 6$ or
$i_{X}=1$.}\end{array}\right.$ (1.2)

The above lower bound is actually sharp when $i_{X}\geq 2$. Indeed, in
Construction 5.2 we use the Serre correspondence to associate certain bundles
of rank $2$ to a suitable set of pairwise disjoint conics, which enables us to
prove the following result in Section 5 extending [29, Example 3.1.2].

######  Theorem 1.5.

Let $X$ be a Fano threefold with $i_{X}\geq 2$ and very ample
${\mathcal{O}}_{X}(h)$.

For each even integer $k$ satisfying Inequality (1.2), Construction 5.2 gives
a $\mu$–stable, non–ordinary instanton bundle ${\mathcal{E}}$ with quantum
number $k$.

When $i_{X}=1$ the existence of even instanton bundles is slightly less
immediate. Indeed Construction 5.2 returns rank $2$ bundles which are
certainly not instanton, because they have non–zero sections. In particular
the approach via Serre correspondence is certainly not possible if $X$ is a
prime Fano threefold, i.e. a Fano threefold such that $\varrho_{X}=1$ as we
show at the beginning of Section 6.

We also describe therein a completely different approach based on a suitable
chain of deformations of the bundles obtained via Construction 5.2, showing
that such an approach leads to instanton bundles and proving the following
result extending [9, Theorem 4.1] at least when $X$ is ordinary, i.e. it
contains a line with normal bundle
${\mathcal{O}}_{{\mathbb{P}^{1}}}\oplus{\mathcal{O}}_{{\mathbb{P}^{1}}}(-1)$.

######  Theorem 1.6.

Let $X$ be an ordinary prime Fano threefold with very ample
${\mathcal{O}}_{X}(h)$.

For each even integer $k\geq 4$, there exists a $\mu$–stable, even instanton
bundle with quantum number $k$.

Again $k=4$ is the minimal admissible value for the quantum number of an even
instanton bundle on an ordinary prime Fano threefold given by Theorem 1.4.
Thus Theorem 1.4 is sharp also if $i_{X}=1$, at least in the case
$\varrho_{X}=1$.

The problem of the existence of instanton bundles when $\varrho_{X}\geq 2$
remains wide open in general, though it is not difficult to show the existence
of even instanton bundles in several specific cases via Serre construction.
Finally, in Remark 6.7 we suggest a completely different possible approach for
constructing minimal instanton bundles when $\varrho_{X}\geq 2$.

An important property of instanton bundles on ${\mathbb{P}^{3}}$, occasionally
assumed in some classical definitions, is that they are trivial when
restricted to the general line, thanks to the Grauert–Mülich theorem (see [46,
Corollary 2 of Theorem II.2.1.4]). A line on $X$ is a curve $L$ with Hilbert
polynomial $t+1$, i.e. such that $\deg(L)=1$ and $p_{a}(L)=0$: if
${\mathcal{O}}_{X}(h)$ is very ample, then lines on $X$ are exactly the curves
which are lines with respect to the embedding induced by
${\mathcal{O}}_{X}(h)$. In what follows we will denote by $\Lambda(X)$ the
Hilbert scheme of lines on $X$.

######  Definition 1.7.

Let $X$ be a Fano threefold.

We say that an instanton bundle ${\mathcal{E}}$ on $X$ with
$c_{1}({\mathcal{E}})=-\varepsilon h$ is generically trivial on the component
$\Lambda(X)_{0}\subseteq\Lambda(X)$, if
$h^{1}\big{(}{\mathcal{E}}((i_{X}+\varepsilon-2q_{X}^{\varepsilon}-1)h)\otimes{\mathcal{O}}_{L}\big{)}=0$
when $L\in\Lambda(X)_{0}$ is general. ${\mathcal{E}}$ is generically trivial,
if it is generically trivial on each component.

When $i_{X}\geq 2$, then each Fano threefold is covered by lines, while this
is no longer true if $i_{X}=1$.

Generically trivial ordinary instanton bundles have been constructed on
several Fano threefolds ([25, 41, 4, 17, 3]). Thus it is quite natural to
analyze the behaviour of the instanton bundles whose existence is guaranteed
by Theorems 1.5 and 1.6 when restricted to general lines: we make such an
analysis in Section 7 where we prove the following result.

######  Theorem 1.8.

Let $X$ be a Fano threefold with very ample ${\mathcal{O}}_{X}(h)$.

Then the bundles obtained via Construction 5.2 are generically trivial.
Moreover, the general bundle as in Theorem 1.6 is generically trivial.

In the same section, we also show that such bundles also behave well when
restricted to smooth and irreducible divisors (see Proposition 7.2).

Theorem 1.2 implies that indecomposable instanton bundles correspond to points
inside the moduli space ${\mathcal{S}}_{X}$ of simple torsion–free sheaves
(see [1] for details about such a space). Moreover, the instantonic condition
implies by semicontinuity that such points fill an open subset
${\mathcal{S}}{\mathcal{I}}_{X}$.

When $X\cong{\mathbb{P}^{3}}$, the moduli space of instanton bundles with
fixed charge is irreducible (see [50, 51]) and smooth (see [37]). In [25], the
author proves that for many other families of Fano threefolds $X$ with
$\varrho_{X}=1$ the moduli space of instanton bundles with fixed charge has a
generically smooth irreducible component. Similar results have been obtained
in [41, 14, 4, 17, 3] for other families of Fano threefolds.

In Section 8 we check that the instanton bundles given by Construction 5.2
represent smooth points of one and the same component of their moduli space
(see Proposition 8.1).

An instanton bundle is called minimal, if its quantum number $k$ is as small
as possible: they typically carry additional properties (e.g. see [14, Remark
4.3]). As a by–product of Theorem 1.5 we infer that the moduli space of
minimal non–ordinary instanton bundles ${\mathcal{E}}$ on Fano threefolds $X$
with $i_{X}=2$ and $\varrho_{X}=1$ is irreducible and smooth, because such
bundles are the ones studied in [49] (see Remark 8.3). In particular such
minimal non–ordinary instanton bundles are aCM (with respect to
${\mathcal{O}}_{X}(h)$), i.e. $h^{i}\big{(}{\mathcal{E}}(th)\big{)}=0$ for
$i=1,2$ and each $t\in\mathbb{Z}$ (see also [6]).

Moreover, we also show that each instanton bundle obtained as in Theorem 1.6
is a smooth point inside the component of the moduli space which contains it
(see Proposition 8.5).

Finally, in the last Section 9, we focus on non-ordinary instanton bundles on
the two Fano threefolds with $i_{X}\geq 3$. In particular we show that non-
ordinary instanton bundles on ${\mathbb{P}^{3}}$ can be realized as cohomology
of a monad, similarly to the classical ordinary case. Using such a monad we
are also able to explicitly describe the locus of jumping lines for
non–ordinary instanton bundles.

###  1.1. Acknowledgements

The authors express their thanks the referee for her/his comments, remarks and
suggestions which have considerably improved the whole exposition correcting
several inaccuracies. The authors also thank R. Notari for a fruitful
discussion about the proof of Proposition 8.2.

##  2\. General results

We list below some general helpful results used throughout the whole paper.
Let $X$ be any smooth variety with canonical line bundle $\omega_{X}$.

Let ${\mathcal{F}}$ be a vector bundle of rank $2$ on $X$ and let $s\in
H^{0}\big{(}{\mathcal{F}}\big{)}$. In general its zero–locus $(s)_{0}\subseteq
X$ is either empty or has codimension at most $2$. We can always write
$(s)_{0}=Z\cup S$ where $Z$ has codimension $2$ (or it is empty) and $S$ has
pure codimension $1$ (or it is empty). In particular ${\mathcal{F}}(-S)$ has a
section vanishing on $Z$, thus we can consider its Koszul complex

$0\longrightarrow{\mathcal{O}}_{X}(S)\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{I}}_{Z|X}(-S)\otimes\det({\mathcal{F}})\longrightarrow
0.$ (2.1)

Moreover we also have the standard exact sequence

$0\longrightarrow{\mathcal{I}}_{Z|X}\longrightarrow{\mathcal{O}}_{X}\longrightarrow{\mathcal{O}}_{Z}\longrightarrow
0$ (2.2)

Sequence 2.1 tensored by ${\mathcal{O}}_{Z}$ yields
${\mathcal{I}}_{Z|X}/{\mathcal{I}}^{2}_{Z|X}\cong{\mathcal{F}}^{\vee}(S)\otimes{\mathcal{O}}_{Z}$,
whence the normal bundle of $Z$ inside $X$ satisfies

${\mathcal{N}}_{Z|X}\cong{\mathcal{F}}(-S)\otimes{\mathcal{O}}_{Z}.$ (2.3)

Thus $Z$ is locally complete intersection inside $X$, because
$\mathrm{rk}({\mathcal{F}})=2$. In particular, it has no embedded components.

The Serre correspondence allows us to revert the above construction as
follows.

######  Theorem 2.1.

Let $Z\subseteq X$ be a local complete intersection subscheme of codimension
$2$.

If $\det({\mathcal{N}}_{Z|X})\cong{\mathcal{O}}_{Z}\otimes\mathcal{L}$ for
some $\mathcal{L}\in\operatorname{Pic}(X)$ such that
$h^{2}\big{(}\mathcal{L}^{\vee}\big{)}=0$, then there exists a vector bundle
${\mathcal{F}}$ of rank $2$ on $X$ such that:

  1. (1)

$\det({\mathcal{F}})\cong\mathcal{L}$;

  2. (2)

${\mathcal{F}}$ has a section $s$ such that $Z$ coincides with the zero–locus
$(s)_{0}$ of $s$.

Moreover, if $H^{1}\big{(}{\mathcal{L}}^{\vee}\big{)}=0$, the above two
conditions determine ${\mathcal{F}}$ up to isomorphism.

###### Proof.

See [5]. ∎

If ${\mathcal{F}}$ and ${\mathcal{G}}$ are coherent sheaves on $X$, then the
Serre duality holds

$\mathrm{Ext}_{X}^{i}\big{(}{\mathcal{F}},{\mathcal{G}}\otimes\omega_{X}\big{)}\cong\mathrm{Ext}_{X}^{\dim(X)-i}\big{(}{\mathcal{G}},{\mathcal{F}}\big{)}^{\vee}$
(2.4)

(see [30, Proposition 7.4]). Moreover, we define

$\chi({\mathcal{G}},{\mathcal{F}}):=\sum_{i=0}^{\dim(X)}\dim\mathrm{Ext}^{i}_{X}\big{(}{\mathcal{G}},{\mathcal{F}}\big{)}.$

Notice that if ${\mathcal{G}}$ is locally free, then
$\chi({\mathcal{G}},{\mathcal{F}})=\chi({\mathcal{F}}\otimes{\mathcal{G}}^{\vee})$.
If ${\mathcal{F}}$ is locally free, Equality (2.4) gives
$\chi({\mathcal{G}},{\mathcal{F}})=(-1)^{\dim(X)}\chi({\mathcal{G}}\otimes{\mathcal{F}}^{\vee}\otimes\omega_{X})$.

Consider a locally complete intersection subscheme $Z\subseteq X$. Then

$\mathcal{E}\kern-1.07639pt{\mathit{x}t}^{i}_{X}\big{(}{\mathcal{O}}_{Z},{\mathcal{O}}_{Z}\big{)}\cong\mathcal{E}\kern-1.07639pt{\mathit{x}t}^{i-1}\big{(}{\mathcal{I}}_{Z|X},{\mathcal{O}}_{Z}\big{)}\cong\wedge^{i}{\mathcal{N}}_{Z|X}$
(2.5)

where
${\mathcal{N}}_{Z|X}^{\vee}:={\mathcal{I}}_{Z|X}/{\mathcal{I}}_{Z|X}^{2}\cong{\mathcal{I}}_{Z|X}\otimes{\mathcal{O}}_{Z}$.

The Riemann–Roch formula for a vector bundle ${\mathcal{F}}$ on a threefold
$X$ is

$\displaystyle\chi({\mathcal{F}})$
$\displaystyle=\mathrm{rk}({\mathcal{F}})\chi({\mathcal{O}}_{X})+{\frac{1}{6}}(c_{1}({\mathcal{F}})^{3}-3c_{1}({\mathcal{F}})c_{2}({\mathcal{F}})+3c_{3}({\mathcal{F}}))$
(2.6)
$\displaystyle-{\frac{1}{4}}(\omega_{X}c_{1}({\mathcal{F}})^{2}-2\omega_{X}c_{2}({\mathcal{F}}))+{\frac{1}{12}}(\omega_{X}^{2}c_{1}({\mathcal{F}})+c_{2}(\Omega_{X})c_{1}({\mathcal{F}}))$

(see [28, Theorem A.4.1]).

We now assume that $X$ is a Fano threefold. The following lemma will be
helpful.

######  Lemma 2.2.

Let $X$ be a Fano threefold.

If ${\mathcal{F}}$ is a simple vector bundle on $X$, then
$\mathrm{Ext}^{3}\big{(}{\mathcal{F}},{\mathcal{F}}\big{)}=0$.

###### Proof.

Equality (2.4) implies
$\mathrm{Ext}^{3}_{X}\big{(}{\mathcal{F}},{\mathcal{F}}\big{)}^{\vee}\cong\mathrm{Hom}_{X}\big{(}{\mathcal{F}},{\mathcal{F}}(-i_{X}h)\big{)}$.
In order to prove the statement, it then suffices to check that the latter
space is zero. On the one hand,
$\varphi\in\mathrm{Hom}_{X}\big{(}{\mathcal{F}},{\mathcal{F}}(-i_{X}h)\big{)}$,
hence $\det(\varphi)\in
H^{0}\big{(}{\mathcal{O}}_{X}(-i_{X}\mathrm{rk}({\mathcal{F}})h)\big{)}=0$. On
the other hand
$\mathrm{Hom}_{X}\big{(}{\mathcal{F}},{\mathcal{F}}(-i_{X}h)\big{)}\subseteq\mathrm{Hom}_{X}\big{(}{\mathcal{F}},{\mathcal{F}}\big{)}$,
because $h^{0}\big{(}{\mathcal{O}}_{X}(h)\big{)}\neq 0$ (see [34, Theorem
2.3.1]). As ${\mathcal{F}}$ is simple, each non zero endomorphism of
${\mathcal{F}}$ is an automorphism. We deduce that necessarily $\varphi=0$. ∎

We are interested in $\mu$–semistable bundles, hence the following lemmas will
be useful.

######  Lemma 2.3.

Let $X$ be a Fano threefold.

If ${\mathcal{F}}$ is a rank $2$ vector bundle on $X$, then ${\mathcal{F}}$ is
$\mu$–stable (resp. $\mu$–semistable) with respect to ${\mathcal{O}}_{X}(h)$
if and only if $h^{0}\big{(}{\mathcal{F}}(-D)\big{)}=0$ for each divisor
$D\subseteq X$ such that $Dh^{2}\geq\mu({\mathcal{F}})$ (resp.
$>\mu({\mathcal{F}})$).

###### Proof.

The group $\operatorname{Pic}(X)$ is free, hence it suffices to apply [36,
Corollary 4]. ∎

The following proposition partially clarifies the importance of the number
$q_{X}^{\varepsilon}$ defined in the introduction.

######  Lemma 2.4.

Let $X$ be a Fano threefold with very ample ${\mathcal{O}}_{X}(h)$.

If ${\mathcal{E}}$ is a rank $2$ $\mu$–semistable bundle on $X$ such that
$c_{1}({\mathcal{E}})=-\varepsilon h$ with
$\varepsilon\in\left\\{0,1\right\\}$ and
$h^{1}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h)\big{)}=0$, then

  1. (1)

$h^{1}\big{(}{\mathcal{E}}(th)\big{)}=0$ for $t\leq-q_{X}^{\varepsilon}$;

  2. (2)

$h^{2}\big{(}{\mathcal{E}}(th)\big{)}=0$ for $t\geq-q_{X}^{\varepsilon}+1$
when $i_{X}-\varepsilon$ is odd and for $t\geq-q_{X}^{\varepsilon}$ when
$i_{X}-\varepsilon$ is even.

###### Proof.

If $D$ is a general surface of degree $-t-q_{X}^{\varepsilon}\geq 1$, then
${\mathcal{E}}\otimes{\mathcal{O}}_{D}$ is $\mu$–semistable, thanks to [42,
Theorem 3.1]. If $i_{X}-\varepsilon\neq 0$, then $q_{X}^{\varepsilon}\geq 1$
and Lemma 2.3 implies

$h^{0}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h)\otimes{\mathcal{O}}_{D}\big{)}=0.$
(2.7)

If $i_{X}-\varepsilon=0$, then $\varepsilon=1$ and $q_{X}^{\varepsilon}=0$. In
this case $\mu({\mathcal{E}}\otimes{\mathcal{O}}_{D})=h^{2}D/2<0$, hence again
Equality (2.7) above holds true. The cohomology of the exact sequence

$0\longrightarrow{\mathcal{O}}_{X}(-D)\longrightarrow{\mathcal{O}}_{X}\longrightarrow{\mathcal{O}}_{D}\longrightarrow
0$ (2.8)

tensored by ${\mathcal{E}}(-q_{X}^{\varepsilon}h)$ yields the first assertion.

Thanks to Equality (2.4) we also obtain
$h^{2}\big{(}{\mathcal{E}}(th)\big{)}=h^{1}\big{(}{\mathcal{E}}(-th)\big{)}=0$
for $t\geq-q_{X}^{\varepsilon}+1$ when $i_{X}-\varepsilon$ is odd and for
$t\geq-q_{X}^{\varepsilon}$ when $i_{X}-\varepsilon$ is even. ∎

A helpful characterization of ordinary instanton bundles on a Fano threefold
is related to the naturality of their cohomology in a certain range. We show
below that such a characterization actually holds also in the non–ordinary
case.

######  Proposition 2.5.

Let $X$ be a Fano threefold.

Every $\mu$–semistable bundle ${\mathcal{E}}$ of rank $2$ on $X$ such that
$(1-\varepsilon)h^{0}\big{(}{\mathcal{E}}\big{)}=0$,
$c_{1}({\mathcal{E}})=-\varepsilon h$ and with natural cohomology in degree
$-q_{X}^{\varepsilon}$ is an instanton bundle.

Every instanton bundle ${\mathcal{E}}$ on $X$ has natural cohomology in degree
$t$ in the range $\varepsilon-i_{X}\leq t\leq 0$. More precisely,
$h^{i}\big{(}{\mathcal{E}}(\lambda h)\big{)}=0$ unless either
$1-q_{X}^{\varepsilon}\leq t\leq 0$ and $i=1$, or $\varepsilon-i_{X}\leq
t\leq\varepsilon-i_{X}-1+q_{X}^{\varepsilon}$ and $i=2$.

###### Proof.

The case when $i_{X}-\varepsilon$ is even is covered by [14, Proposition 4.1],
hence we can restrict our attention to the odd $i_{X}-\varepsilon$ case. In
this case $2q_{X}^{\varepsilon}=i_{X}+1-\varepsilon\geq 2$, hence
$q_{X}^{\varepsilon}\geq 1$. If $\varepsilon=1$, then we know that
$h^{0}\big{(}{\mathcal{E}}\big{)}=0$, because ${\mathcal{E}}$ is
$\mu$–semistable. If $\varepsilon=0$ the same vanishing holds by definition.

Assume that ${\mathcal{E}}$ has natural cohomology in degree
$-q_{X}^{\varepsilon}$. We have

$\displaystyle h^{0}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h)\big{)}\leq
h^{0}\big{(}{\mathcal{E}}\big{)}=0,\qquad
h^{3}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h)\big{)}=h^{0}\big{(}{\mathcal{E}}((1-q_{X}^{\varepsilon})h)\big{)}\leq
h^{0}\big{(}{\mathcal{E}}\big{)}=0.$

If ${\mathcal{O}}_{X}(dh)$ is very ample, then
${\mathcal{E}}\otimes{\mathcal{O}}_{S}$ is $\mu$–semistable for general
$S\in|dh|$, thanks to [42, Theorem 3.1]. Thus
$4hc_{2}({\mathcal{E}})\geq\varepsilon^{2}h^{3}$ by Bogomolov inequality. A
case by case computation yields
$\chi({\mathcal{E}}(-q_{X}^{\varepsilon}h))\geq 0$. We deduce that
$h^{1}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h)\big{)}=0$ necessarily, hence
${\mathcal{E}}$ is an instanton bundle.

Conversely, let ${\mathcal{E}}$ be an instanton bundle. Equality (2.4) implies

$h^{i}\big{(}{\mathcal{E}}(th)\big{)}=h^{3-i}\big{(}{\mathcal{E}}(-(i_{X}-\varepsilon+t)h)\big{)}=h^{3-i}\big{(}{\mathcal{E}}((1-2q_{X}^{\varepsilon}-t)h)\big{)}.$
(2.9)

Thus the cohomology of ${\mathcal{E}}$ in the range $\varepsilon-
i_{X}=1-2q_{X}^{\varepsilon}\leq t\leq 0$ is known when it is known in the
range $1-q_{X}^{\varepsilon}\leq t\leq 0$.

If $i_{X}-\varepsilon=1$, then $q_{X}^{\varepsilon}=1$, hence $t=0$. The
assertion then follows from the hypothesis. If $i_{X}-\varepsilon=3$, then
$q_{X}^{\varepsilon}=2$, hence $t=0,1$. Moreover, in this case $X$ is either
${\mathbb{P}^{3}}$ or $Q$, hence ${\mathcal{O}}_{X}(h)$ is very ample. Thus
the statement follows by combining the definition, Equality (2.9) and Lemma
2.4.∎

##  3\. Endomorphism group of instanton bundles

In this section we deal with the endomorphism group of an instanton bundle.
The lemma below is well–known.

######  Lemma 3.1.

Let $X$ be a smooth variety of dimension $n\geq 2$ endowed with an ample line
bundle ${\mathcal{O}}_{X}(h)$.

If ${\mathcal{E}}$ is a strictly $\mu$–semistable bundle of rank $2$ on $X$,
then it fits into an exact sequence

$0\longrightarrow{\mathcal{O}}_{X}(D)\longrightarrow{\mathcal{E}}\longrightarrow{\mathcal{I}}_{Z|X}(-D)\otimes\det({\mathcal{E}})\longrightarrow
0$ (3.1)

where $\mu({\mathcal{O}}_{X}(D))=\mu({\mathcal{E}})$ and $Z$ is either empty
or of pure codimension $2$.

###### Proof.

By definition, there is a subsheaf $\mathcal{L}\subseteq{\mathcal{E}}$ of rank
$1$ such that $\mu(\mathcal{L})=\mu({\mathcal{E}})$.

Let $\mathcal{Q}:={\mathcal{E}}/\mathcal{L}$ and denote by ${\mathcal{M}}$ the
kernel of the natural morphism ${\mathcal{E}}\to\mathcal{Q}/\mathcal{T}$,
where $\mathcal{T}\subseteq\mathcal{Q}$ is the torsion subsheaf. By
construction ${\mathcal{M}}$ is torsion–free and of rank $1$: moreover,
$\mu({\mathcal{M}})\geq\mu(\mathcal{L})=\mu({\mathcal{E}})$, hence
$\mu({\mathcal{M}})=\mu({\mathcal{E}})$ because ${\mathcal{E}}$ is
$\mu$–semistable. Since ${\mathcal{E}}/{\mathcal{M}}$ is torsion–free, it
follows that ${\mathcal{M}}$ is also normal thanks to [46, Lemma II.1.1.16].
Thus [46, Lemma II.1.1.12] implies that ${\mathcal{M}}$ is reflexive, hence it
is a line bundle, thanks to [46, Lemma II.1.1.15]. We deduce that
${\mathcal{M}}\cong{\mathcal{O}}_{X}(D)$ for some divisor $D$ on $X$.

Let $s\in H^{0}\big{(}{\mathcal{E}}(-D)\big{)}$ be a non–zero section: with
the notation introduced in the previous section we write $(s)_{0}=Z\cup S$ and
we can construct Sequence (2.1) with ${\mathcal{F}}:={\mathcal{E}}(-D)$. In
particular we have an injective map ${\mathcal{O}}_{S}(D+S)\to{\mathcal{E}}$
and we can write the chain of inequalities

$0\leq
Sh^{n-1}=(D+S)h^{n-1}-Dh^{n-1}\leq\mu({\mathcal{E}})-\mu({\mathcal{O}}_{X}(D))=0,$

i.e. $S=0$, because ${\mathcal{O}}_{X}(h)$ is ample (we are using the
Nakai–Moishezon criterion: see [28, Theorem A.5.1] and the references
therein). Thus tensoring Sequence (2.1) with
${\mathcal{F}}:={\mathcal{E}}(-D)$ by ${\mathcal{O}}_{X}(D)$ we obtain
Sequence (3.1). ∎

Theorem 1.2 stated in the Introduction is an immediate by–product of the
following proposition.

######  Proposition 3.2.

Let $X$ be a smooth variety of dimension $n\geq 2$ endowed with an ample line
bundle ${\mathcal{O}}_{X}(h)$. Assume that $\operatorname{Pic}(X)$ has no
non–trivial $2$–torsion elements and ${\mathcal{O}}_{X}(h)$ is not
$2$–divisible.

If ${\mathcal{E}}$ is a rank $2$ indecomposable $\mu$–semistable bundle on $X$
such that $c_{1}({\mathcal{E}})=-\varepsilon h$ with $\varepsilon\in\\{\ 0,1\
\\}$ and $(1-\varepsilon)h^{0}\big{(}{\mathcal{E}}\big{)}=0$, then
${\mathcal{E}}$ is simple.

###### Proof.

Trivially $h^{0}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}\geq 1$,
hence we have to prove the opposite inequality assuming ${\mathcal{E}}$
indecomposable. If ${\mathcal{E}}$ is $\mu$–stable, then it is simple thanks
to [46, Theorem II.1.2.9]. Thus we will assume that ${\mathcal{E}}$ is
strictly $\mu$–semistable.

Lemma 3.1 yields the existence of a divisor $D$ on $X$ with

$Dh^{n-1}=\mu({\mathcal{M}})=\mu({\mathcal{E}})=\frac{-\varepsilon h^{n}}{2}.$
(3.2)

and of Sequence (3.1). Tensoring it with
${\mathcal{E}}^{\vee}\cong{\mathcal{E}}(\varepsilon h)$ and taking its
cohomology we obtain

$h^{0}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}\leq
h^{0}\big{(}{\mathcal{E}}(D+\varepsilon
h)\big{)}+h^{0}\big{(}{\mathcal{E}}\otimes{\mathcal{I}}_{Z|X}(-D)\big{)}.$
(3.3)

We first show that $h^{0}\big{(}{\mathcal{E}}(D+\varepsilon h)\big{)}=0$. The
cohomology of Sequence (3.1) tensored by ${\mathcal{O}}_{X}(D+\varepsilon h)$
yields the exact sequence

$\displaystyle 0\longrightarrow H^{0}\big{(}{\mathcal{O}}_{X}(2D+\varepsilon
h)\big{)}$ $\displaystyle\longrightarrow
H^{0}\big{(}{\mathcal{E}}(D+\varepsilon h)\big{)}$
$\displaystyle\longrightarrow
H^{0}\big{(}{\mathcal{I}}_{Z|X}\big{)}\mathbin{\smash{\mathop{\longrightarrow}\limits^{\partial}}}H^{1}\big{(}{\mathcal{O}}_{X}(2D+\varepsilon
h)\big{)}.$

Equality (3.2) implies $(2D+\varepsilon h)h^{n-1}=0$.

If $h^{0}\big{(}{\mathcal{O}}_{X}(2D+\varepsilon h)\big{)}\neq 0$, then the
Nakai–Moishezon criterion would return $2D+\varepsilon h=0$, because
${\mathcal{O}}_{X}(h)$ is ample. If $\varepsilon=0$, hence $2D=2D+\varepsilon
h=0$, i.e. $D=0$, because $\operatorname{Pic}(X)$ has no non–trivial
$2$–torsion elements. The cohomology of Sequence (3.1) implies

$h^{0}\big{(}{\mathcal{E}}\big{)}=(1-\varepsilon)h^{0}\big{(}{\mathcal{E}}\big{)}\geq
h^{0}\big{(}{\mathcal{O}}_{X}(D)\big{)}=1,$

a contradiction. If $\varepsilon=1$, then $h=-2D$, contradicting that
${\mathcal{O}}_{X}(h)$ is not $2$–divisible. Thus

$h^{0}\big{(}{\mathcal{O}}_{X}(2D+\varepsilon h)\big{)}=0,$ (3.4)

hence $H^{0}\big{(}{\mathcal{E}}(D+\varepsilon
h)\big{)}\subseteq\ker(\partial)\subseteq
H^{0}\big{(}{\mathcal{I}}_{Z|X}\big{)}$.

If $Z\neq\emptyset$, then $h^{0}\big{(}{\mathcal{I}}_{Z|X}\big{)}=0$. If
$Z=\emptyset$, then
$h^{0}\big{(}{\mathcal{I}}_{Z|X}\big{)}=h^{0}\big{(}{\mathcal{O}}_{X}\big{)}=1$.
Since $\partial$ must be non–zero, because ${\mathcal{E}}$ is indecomposable,
it follows that $\ker(\partial)=0$. In both cases we finally infer the
vanishing

$h^{0}\big{(}{\mathcal{E}}(D+\varepsilon h)\big{)}=0.$ (3.5)

We now prove
$h^{0}\big{(}{\mathcal{E}}\otimes{\mathcal{I}}_{Z|X}(-D)\big{)}\leq 1$. The
cohomology of Sequence (3.1) tensored by ${\mathcal{O}}_{X}(-D)$ yields

$\displaystyle
h^{0}\big{(}{\mathcal{E}}\otimes{\mathcal{I}}_{Z|X}(-D)\big{)}\leq
h^{0}\big{(}{\mathcal{E}}(-D)\big{)}$ $\displaystyle\leq
h^{0}\big{(}{\mathcal{O}}_{X}\big{)}+h^{0}\big{(}{\mathcal{I}}_{Z|X}(-2D-\varepsilon
h)\big{)}$ (3.6) $\displaystyle\leq
1+h^{0}\big{(}{\mathcal{O}}_{X}(-2D-\varepsilon h)\big{)}.$

We have $(-2D-\varepsilon h)h^{n-1}=0$, hence as above we again deduce

$h^{0}\big{(}{\mathcal{O}}_{X}(-2D-\varepsilon h)\big{)}=0.$ (3.7)

By combining Inequalities (3.3) and (3.6) with Equalities (3.5) and (3.7), we
finally obtain
$h^{0}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}\leq 1$. ∎

The above proposition leads to the following proof of Theorem 1.2 stated in
the Introduction.

Proof of Theorem 1.2. If ${\mathcal{E}}$ is indecomposable, then the statement
follows from Proposition 3.2, because on a Fano threefold $X$ the fundamental
line bundle ${\mathcal{O}}_{X}(h)$ is not divisible by definition and
$\operatorname{Pic}(X)$ is torsion–free (see [34]).

Let ${\mathcal{E}}$ be decomposable. Thus there is a divisor $D\subseteq X$
such that
${\mathcal{E}}\cong{\mathcal{O}}_{X}(D)\oplus{\mathcal{O}}_{X}(-D-\varepsilon
h)$: necessarily $Dh^{2}=-\varepsilon h^{3}/2$, otherwise ${\mathcal{E}}$
would be unstable. Moreover, Equalities (1.1) follow from the vanishing
$h^{0}\big{(}{\mathcal{E}}\big{)}=0$ and the instantonic condition.

Equalities (3.4) and (3.7) imply that
$\mathrm{Hom}_{X}\big{(}{\mathcal{E}},{\mathcal{E}}\big{)}$ contains only
scalar multiples of the identity. ∎

Notice that the existence of decomposable instanton bundles strongly depends
on the choice of the Fano threefold $X$, as the following example shows.

######  Example 3.3.

Decomposable instanton bundles exist on the flag threefold (see [41,
Proposition 2.5]): it is easy to check that that the same is true on
${\mathbb{P}^{1}}\times{\mathbb{P}^{1}}\times{\mathbb{P}^{1}}$ (see [4]). On
the other hand, the blow up of ${\mathbb{P}^{3}}$ at a point does not support
any decomposable instanton bundle (see [14, Proposition 6.5]).

If ${\mathcal{E}}$ is an indecomposable instanton bundle on a Fano threefold
$X$ we then have
$h^{0}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=1$. Moreover,
$h^{3}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=\dim\mathrm{Ext}^{3}_{X}\big{(}{\mathcal{E}},{\mathcal{E}}\big{)}=0$
thanks to Lemma 2.2. Thus Equality (2.6) for
${\mathcal{E}}\otimes{\mathcal{E}}^{\vee}$ yields

$h^{1}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}-h^{2}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=2i_{X}c_{2}({\mathcal{E}})h-\frac{i_{X}\varepsilon\deg(X)}{2}-3.$
(3.8)

##  4\. Quantum numbers of non–ordinary instanton bundles

Before proving Theorem 1.4 we first briefly recall the classification of Fano
threefolds with $i_{X}\geq 2$ and very ample ${\mathcal{O}}_{X}(h)$ fixing the
notation (see [34] and the references therein).

  * •

$X\cong{\mathbb{P}^{3}}$. In this case $h^{3}=1$, $i_{X}=4$ and
$\varrho_{X}=1$.

  * •

$X$ is a smooth quadric $Q\subseteq{\mathbb{P}^{4}}$. In this case $h^{3}=2$,
$i_{X}=3$ and $\varrho_{X}=1$.

  * •

$X$ is a smooth cubic in $F_{3}\subseteq{\mathbb{P}^{4}}$. In this case
$h^{3}=3$, $i_{X}=2$ and $\varrho_{X}=1$.

  * •

$X$ is the smooth complete intersection of two quadrics in
$F_{4}\subseteq{\mathbb{P}^{5}}$. In this case $h^{3}=4$, $i_{X}=2$ and
$\varrho_{X}=1$.

  * •

$X$ is the smooth and proper intersection of the grassmannian of lines in
${\mathbb{P}^{4}}$, $F_{5}:=G(2,5)\subseteq{\mathbb{P}^{9}}$, with a linear
subspace in ${\mathbb{P}^{9}}$ of dimension $3$. In this case $h^{3}=5$,
$i_{X}=2$ and $\varrho_{X}=1$.

  * •

$X$ is any smooth intersection $F_{6,2}$ of the image of the Segre embedding
of ${\mathbb{P}^{2}}\times{\mathbb{P}^{2}}$ inside ${\mathbb{P}^{8}}$ with a
hyperplane $H$. In this case $h^{3}=6$, $i_{X}=2$ and $\varrho_{X}=2$.

  * •

$X$ is the image $F_{6,3}$ of the Segre embedding of
${\mathbb{P}^{1}}\times{\mathbb{P}^{1}}\times{\mathbb{P}^{1}}$ inside
${\mathbb{P}^{7}}$. In this case $h^{3}=6$, $i_{X}=2$ and $\varrho_{X}=3$.

  * •

$X$ is the blow up $F_{7}$ of ${\mathbb{P}^{3}}$ at a point $p$ embedded in
${\mathbb{P}^{8}}$ via the linear system induced by the quadrics through $p$.
In this case $h^{3}=7$, $i_{X}=2$ and $\varrho_{X}=2$.

Recall that we already introduced in the Introduction the notion of line on a
Fano threefold and we denoted by $\Lambda(X)$ their Hilbert scheme. A conic
$C\subseteq X$ is a curve with Hilbert polynomial $2t+1$ or, in other words,
such that $\deg(C)=2$ and $p_{a}(C)=0$. From now on $\Gamma(X)$ will denote
the Hilbert scheme of conics in $X$.

It is helpful to identify lines and conics on $X$. We exploit such an
identification in the following remarks that will be used several times in the
paper. In what follows $H^{*}(X)$ denotes the integral cohomology ring of $X$.

######  Remark 4.1.

Let $\varrho_{X}=1$.

If $X\cong{\mathbb{P}^{3}}$, then $H^{*}(X)\cong\mathbb{Z}[h]/(h^{4})$, where
$h$ is the class of a plane. The classes of lines and conics on $X$ are
$\ell:=h^{2}$ and $2h^{2}$ respectively. By definition $\Lambda(X)$ is the
Grassmannian $G(2,4)$ of lines in ${\mathbb{P}^{3}}$. The scheme $\Gamma(X)$
is a projective bundle on $|h|$ with fibre isomorphic to the
${\mathbb{P}^{5}}$ of conics in a plane.

If $X$ is the smooth quadric $Q\subseteq{\mathbb{P}^{4}}$, then
$H^{*}(X)\cong\mathbb{Z}[h]/(h^{2}-2\ell,h^{4})$, where
${\mathcal{O}}_{X}(h):={\mathcal{O}}_{{\mathbb{P}^{4}}}(1)\otimes{\mathcal{O}}_{X}$
and $\ell$ is the class of a line. The class of a conic on $X$ is $h^{2}$:
each conic in $X$ is the intersection of $X$ with a unique plane in
${\mathbb{P}^{4}}$, hence $\Gamma(X)$ is the grassmannian $G(3,5)$ of planes
in ${\mathbb{P}^{4}}$. On $X$ there is also an indecomposable rank $2$ bundle
${\mathcal{S}}$ called spinor bundle whose sections vanish exactly on the
lines inside $X$. We have $c_{1}({\mathcal{S}})=-h$,
$c_{2}({\mathcal{S}})=\ell$, $h^{i}\big{(}{\mathcal{S}}(th)\big{)}=0$ for
$t\in\mathbb{Z}$ and $i=1,2$, $h^{0}\big{(}{\mathcal{S}}\big{)}=0$, hence
${\mathcal{S}}$ is $\mu$–stable. Moreover, ${\mathcal{S}}$ fits into an exact
sequence of the form

$0\longrightarrow{\mathcal{S}}\longrightarrow{\mathcal{O}}_{X}^{\oplus
4}\longrightarrow{\mathcal{S}}^{\vee}\longrightarrow 0$

(see [47] for all the details on spinor bundles on $X$). It follows that
$\Lambda(X)\cong{\mathbb{P}^{3}}$.

Let $\delta:=h^{3}\in\\{\ 3,4,5\ \\}$. Thus $X$ contains a line: if $\ell$ is
its class in $H^{4}(X)$, then $h^{2}=\delta\ell$ and the class of a conic on
$X$ is $2\ell$. In this case $\Lambda(X)$ and $\Gamma(X)$ are respectively a
smooth surface and a smooth fourfold: for more details see [39, Propositions
2.2.10 and 2.3.8] and the references therein.

######  Remark 4.2.

Let $h^{3}=6$ and $\varrho_{X}=2$, i.e. $X=F_{6,2}$.

If $\pi_{i}\colon F_{6,2}\to{\mathbb{P}^{2}}$ denotes the projection to the
$i^{\mathrm{th}}$ factor, then
${\mathcal{O}}_{F_{6,2}}(h_{i}):=\pi_{i}^{*}{\mathcal{O}}_{{\mathbb{P}^{2}}}(1)$
is globally generated, $h=h_{1}+h_{2}$ and

$H^{*}({F_{6,2}})\cong\frac{\mathbb{Z}[h_{1},h_{2}]}{(h_{1}^{2}-h_{1}h_{2}+h_{2}^{2},h_{1}^{3},h_{2}^{3})}.$

The fibres of $\pi_{i}$ are lines with class $h_{i}^{2}$. Conversely, let
$\beta_{1}h_{2}^{2}+\beta_{2}h_{1}^{2}\in H^{4}({F_{6,2}})$ be the class of a
line $L$. Thus $\beta_{1}+\beta_{2}=hL=1$ and $\beta_{i}=Lh_{i}\geq 0$, hence
the class of $L$ is $h_{i}^{2}$ and $L$ is a fibre of $\pi_{i}$. In particular
two distinct lines in the same class do not intersect. We conclude that
$\Lambda({F_{6,2}})$ has two components both isomorphic to ${\mathbb{P}^{2}}$.

Now let $\beta_{1}h_{2}^{2}+\beta_{2}h_{1}^{2}\in H^{4}({F_{6,2}})$ be the
class of a conic $C$, i.e. a curve of degree $2$ with $p_{a}(C)=0$. Thus
$\beta_{1}+\beta_{2}=hC=2$ and $\beta_{i}=Ch_{i}\geq 0$. We then infer that
the class of $C$ is either $h_{1}h_{2}$ (coinciding with $h_{1}^{2}+h_{2}^{2}$
as cycles) or $2h_{i}^{2}$.

Let us consider the latter case. If $C$ is reducible, then it is the union of
two fibres of $\pi_{i}$ which are skew lines, hence $p_{a}(C)=-1$, a
contradiction. Thus $C$ must be a double structure on a line $L$ with class
$h_{i}^{2}$. Since $h_{i}^{3}=0$, it follows that each divisor in $|h_{i}|$
intersecting $L$ actually contains it, hence there is an exact sequence

$0\longrightarrow{\mathcal{O}}_{F_{6,2}}(-2h_{i})\longrightarrow{\mathcal{O}}_{F_{6,2}}(-h_{i})^{\oplus
2}\longrightarrow{\mathcal{I}}_{L|F_{6,2}}\longrightarrow 0.$ (4.1)

We have
${\mathcal{I}}_{L|F_{6,2}}^{2}\subseteq{\mathcal{I}}_{C|F_{6,2}}\subseteq{\mathcal{I}}_{L|F_{6,2}}$,
hence a surjective morphism

$\nu\colon{\mathcal{N}}_{L|F_{6,2}}^{\vee}\cong{\mathcal{I}}_{L|F_{6,2}}/{\mathcal{I}}_{L|F_{6,2}}^{2}\longrightarrow{\mathcal{I}}_{L|F_{6,2}}/{\mathcal{I}}_{C|F_{6,2}}\cong{\mathcal{I}}_{L|C}.$

On the one hand, Sequence (4.1) restricted to ${\mathcal{O}}_{L}$ yields
${\mathcal{N}}_{L|F_{6,2}}\cong{\mathcal{O}}_{{\mathbb{P}^{1}}}^{\oplus 2}$.
On the other hand, the latter sheaf is the kernel of the natural map
${\mathcal{O}}_{C}\to{\mathcal{O}}_{L}$, hence its Hilbert polynomial is $t$.
Since ${\mathcal{I}}_{L|F_{6,2}}^{2}\subseteq{\mathcal{I}}_{C|F_{6,2}}$, it
follows that ${\mathcal{I}}_{L|C}$ is a sheaf of rank $1$ on $L$. We deduce
that ${\mathcal{I}}_{L|C}\cong{\mathcal{O}}_{{\mathbb{P}^{1}}}(-1)$, hence the
morphism $\nu$ cannot exist. In particular, this case cannot occur.

Thus the class of $C$ is necessarily $h_{1}h_{2}$. If $C$ is integral, then
the equalities $h_{1}h_{2}h_{i}=1$ and
$h^{0}\big{(}{\mathcal{O}}_{F_{6,2}}(h_{i})\big{)}=3$ imply that $C$ is
contained in the complete intersection of two divisors in $|h_{1}|$ and
$|h_{2}|$, hence it must coincide with it by degree reasons. This implies the
existence of an exact sequence of the form

$0\longrightarrow{\mathcal{O}}_{F_{6,2}}(-h)\longrightarrow{\mathcal{O}}_{F_{6,2}}(-h_{1})\oplus{\mathcal{O}}_{F_{6,2}}(-h_{2})\longrightarrow{\mathcal{I}}_{C|{F_{6,2}}}\longrightarrow
0.$ (4.2)

If $C$ is not integral, then it must be the union of a fibre $L_{1}$ of
$\pi_{1}$ and a fibre $L_{2}$ of $\pi_{2}$. In particular there are pencils
$\Sigma_{i}\subseteq|h_{i}|$ with base locus $L_{i}$. Thus we can find a
divisor inside $\Sigma_{1}$ containing a second point in $L_{2}$ besides
$L_{1}\cap L_{2}$, hence the whole $C$ because $h_{1}h_{2}^{2}=1$. Similarly
there is a divisor in $\Sigma_{2}$ containing $C$ and we get the above
sequence again.

Notice that the Cohomology of Sequence (4.2) tensored by
${\mathcal{O}}_{F_{6,2}}(h_{i})$ yields the uniqueness of the divisors in
$|h_{i}|$ containing $C$. In particular, $\Gamma({F_{6,2}})$ is
$|h_{1}|\times|h_{2}|\cong{\mathbb{P}^{2}}\times{\mathbb{P}^{2}}$.

######  Remark 4.3.

Let $h^{3}=6$ and $\varrho_{X}=3$, i.e. $X=F_{6,3}$.

If $\pi_{i}\colon{F_{6,3}}\to{\mathbb{P}^{1}}$ denote the
$i^{\mathrm{th}}$–projection, then
${\mathcal{O}}_{F_{6,3}}(h_{i}):=\pi_{i}^{*}{\mathcal{O}}_{{\mathbb{P}^{1}}}(1)$
is globally generated, $h=h_{1}+h_{2}+h_{3}$ and

$H^{*}({F_{6,3}})\cong\frac{\mathbb{Z}[h_{1},h_{2},h_{3}]}{(h_{1}^{2},h_{2}^{2},h_{3}^{2})}.$

The fibres of $\pi_{i}$ are smooth quadrics with class $h_{i}$. Thus the
fibres of $\pi_{i}\times\pi_{j}$ have class $h_{i}h_{j}$, hence they are
lines. Conversely, let
$\beta_{1}h_{2}h_{3}+\beta_{2}h_{1}h_{3}+\beta_{3}h_{1}h_{2}\in
A^{2}({F_{6,3}})$ be the class of a line $L$. Thus
$\beta_{1}+\beta_{2}+\beta_{3}=hL=1$ and $\beta_{i}=Lh_{i}\geq 0$, hence the
class of $L$ is $h_{i}h_{j}$ and $L$ is a fibre of $\pi_{i}\times\pi_{j}$. In
particular two lines in the same class do not intersect. In particular,
$\Lambda({F_{6,3}})$ has three components isomorphic to
${\mathbb{P}^{1}}\times{\mathbb{P}^{1}}$.

Now let $\beta_{1}h_{2}h_{3}+\beta_{2}h_{1}h_{3}+\beta_{3}h_{1}h_{2}\in
A^{2}({F_{6,3}})$ be the class of a conic $C$. Thus
$\beta_{1}+\beta_{2}+\beta_{3}=hC=2$ and $\beta_{i}=Ch_{i}\geq 0$. Up to
permutations of the indices, we then infer that the class of $C$ is either
$h_{1}(h_{2}+h_{3})$, or $2h_{1}h_{2}$. As in Remark 4.2, one can exclude the
latter case and find an exact sequence of the form

$0\longrightarrow{\mathcal{O}}_{F_{6,3}}(-h)\longrightarrow{\mathcal{O}}_{F_{6,3}}(-h_{1})\oplus{\mathcal{O}}_{F_{6,3}}(-h_{2}-h_{3})\longrightarrow{\mathcal{I}}_{C|{F_{6,3}}}\longrightarrow
0.$ (4.3)

As in Remark 4.2 computing the Cohomology of Sequence (4.3) tensored by
${\mathcal{O}}_{F_{6,3}}(h_{1})$ and ${\mathcal{O}}_{F_{6,3}}(h_{2}+h_{3})$
one checks that
$\Gamma({F_{6,3}})\cong{\mathbb{P}^{1}}\times{\mathbb{P}^{3}}$.

######  Remark 4.4.

Let $h^{3}=7$, hence $X=F_{7}$.

We have two natural projections, i.e. the blow up map
$\sigma\colon{F_{7}}\to{\mathbb{P}^{3}}$ and the map
$\pi\colon{F_{7}}\to{\mathbb{P}^{2}}$ induced by the projection from $p$ onto
a plane ${\mathbb{P}^{2}}$: we define
${\mathcal{O}}_{F_{7}}(\xi):=\sigma^{*}{\mathcal{O}}_{{\mathbb{P}^{3}}}(1)$
and ${\mathcal{O}}_{F_{7}}(f):=\pi^{*}{\mathcal{O}}_{{\mathbb{P}^{2}}}(1)$.
Notice that both ${\mathcal{O}}_{F_{7}}(\xi)$ and ${\mathcal{O}}_{F_{7}}(f)$
are globally generated. The map $\pi$ induces an isomorphism
${F_{7}}\cong\mathbb{P}({\mathcal{O}}_{{\mathbb{P}^{2}}}\oplus{\mathcal{O}}_{{\mathbb{P}^{2}}}(1))$,
$h=\xi+f$ and

$H^{*}({F_{7}})\cong\frac{\mathbb{Z}[\xi,f]}{(f^{3},\xi^{2}-\xi f)}.$

The fibres of $\pi$ are lines with class $f^{2}$. Moreover, ${F_{7}}$ also
contains the exceptional divisor $E\cong{\mathbb{P}^{2}}$ of $\sigma$. We have
$\\{\ E\ \\}=|\xi-f|$, hence $E$ is also embedded in ${\mathbb{P}^{8}}$ as a
plane, because $Eh^{2}=1$. Trivially each line inside $E$ is also a line in
${F_{7}}$ whose image via $\pi$ is a line. Conversely, arguing as in the
previous remarks one deduces that all the lines on $X$ are of the above form.
In particular, $\Lambda({F_{7}})$ has two components both isomorphic to
${\mathbb{P}^{2}}$.

Now let $\beta_{1}\xi f+\beta_{2}f^{2}\in A^{2}({F_{7}})$ be the class of a
conic $C$. Thus $2\beta_{1}+\beta_{2}=hC=2$, $\beta_{1}=Cf\geq 0$ and
$\beta_{1}+\beta_{2}=C\xi\geq 0$. It is easy to check that the class of $C$ is
$\xi f$, or $2(\xi-f)f$, or $f^{2}$ and one can exclude the latter case as in
Remark 4.2. Notice that in the first case $C$ does not intersect $E$, while in
the second $C\subseteq E$. Again as in Remark 4.2 one obtains the existence of
exact sequences

$0\longrightarrow{\mathcal{O}}_{F_{7}}(-h)\longrightarrow{\mathcal{O}}_{F_{7}}(-\xi)\oplus{\mathcal{O}}_{F_{7}}(-f)\longrightarrow{\mathcal{I}}_{C|{F_{7}}}\longrightarrow
0,$ (4.4)

if the class of $C$ is $\xi f$, and

$0\longrightarrow{\mathcal{O}}_{F_{7}}(-h)\longrightarrow{\mathcal{O}}_{F_{7}}(-\xi+f)\oplus{\mathcal{O}}_{F_{7}}(-2f)\longrightarrow{\mathcal{I}}_{C|{F_{7}}}\longrightarrow
0,$ (4.5)

if the class of $C$ is $2(\xi-f)f$.

Thus $\Gamma({F_{7}})$ is the union of two components. One of them consists of
conics in $E\cong{\mathbb{P}^{2}}$, hence it is isomorphic to
${\mathbb{P}^{5}}$. The other one is given by the complete intersections of
divisors in $|f|$ and $|\xi|$. As in Remark 4.2, one deduces from the analysis
of Sequences (4.4) and (4.5) that the divisor in $A\in|f|$ through $C$ is
unique and that
$h^{0}\big{(}{\mathcal{O}}_{F_{7}}(\xi)\otimes{\mathcal{O}}_{A}\big{)}=3$,
hence the second component is a projective bundle over
$|f|\cong{\mathbb{P}^{2}}$ with fibre ${\mathbb{P}^{2}}$. In particular it has
dimension $4$.

We finally prove the main theorem of the section.

Proof of Theorem 1.4. We know that $h^{0}\big{(}{\mathcal{E}}\big{)}=0$ by
definition. Moreover, if ${\mathcal{E}}$ is non–ordinary, then
$q_{X}^{\varepsilon}\geq 1$. Table 1 will be sometimes helpful in what
follows.

$i_{X}$ | 4 | 4 | 3 | 3 | 2 | 2 | 1 | 1
---|---|---|---|---|---|---|---|---
$\varepsilon$ | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1
$i_{X}-\varepsilon$ | 4 | 3 | 3 | 2 | 2 | 1 | 1 | 0
$q_{X}^{\varepsilon}$ | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 0
ordinary | yes | no | no | yes | yes | no | no | yes
Table 1. Values of $i_{X}$, $\varepsilon$, $i_{X}-\varepsilon$,
$q_{X}^{\varepsilon}$.

We have
$h^{i}\big{(}{\mathcal{E}}\big{)}=h^{3-i}\big{(}{\mathcal{E}}((\varepsilon-
i_{X})h)\big{)}$, thanks to Equality (2.4). One checks directly that
$\varepsilon-i_{X}\leq-q_{X}^{\varepsilon}\leq 0$, hence
$h^{3}\big{(}{\mathcal{E}}\big{)}=h^{0}\big{(}{\mathcal{E}}((\varepsilon-
i_{X})h)\big{)}=0$. Lemma 2.4 implies $h^{2}\big{(}{\mathcal{E}}\big{)}=0$ if
$i_{X}-\varepsilon\neq 0$, i.e. if $(\varepsilon,i_{X})\neq(1,1)$.

Thanks to Equality (2.6), when $(\varepsilon,i_{X})\neq(1,1)$, we then obtain

$\varepsilon-2+\frac{i_{X}-\varepsilon}{2}hc_{2}({\mathcal{E}})=-\chi({\mathcal{E}})=h^{1}\big{(}{\mathcal{E}}\big{)}\geq
0.$ (4.6)

Thus $hc_{2}({\mathcal{E}})$ must be even and positive when
$i_{X}-\varepsilon$ is odd, i.e. when ${\mathcal{E}}$ is non–ordinary.
Moreover, in this case $i_{X}-\varepsilon\neq 0$, hence Inequality (4.6)
returns

$hc_{2}({\mathcal{E}})\geq\left\\{\begin{array}[]{ll}2&\text{if $i_{X}\geq
2$,}\\\ 4&\text{if $i_{X}=1$.}\end{array}\right.$

Thus, in order to complete the proof of the statement, we have only to check
that there is no non–ordinary indecomposable instanton bundle ${\mathcal{E}}$
such that $c_{2}({\mathcal{E}})h=2$ when $i_{X}=2$ and $h^{3}\geq 6$. If it
exists, then $h^{2}\big{(}{\mathcal{E}}(h)\big{)}=0$ thanks to Lemma 2.4. It
follows that
$h^{0}\big{(}{\mathcal{E}}(h)\big{)}\geq\chi({\mathcal{E}}(h))=h^{3}$, thanks
to Equality (2.6). Let $s\in H^{0}\big{(}{\mathcal{E}}(h)\big{)}$ be a
non–zero section.

We first assume that the zero–locus of the section $s$ is the union of a curve
$Z$ and an effective divisor $D\subseteq X$. Thus, there is a non–zero section
of ${\mathcal{E}}(h-D)$ vanishing exactly along $Z$ whose class inside
$A^{2}(X)$ is $c_{2}({\mathcal{E}}(h-D))=c_{2}({\mathcal{E}})-hD+D^{2}$. We
know that

$Dh^{2}-h^{3}=(D-h)h^{2}=\mu({\mathcal{O}}_{X}(D-h))\leq\mu({\mathcal{E}})=-\frac{h^{3}}{2},$

because ${\mathcal{E}}$ is $\mu$–semistable, thanks to Lemma 2.3. We deduce
that

$2Dh^{2}\leq h^{3}.$ (4.7)

Let $X=F_{6,2}$. If $D\in|\alpha_{1}h_{1}+\alpha_{2}h_{2}|$, then
$\alpha_{1}=Dh_{2}^{2}\geq 0$, because $D$ is effective: similarly
$\alpha_{2}\geq 0$. Inequality (4.7) yields $\alpha_{1}+\alpha_{2}\leq 1$,
hence without restricting we can then assume $D=h_{1}$. It then follows
$hc_{2}({\mathcal{E}}(h-D))=0$, hence $Z=\emptyset$ and the Koszul complex of
$s$ becomes

$0\longrightarrow{\mathcal{O}}_{F_{6,2}}\longrightarrow{\mathcal{E}}(h_{2})\longrightarrow{\mathcal{O}}_{F_{6,2}}(h_{2}-h_{1})\longrightarrow
0.$

Such an exact sequence splits, because
$\mathrm{Ext}^{1}_{F_{6,2}}\big{(}{\mathcal{O}}_{F_{6,2}}(h_{2}-h_{1}),{\mathcal{O}}_{F_{6,2}}\big{)}=0$
(see [15, Proposition 2.5]), hence we infer
${\mathcal{E}}\cong{\mathcal{O}}_{F_{6,2}}(-h_{1})\oplus{\mathcal{O}}_{F_{6,2}}(-h_{2})$.

Let $X={F_{6,3}}$. Arguing as in the previous case, we can still assume that
$D=h_{1}$ and we still deduce $Z=\emptyset$. The cohomology of the Koszul
complex of $s$ and the Künneth formulas imply
$h^{0}\big{(}{\mathcal{E}}(h_{1})\big{)}\neq 0$, contradicting the
$\mu$–semistability of ${\mathcal{E}}$, hence this case cannot occur.

Let $X=F_{7}$. If $D\in|\alpha_{1}\xi+\alpha_{2}f|$, then
$\alpha_{1},\alpha_{1}+\alpha_{2}\geq 0$, because $D$ is effective, and
Inequality (4.7) yields $8\alpha_{1}+6\alpha_{2}\leq 7$. It follows that
either $D=f$ or $D=\xi-f$ and we can easily deduce as in the previous case
that such a case cannot occur too.

We now assume that the zero–locus of $s$ is a curve $Z$. The degree of $Z$ is
$2$: the Koszul complex of $s$ and the inequality
$h^{0}\big{(}{\mathcal{E}}(h)\big{)}\geq h^{3}$ imply that $Z$ is contained in
a plane. We deduce that $Z$ is a possibly non–integral conic, hence
$\omega_{Z}\cong{\mathcal{O}}_{X}(-h)\otimes{\mathcal{O}}_{Z}$. In what
follows we will again make a case by case analysis.

If $X={F_{6,2}}$, then we have Sequence (4.2). Recall that we also have the
Kosxul complex of $s$, i.e.

$0\longrightarrow{\mathcal{O}}_{F_{6,2}}\longrightarrow{\mathcal{E}}(h)\longrightarrow{\mathcal{I}}_{Z|{F_{6,2}}}(h)\longrightarrow
0.$

Equality (2.4) and the cohomology of Sequence (2.2) tensored by
${\mathcal{O}}_{F_{6,2}}(-h)$ yield

$\displaystyle\mathrm{Ext}^{1}_{F_{6,2}}\big{(}{\mathcal{I}}_{Z|{F_{6,2}}}(h),{\mathcal{O}}_{F_{6,2}})^{\vee}$
$\displaystyle\cong\mathrm{Ext}^{2}_{F_{6,2}}\big{(}{\mathcal{O}}_{F_{6,2}},{\mathcal{I}}_{Z|{F_{6,2}}}(-h)\big{)}$
$\displaystyle\cong
H^{1}\big{(}{\mathcal{O}}_{F_{6,2}}(-h)\otimes{\mathcal{O}}_{Z}\big{)}\cong
H^{0}\big{(}{\mathcal{O}}_{Z}\big{)}\cong\mathbb{C}.$

Taking account of the equality
$h^{1}\big{(}{\mathcal{O}}_{F_{6,2}}(-h)\big{)}=0$, Theorem 2.1 and Sequence
4.2, we deduce that
${\mathcal{E}}\cong{\mathcal{O}}_{F_{6,2}}(-h_{1})\oplus{\mathcal{O}}_{F_{6,2}}(-h_{2})$
taking into account Sequence 4.2, because there is only one non–trivial
extension of ${\mathcal{I}}_{Z|{F_{6,2}}}(h)$ by ${\mathcal{O}}_{X}$.

We exclude the two remaining cases again using the $\mu$–semistability of
${\mathcal{E}}$. If $X={F_{6,3}}$ and $Z$ is the complete intersection of
divisors in $|h_{1}|$ and $|h_{2}+h_{3}|$, the cohomology of Sequence (4.3)
tensored by ${\mathcal{O}}_{F_{6,3}}(h_{1})$ yields
$h^{0}\big{(}{\mathcal{E}}(h_{1})\big{)}=h^{0}\big{(}{\mathcal{I}}_{Z|{F_{6,3}}}(h_{1})\big{)}\neq
0$. Since $\mu({\mathcal{O}}_{F_{6,3}}(-h_{1}))=-2>-3=\mu({\mathcal{E}})$, it
follows from Lemma 2.3 that this case does not occur.

A similar argument can be applied also to the case $h^{3}=7$, i.e.
$X:={F_{7}}$, by computing the cohomology of Sequences (4.4) and (4.5)
respectively tensored by ${\mathcal{O}}_{F_{7}}(f)$ and
${\mathcal{O}}_{F_{7}}(\xi-f)$. ∎

######  Remark 4.5.

As shown in the above proof, when $i_{X}=2$ and $\varrho_{X}\geq 2$ the only
instanton bundle with quantum number $k=2$ is
${\mathcal{O}}_{F_{6,2}}(-h_{1})\oplus{\mathcal{O}}_{F_{6,2}}(-h_{2})$.

##  5\. Non–ordinary instanton bundles when $i_{X}\geq 2$

The existence of ordinary instanton bundles on a Fano threefold $X$ with very
ample ${\mathcal{O}}_{X}(h)$ has been the object of several papers. More
precisely ordinary instanton bundles always exist when $i_{X}\geq 2$ and in
several cases when $i_{X}=1$ (see the introduction and the references therein
for further details).

In this section we will show that $X$ also supports non–ordinary instanton
bundles when $i_{X}\geq 2$. The first step in this direction is the following
result.

######  Lemma 5.1.

Let $X$ be a Fano threefold with $i_{X}\geq 2$ and very ample
${\mathcal{O}}_{X}(h)$.

For each non–negative integer $s$, there is a curve $Z\subseteq X$ whose
connected components are $s$ pairwise disjoint integral conics.

###### Proof.

The statement is trivial if $i_{X}\geq 3$, hence we only need to examine the
case $i_{X}=2$ in what follows. Let $d:=h^{3}$ be the degree of $X$: we know
that $3\leq d\leq 7$.

In this case the general hyperplane section $H$ of $X$ is a del Pezzo surface
obtained by blowing up $r=9-d$ points in general position on
${\mathbb{P}^{2}}$. Thus $\operatorname{Pic}(H)$ is generated by the pull–back
$\ell$ of the class of a general line in ${\mathbb{P}^{2}}$ and by the classes
$e_{1},\dots,e_{r}$ of the exceptional divisors. Moreover,
$\ell^{2}=-e_{i}^{2}=1$, $\ell e_{i}=e_{i}e_{j}=0$ for $1\leq i<j\leq r$.

Since the class of the hyperplane section of $H$ is
$3\ell-\sum_{k=1}^{r}e_{k}$, it follows that that every element in $|\ell-
e_{1}|$ is a conic. Notice that two such conics do not intersect. ∎

Now let us fix a Fano threefold $X$ with $i_{X}\geq 2$ and consider
$\varepsilon\in\\{\ 0,1\ \\}$ such that $i_{X}-\varepsilon$ is odd. The
following construction extends [29, Example 3.1.2] to all Fano threefolds with
$i_{X}\geq 2$.

######  Construction 5.2.

Let $X$ be a Fano threefold with very ample ${\mathcal{O}}_{X}(h)$. For each
even integer $k\geq 2$, we consider the union $Z$ of
$s:=(k+2q_{X}^{\varepsilon}-2)/2$ pairwise disjoint integral conics on $X$.
The adjunction formula on $X$ and the definition of $Z$ imply
$\det({\mathcal{N}}_{Z|X})\cong{\mathcal{O}}_{X}((i_{X}-1)h)\otimes{\mathcal{O}}_{Z}$.
Since $h^{i}\big{(}{\mathcal{O}}_{X}(th)\big{)}=0$ for $i=1,2$ and each
$t\in\mathbb{Z}$, it follows the existence of an exact sequence

$0\longrightarrow{\mathcal{O}}_{X}\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{I}}_{Z|X}((i_{X}-1)h)\longrightarrow
0,$ (5.1)

thanks to Theorem 2.1. The definition of $q_{X}^{\varepsilon}$ implies
$i_{X}-1=2(q_{X}^{\varepsilon}-1)+\varepsilon$, hence the above sequence
yields

$0\longrightarrow{\mathcal{O}}_{X}((1-q_{X}^{\varepsilon}-\varepsilon)h)\longrightarrow{\mathcal{E}}\longrightarrow{\mathcal{I}}_{Z|X}((q_{X}^{\varepsilon}-1)h)\longrightarrow
0.$ (5.2)

On the one hand, if $i_{X}=1$, then the above construction does not lead to
any instanton bundle ${\mathcal{E}}$ for every value of $k$, because
$h^{0}\big{(}{\mathcal{E}}\big{)}=1$ in this case. On the other hand, in what
follows we will show that the construction above leads to non–ordinary
instanton bundles for all the admissible values of the quantum number $k$
given in Bound (1.2) when $i_{X}\geq 2$.

We are now ready to prove Theorem 1.5. We first start by proving that all the
admissible values of the quantum number $k$ can be obtained when
$\varrho_{X}=1$.

Proof of Theorem 1.5 when $\varrho_{X}=1$. Since $\varrho_{X}=1$ we know that
each divisor $D$ on $X$ is linearly equivalent to $dh$. Moreover, such a $D$
is effective if and only if $d\geq 1$.

It is immediate from Construction 5.2 that $c_{1}({\mathcal{E}})=-\varepsilon
h$ and $hc_{2}({\mathcal{E}})=k$. The cohomology of Sequence (5.2) implies
$h^{0}\big{(}{\mathcal{E}}\big{)}\leq
h^{0}\big{(}{\mathcal{I}}_{Z|X}((q_{X}^{\varepsilon}-1)h)\big{)}$. The above
vanishing is trivial if $q_{X}^{\varepsilon}=1$. Thus we have to deal with
${\mathbb{P}^{3}}$ and the smooth quadric $Q\subseteq{\mathbb{P}^{4}}$: in
this case $q_{X}^{\varepsilon}=2$.

In both cases $Z$ is the union of $s=(k+2)/2\geq 2$ pairwise disjoint conics.
Thus, if $X\cong{\mathbb{P}^{3}}$, then $Z$ is not contained in any plane,
hence $h^{0}\big{(}{\mathcal{I}}_{Z|X}(h)\big{)}=0$. Assume $X\cong Q$ and let
$h^{0}\big{(}{\mathcal{I}}_{Z|X}(h)\big{)}\neq 0$. Each hyperplane
$H\subseteq{\mathbb{P}^{4}}$ intersects $Q$ along a quadric $Q_{H}\subseteq
H\cong{\mathbb{P}^{3}}$ which has rank at least $3$, hence it is integral as
well. It is easy to check that two conics on an integral quadric in
${\mathbb{P}^{3}}$ always intersect. It follows that a
$h^{0}\big{(}{\mathcal{I}}_{Z|X}(h)\big{)}=0$ necessarily. Consequently,
$h^{0}\big{(}{\mathcal{E}}\big{)}=0$ also when $X$ is either
${\mathbb{P}^{3}}$ or $Q$.

Let $D$ be a divisor on $X$ such that
$\mu({\mathcal{O}}_{X}(D))\geq\mu({\mathcal{E}})$. If $dh$ is its class, then
such a condition forces $d\geq-\varepsilon/2\geq-1/2$, hence $d\geq 0$
necessarily. Thus $h^{0}\big{(}{\mathcal{E}}(-D)\big{)}\leq
h^{0}\big{(}{\mathcal{E}}\big{)}=0$. Thanks to Lemma 2.3, we deduce that
${\mathcal{E}}$ is $\mu$–stable. ∎

We now deal with the case $\varrho_{X}\geq 2$ distinguishing two cases.

Proof of Theorem 1.5 when $\varrho_{X}\geq 2$. The proof is similar to the
previous one. We have only to put some more care in showing that
${\mathcal{E}}$ is $\mu$–(semi)stable.

We have three distinct cases to deal with. The threefold $X$ can be
$F_{6,2}\subseteq{\mathbb{P}^{8}}$, or $F_{6,3}\subseteq{\mathbb{P}^{7}}$, or
$F_{7}\subseteq{\mathbb{P}^{8}}$.

Let $X:=F_{6,2}$. Recall that conics on ${F_{6,2}}$ were described in Remark
4.2, where we showed that $\Gamma({F_{6,2}})\cong|h_{1}|\times|h_{2}|$. Thus
two distinct components of $Z$ cannot be contained in the same divisor in
either $|h_{1}|$ or $|h_{2}|$.

We show that ${\mathcal{E}}$ is $\mu$–(semi)stable using Lemma 2.3 and [15,
Proposition 2.5]. To this purpose let
${\mathcal{O}}_{F_{6,2}}(D)\cong{\mathcal{O}}_{F_{6,2}}(\alpha_{1}h_{1}+\alpha_{2}h_{2})$:
we have $\mu({\mathcal{O}}_{F_{6,2}}(D))\geq\mu({\mathcal{E}})$ if and only if
$\alpha_{1}+\alpha_{2}\geq-1$. Thus either both $\alpha_{i}$’s are
non–negative, or one of them is strictly negative and the other one
non–negative. In both cases
$h^{0}\big{(}{\mathcal{O}}_{F_{6,2}}(-h-D)\big{)}=0$ thanks to [15,
Proposition 2.5]. The cohomology of Sequence (5.2) yields

$h^{0}\big{(}{\mathcal{E}}(-D)\big{)}\leq
h^{0}\big{(}{\mathcal{I}}_{Z|{F_{6,2}}}(-D)\big{)}.$ (5.3)

If $\alpha_{i}\geq 1$ for some $i$, then the dimension on the right–hand side
of Inequality (5.3) vanishes. Thus we must deal with the case
$\alpha_{1},\alpha_{2}\leq 0$. If equality holds, then again the dimension on
the right vanishes. In particular ${\mathcal{E}}$ is certainly at least
$\mu$–semistable. It then remains to deal with the case $-D=h_{i}$ for some
$i$. In this case [15, Proposition 2.5] yields
$h^{1}\big{(}{\mathcal{O}}_{F_{6,2}}(-h+h_{i})\big{)}=0$, hence
$h^{0}\big{(}{\mathcal{E}}(h_{i})\big{)}=h^{0}\big{(}{\mathcal{I}}_{Z|{F_{6,2}}}(h_{i})\big{)}$.
Since $k\geq 4$, it follows that $Z$ has at least two connected components: as
we showed above such components cannot be contained in the same divisor in
$|h_{i}|$, hence $h^{0}\big{(}{\mathcal{I}}_{Z|{F_{6,2}}}(h_{i})\big{)}=0$. It
follows that ${\mathcal{E}}$ is $\mu$–stable in this range, hence it is also
indecomposable.

Let $X:=F_{6,3}$. Recall that each irreducible component in $Z$ is the
complete intersection of divisors in $|h_{i}|$ and $|h-h_{i}|$ and that two
distinct components of $Z$ cannot be contained in the same divisors in either
$|h_{i}|$ or $|h-h_{i}|$. We will deal with the $\mu$–semistability of
${\mathcal{E}}$ as in the previous case using Lemma 2.3 and Künneth formulas.
In this case if
${\mathcal{O}}_{F_{6,3}}(D)\cong{\mathcal{O}}_{F_{6,3}}(\alpha_{1}h_{1}+\alpha_{2}h_{2}+\alpha_{3}h_{3})$
we have $\mu({\mathcal{O}}_{F_{6,3}}(D))\geq\mu({\mathcal{E}})$ if and only if
$\alpha_{1}+\alpha_{2}+\alpha_{3}\geq-3/2$. Thus either all the $\alpha_{i}$’s
are non–negative, or one of them is strictly negative and one non–negative. In
both cases $h^{0}\big{(}{\mathcal{O}}_{F_{6,3}}(-h-D)\big{)}=0$, hence we can
use the same argument used for proving the $\mu$–stability in the previous
case.

Finally we examine the $\mu$–stability of ${\mathcal{E}}$ when $X:=F_{7}$
using Lemma 2.3 and [16, Proposition 2.7]. The argument is essentially the
same used before, but we need some more care because the geometry of ${F_{7}}$
is slightly richer. Indeed in this case $\Gamma({F_{7}})$ is the union of the
fourfold of conics which are complete intersection of divisors in $|\xi|$ and
$|f|$, with the ${\mathbb{P}^{5}}$ of conics inside $E$, as we showed in
Remark 4.4. Anyhow there are still no divisors in $|f|$ containing more than
one component of $Z$. In this case if
${\mathcal{O}}_{F_{7}}(D)\cong{\mathcal{O}}_{F_{7}}(\alpha_{1}\xi+\alpha_{2}f)$,
then $\mu({\mathcal{O}}_{F_{7}}(D))\geq\mu({\mathcal{E}})$ if and only if

$4\alpha_{1}+3\alpha_{2}\geq-7/2.$ (5.4)

The cohomology of Sequence (5.2) yields

$\displaystyle h^{0}\big{(}{\mathcal{E}}(-\alpha_{1}\xi-\alpha_{2}f)\big{)}$
$\displaystyle\leq
h^{0}\big{(}{\mathcal{O}}_{{F_{7}}}(-(\alpha_{1}+1)\xi-(\alpha_{2}+1)f)\big{)}$
$\displaystyle+h^{0}\big{(}{\mathcal{I}}_{Z|{F_{7}}}(-\alpha_{1}\xi-\alpha_{2}f)\big{)}.$

If $\alpha_{1}\geq 1$ the last member is zero, because
$-\alpha_{1}=(-\alpha_{1}\xi-\alpha_{2}f)f^{2}$. Thus we have to deal with the
case $\alpha_{1}\leq 0$. In this case Inequality (5.4) yields

$-(\alpha_{1}+1)-(\alpha_{2}+1)\leq-\alpha_{1}+\frac{8\alpha_{1}+7}{6}-2=\frac{2\alpha_{1}-5}{6}<0$

hence
$h^{0}\big{(}{\mathcal{O}}_{{F_{7}}}(-(\alpha_{1}+1)\xi-(\alpha_{2}+1)f)\big{)}=0$.
Moreover,

$-\alpha_{1}-\alpha_{2}\leq-\alpha_{1}+\frac{8\alpha_{1}+7}{6}=\frac{2\alpha_{1}+7}{6}.$

If $-\alpha_{1}-\alpha_{2}\leq-1$, then
$h^{0}\big{(}{\mathcal{I}}_{Z|{F_{7}}}(-\alpha_{1}\xi-\alpha_{2}f)\big{)}\leq
h^{0}\big{(}{\mathcal{O}}_{{F_{7}}}(-\alpha_{1}\xi-\alpha_{2}f)\big{)}=0$.
Thus we have to check the cases $-\alpha_{1}-\alpha_{2}=1$ and
$-\alpha_{1}-\alpha_{2}=0$.

In the former case $\alpha_{2}=-1-\alpha_{1}$, hence Inequality (5.4) returns
$2\alpha_{1}\geq-1$. The hypothesis $\alpha_{1}\leq 0$ then forces
$\alpha_{1}=0$ and we have to check if
$h^{0}\big{(}{\mathcal{I}}_{Z|{F_{7}}}(f)\big{)}=0$. In the latter case
$\alpha_{2}=-\alpha_{1}$ and we have to check if
$h^{0}\big{(}{\mathcal{I}}_{Z|{F_{7}}}(-\alpha_{1}\xi+\alpha_{1}f)\big{)}=0$.
This is true if $\alpha_{1}=0$, hence we consider the case $\alpha_{1}\leq-1$
in what follows. In this case
$h^{0}\big{(}{\mathcal{O}}_{{F_{7}}}(-\alpha_{1}\xi+\alpha_{1}f)\big{)}=1$,
hence the linear system $|-\alpha_{1}\xi+\alpha_{1}f|$ only contains the
divisor $-\alpha_{1}E$. Since we are assuming $k\geq 4$, then $Z$ has at least
two connected components, which cannot be contained simultaneously in either a
divisor in $|f|$ or in $E$, as explained above. Thus the proof of the
$\mu$–stability of ${\mathcal{E}}$ is complete. ∎

##  6\. Non–ordinary instanton bundles on ordinary prime Fano threefolds

In this section we will give some results on instanton bundles on an ordinary
prime Fano threefold $X$, proving Theorem 1.6. We will also assume that
${\mathcal{O}}_{X}(h)$ is very ample.

In this case $H^{2}(X)$ and $H^{4}(X)$ are respectively generated by the
hyperplane class $h$ and by the class $\ell$ of a line. Equality (2.6) returns

$\chi({\mathcal{O}}_{X}(h))=\frac{h^{3}}{2}+3.$

Thus $\deg(X)=h^{3}$ is an even integer. The number
$g_{X}:=\chi({\mathcal{O}}_{X}(h))-2$ is usually called the genus of $X$. If
$\varrho_{X}=1$, then $3\leq g_{X}\leq 12$ and $g_{X}\neq 11$.

In this case the irreducible components of $\Lambda(X)$ and $\Gamma(X)$ have
dimensions $1$ and $2$ respectively (see [39, Lemmas 2.2.3 and 2.3.4]).
Moreover, there is component $\Gamma(X)_{0}\subseteq\Gamma(X)$ whose general
point is an integral conic $Z$ by [35, Theorem 4.4]. The conics in
$\Gamma(X)_{0}$ cover $X$ by [39, Lemma 2.3.4], hence the normal bundle of $Z$
satisfies ${\mathcal{N}}_{Z|X}\cong{\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus 2}$
by [35, Proposition 4.3].

######  Lemma 6.1.

Let $X$ be an ordinary prime Fano threefold with very ample
${\mathcal{O}}_{X}(h)$.

If $Z\in\Gamma(X)_{0}$ is general, then the general conic in $\Gamma(X)_{0}$
and the general line in each component of $\Lambda(X)$ do not intersect $Z$.

###### Proof.

There exists a conic $C$ not intersecting $Z$: see [9, Proof of Step 1 of the
proof of Theorem 4.1] where it suffices to replace $\mathcal{H}^{0}_{2}(X)$
with $\Gamma(X)_{0}$. Thus the same is true for the general conic in
$\Gamma(X)_{0}$.

Let us now prove that there is a also a line in each component of
$\Lambda(X)_{i}\subseteq\Lambda(X)$ not intersecting $Z$. Assume that all the
conics in $\Gamma(X)_{0}$ intersect each line in $\Lambda(X)_{i}$. The scheme
$R_{i}:=\bigcup_{L\in\Lambda(X)_{i}}L\subseteq X$ is an irreducible surface.
If $Z\in\Gamma(X)_{0}$ is general, then $Z\cap R_{i}$ is a finite set of
points, because $X$ is dominated by $\Gamma(X)_{0}$. Since the surfaces
$R_{i}$ are irreducible, it follows that we can find a point $v_{i}\in Z\cap
R_{i}$ lying on all the lines in $R_{i}$. Thus each $R_{i}$ is a cone with
vertex $v_{i}$ and the general conic in $\Gamma(X)_{0}$ passes necessarily
through all such points, hence it intersects $Z$, contradicting what we proved
above. ∎

In what follows we want to construct instanton bundles on each prime Fano
threefold. On the one hand, the only line bundle we can actually use in Serre
construction for obtaining instanton bundles is ${\mathcal{O}}_{X}$. Thus the
only admissible construction by means of Serre correspondence is actually
Construction 5.2. On the other hand, such a construction does not return
instanton bundles ${\mathcal{E}}$ because in this case
$h^{0}\big{(}{\mathcal{E}}\big{)}=1$, as pointed out in the previous section.
For this reason we need a different and smarter approach for prime Fano
threefolds.

The existence of minimal instanton bundles on a Fano threefold $X$ follows
from [9, Theorem 4.1]. Thanks to an easy induction on the quantum number, it
is possible to prove the existence of even instanton bundles on each prime
Fano threefold for every admissible quantum number.

We will prove the existence of instanton bundles by induction on $k$. The
proof of the inductive step essentially coincides with few differences with
the proof of [9, Theorem 4.1] which suffices to prove the existence of minimal
instanton bundles. Anyhow, in order to start with our induction we need some
more properties of such minimal bundles. The proofs of these properties can be
found in [9], but they are a little hidden in the proofs and not explicitly
indicated in the statements. Thus, for reader’s benefit we explain how to
recover them in what follows and how to adapt the proof in [9] to get the
inductive step.

To this purpose we introduce a new construction.

######  Construction 6.2.

Let $X$ be a prime Fano threefold with very ample ${\mathcal{O}}_{X}(h)$.

If $k=2$, let ${\mathcal{E}}_{2}$ be the bundle ${\mathcal{F}}$ obtained via
Construction 5.2 from a general $Z\in\Gamma(X)_{0}$ and
$\eta\colon{\mathcal{O}}_{X}\to{\mathcal{E}}_{2}$ the corresponding inclusion
map. By construction the bundle ${\mathcal{E}}_{2}$ is strictly
$\mu$–semistable: moreover, restricting Sequence (5.1) to a general integral
conic $Y\in\Gamma(X)_{0}$, we deduce that
${\mathcal{E}}_{2}\otimes{\mathcal{O}}_{Y}\cong{\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus
2}$.

If $k\geq 4$ is even, let ${\mathcal{E}}_{k}$ be a $\mu$–stable even instanton
bundle with quantum number $k$, such that
${\mathcal{E}}_{k}\otimes{\mathcal{O}}_{Y}\cong{\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus
2}$ for the general $Y\in\Gamma(X)_{0}$ and
$\mathrm{Ext}_{X}^{2}\big{(}{\mathcal{E}}_{k},{\mathcal{E}}_{k}\big{)}=0$.

Thus, if $C\in\Gamma(X)_{0}$ is general, then the general
$\varphi\colon{\mathcal{E}}_{k}\to{\mathcal{O}}_{C}$ is surjective. We define
${\mathcal{E}}_{k,\varphi}:=\ker(\varphi)$. By construction there is the exact
sequence

$0\longrightarrow{\mathcal{E}}_{k,\varphi}\longrightarrow{\mathcal{E}}_{k}\longrightarrow{\mathcal{O}}_{C}\longrightarrow
0.$ (6.1)

While the existence of ${\mathcal{E}}_{2}$ is clear, the existence of
${\mathcal{E}}_{k}$ when $k\geq 4$ will be proved in what follows. Anyhow
${\mathcal{E}}_{k,\varphi}$, if it exists, is not locally free at the points
of $C$, thus it cannot be an instanton bundle, though it enjoys many
properties of instanton bundles as we will show in the following proposition.

######  Proposition 6.3.

Let $X$ be an ordinary prime Fano threefold with very ample
${\mathcal{O}}_{X}(h)$.

For each even $k\geq 2$, the sheaf ${\mathcal{E}}_{k,\varphi}$ defined in
Construction 6.2 satisfies the following properties.

  * •

${\mathcal{E}}_{k,\varphi}$ is torsion–free and
$c_{1}({\mathcal{E}}_{k,\varphi})=0$, $hc_{2}({\mathcal{E}}_{k,\varphi})=k+2$,
$c_{3}({\mathcal{E}}_{k,\varphi})=0$.

  * •

$h^{0}\big{(}{\mathcal{E}}_{k,\varphi}\big{)}=h^{1}\big{(}{\mathcal{E}}_{k,\varphi}(-h)\big{)}=0$.

  * •

${\mathcal{E}}_{k,\varphi}$ is simple.

  * •

For the general $Y\in\Gamma(X)_{0}$, then
${\mathcal{E}}_{k,\varphi}\otimes{\mathcal{O}}_{Y}\cong{\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus
2}$.

  * •

$\dim\mathrm{Ext}^{1}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k,\varphi}\big{)}=2k+1$,
$\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k,\varphi}\big{)}=\mathrm{Ext}^{3}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k,\varphi}\big{)}=0$.

###### Proof.

Trivially ${\mathcal{E}}_{k,\varphi}$ is torsion free, because it is contained
in the vector bundle ${\mathcal{E}}_{k}$. By combining Sequences (5.1) and
(2.2) we obtain

$0\longrightarrow{\mathcal{O}}_{X}\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{O}}_{X}\longrightarrow{\mathcal{O}}_{C}\longrightarrow
0.$ (6.2)

We deduce that the product of the Chern polynomials of ${\mathcal{F}}$ and
${\mathcal{O}}_{C}$ must be $1$, hence $c_{1}({\mathcal{O}}_{C})=0$,
$c_{2}({\mathcal{O}}_{C})=-C$ and $c_{3}({\mathcal{O}}_{C})=0$. Using these
informations we compute the Chern classes of ${\mathcal{E}}_{k,\varphi}$ from
Sequence (6.1) obtaining $c_{1}({\mathcal{E}}_{k,\varphi})=0$,
$c_{2}({\mathcal{E}}_{k,\varphi})=c_{2}({\mathcal{E}}_{k})+C$, hence
$hc_{2}({\mathcal{E}}_{k,\varphi})=hc_{2}({\mathcal{E}}_{k})+2$, and
$c_{3}({\mathcal{E}}_{k,\varphi})=0$.

When $k\geq 4$, the vanishings
$h^{0}\big{(}{\mathcal{E}}_{k,\varphi}\big{)}=h^{1}\big{(}{\mathcal{E}}_{k,\varphi}(-h)\big{)}=0$
follow trivially from the cohomology of Sequence (6.1), because the same
vanishings hold for ${\mathcal{E}}_{k}$ by induction and
$h^{0}\big{(}{\mathcal{O}}_{C}\otimes{\mathcal{O}}_{X}(-h)\big{)}=0$.

Let $k=2$. Notice that
$\mathrm{im}(\eta)\not\subseteq{\mathcal{E}}_{2,\varphi}$, hence the
restriction $\varphi\circ\eta\colon{\mathcal{O}}_{X}\to{\mathcal{O}}_{C}$ is
surjective. Combining Sequences (5.1) and (6.1), the Snake lemma implies the
existence of an exact sequence

$0\longrightarrow{\mathcal{I}}_{C|X}\longrightarrow{\mathcal{E}}_{2,\varphi}\longrightarrow{\mathcal{I}}_{Z|X}\longrightarrow
0,$ (6.3)

whence we deduce
$h^{0}\big{(}{\mathcal{E}}_{2,\varphi}\big{)}=h^{1}\big{(}{\mathcal{E}}_{2,\varphi}(-h)\big{)}=0$.

Let us prove that ${\mathcal{E}}_{k,\varphi}$ is simple. If $k=2$ then it
follows from [9, Lemma 4.5]. If $k\geq 4$, then ${\mathcal{E}}_{k,\varphi}$ is
actually $\mu$–stable, because each sheaf destabilizing
${\mathcal{E}}_{k,\varphi}$, also destabilizes ${\mathcal{E}}_{k}$, because
$\mu({\mathcal{E}}_{k,\varphi})=\mu({\mathcal{E}}_{k})$. Thus
${\mathcal{E}}_{k,\varphi}$ is also simple thanks to [33, Corollary 1.2.8].

By Lemma 6.1, a general $Y\in\Gamma(X)_{0}$ does not intersect $C$.
Restricting Sequence (6.1) to $Y$ we then obtain
${\mathcal{E}}_{k,\varphi}\otimes{\mathcal{O}}_{Y}\cong{\mathcal{E}}\otimes{\mathcal{O}}_{Y}\cong{\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus
2}$. In the remaining part of the proof, we will compute
$\dim\mathrm{Ext}^{i}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k,\varphi}\big{)}$,
for $i\geq 1$.

Recall that
$\mathrm{Ext}^{3}_{X}\big{(}{\mathcal{E}}_{k},{\mathcal{E}}_{k}\big{)}=0$ for
each $k\geq 4$, thanks to Lemma 2.2, because ${\mathcal{E}}_{k}$ is simple. If
$k=2$, then the isomorphism ${\mathcal{E}}_{2}\cong{\mathcal{E}}_{2}^{\vee}$
and Equality (2.4) yield

$\dim\mathrm{Ext}^{3}_{X}\big{(}{\mathcal{E}}_{2},{\mathcal{E}}_{2}\big{)}=h^{0}\big{(}{\mathcal{E}}_{2}\otimes{\mathcal{E}}_{2}(-h)\big{)}.$

The cohomologies of Sequences (5.2) and (2.2) tensored by
${\mathcal{E}}_{2}(-h)$ imply

$h^{0}\big{(}{\mathcal{E}}_{2}\otimes{\mathcal{E}}_{2}(-h)\big{)}\leq
2h^{0}\big{(}{\mathcal{E}}_{2}(-h)\big{)}.$

Again the cohomologies of the same sequences tensored by
${\mathcal{O}}_{X}(-h)$ imply that the dimension on the right is zero.
Similarly,
$\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}}_{k},{\mathcal{E}}_{k}\big{)}=0$ when
$k\geq 4$ by hypothesis. If $k=2$, then
$\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}}_{2},{\mathcal{E}}_{2}\big{)}=0$
thanks to [9, Lemma 4.4].

If we apply $\mathrm{Hom}_{X}\big{(}{\mathcal{E}}_{k},-\big{)}$ to Sequence
(6.1), then
$\mathrm{Ext}^{i}_{X}\big{(}{\mathcal{E}}_{k},{\mathcal{E}}_{k,\varphi}\big{)}\cong\mathrm{Ext}^{i}_{X}\big{(}{\mathcal{E}}_{k},{\mathcal{E}}_{k}\big{)}$
for $i\geq 2$, because
$\mathrm{Ext}^{i}_{X}\big{(}{\mathcal{E}}_{k},{\mathcal{O}}_{C}\big{)}\cong
H^{i}\big{(}{\mathcal{O}}_{C}^{\oplus 2}\big{)}=0$ for $i\geq 1$. It follows
that
$\mathrm{Ext}^{3}_{X}\big{(}{\mathcal{E}}_{k},{\mathcal{E}}_{k,\varphi}\big{)}=\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}}_{k},{\mathcal{E}}_{k,\varphi}\big{)}=0$
for each $k\geq 2$. Thus,

$\mathrm{Ext}^{3}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k,\varphi}\big{)}=0,\qquad\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k,\varphi}\big{)}\subseteq\mathrm{Ext}^{3}_{X}\big{(}{\mathcal{O}}_{C},{\mathcal{E}}_{k,\varphi}\big{)},$

for each $k\geq 2$, by applying
$\mathrm{Hom}_{X}\big{(}-,{\mathcal{E}}_{k,\varphi}\big{)}$ to Sequence (6.1).

Equality (2.4) implies
$\mathrm{Ext}^{3}_{X}\big{(}{\mathcal{O}}_{C},{\mathcal{E}}_{k,\varphi}\big{)}\cong\mathrm{Hom}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{O}}_{C}\otimes{\mathcal{O}}_{X}(-h)\big{)}$:
notice that
${\mathcal{O}}_{C}\otimes{\mathcal{O}}_{X}(-h)\cong{\mathcal{O}}_{{\mathbb{P}^{1}}}(-2)$.
Thus from ${\mathcal{N}}_{C|X}\cong{\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus 2}$
we deduce

$\mathcal{E}\kern-1.07639pt{\mathit{x}t}^{1}_{X}\big{(}{\mathcal{O}}_{C},{\mathcal{O}}_{C}\otimes{\mathcal{O}}_{X}(-h)\big{)}\cong{\mathcal{O}}_{{\mathbb{P}^{1}}}(-2)^{\oplus
2}$

(see Equalities (2.5)). By the above equalities, applying
$\mathcal{H}\kern-1.07639pt{\mathit{o}m}_{X}\big{(}-,{\mathcal{O}}_{{\mathbb{P}^{1}}}(-2)\big{)}$
to Sequence (6.1) we obtain the exact sequence

$\displaystyle
0\longrightarrow{\mathcal{O}}_{{\mathbb{P}^{1}}}(-2)\longrightarrow{\mathcal{O}}_{{\mathbb{P}^{1}}}(-2)^{\oplus
2}\longrightarrow\mathcal{H}\kern-1.07639pt{\mathit{o}m}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{O}}_{{\mathbb{P}^{1}}}(-2)\big{)}\longrightarrow{\mathcal{O}}_{{\mathbb{P}^{1}}}(-2)^{\oplus
2}\longrightarrow 0,$

because, being ${\mathcal{E}}_{k}$ locally free, we have
$\mathcal{E}\kern-1.07639pt{\mathit{x}t}^{1}_{X}\big{(}{\mathcal{E}}_{k},{\mathcal{O}}_{{\mathbb{P}^{1}}}(-2)\big{)}=0$.
Trivially the injective map on the left has a section. Thus taking the
cohomology of the induced short exact sequence yields

$\mathrm{Hom}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{O}}_{{\mathbb{P}^{1}}}(-2)\big{)}=H^{0}\big{(}\mathcal{H}\kern-1.07639pt{\mathit{o}m}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{O}}_{{\mathbb{P}^{1}}}(-2)\big{)}\big{)}=0,$

hence
$\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k,\varphi}\big{)}=0$.
Thus
$\dim\mathrm{Ext}^{1}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k,\varphi}\big{)}=1-\chi({\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k,\varphi})$.
By applying $\mathrm{Hom}_{X}({\mathcal{E}}_{k,\varphi},-\big{)}$,
$\mathrm{Hom}_{X}(-,{\mathcal{E}}_{k}\big{)}$ and
$\mathrm{Hom}_{X}(-,{\mathcal{O}}_{C}\big{)}$ to Sequence (6.1) we deduce that

$\chi({\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k,\varphi})=\chi({\mathcal{E}}_{k},{\mathcal{E}}_{k})-\chi({\mathcal{E}}_{k},{\mathcal{O}}_{C})-\chi({\mathcal{O}}_{C},{\mathcal{E}}_{k})+\chi({\mathcal{O}}_{C},{\mathcal{O}}_{C}).$

We have $\chi({\mathcal{E}}_{k},{\mathcal{O}}_{C})=2\chi({\mathcal{O}}_{C})=2$
and
$\chi({\mathcal{O}}_{C},{\mathcal{E}}_{k})=-2\chi({\mathcal{O}}_{C}\otimes{\mathcal{O}}_{X}(-h))=2$.
Equality (2.6) returns $\chi({\mathcal{E}}_{k},{\mathcal{E}}_{k})=4-2k$. From
Sequence (6.2) we obtain
$\chi({\mathcal{O}}_{C},{\mathcal{O}}_{C})=2\chi({\mathcal{O}}_{X},{\mathcal{O}}_{C})-\chi({\mathcal{F}},{\mathcal{O}}_{C})=0$,
because
${\mathcal{F}}\otimes{\mathcal{O}}_{C}\cong{\mathcal{N}}_{C|X}\cong{\mathcal{O}}_{C}^{\oplus
2}$. We deduce from the above computations that
$\dim\mathrm{Ext}^{1}_{X}\big{(}{\mathcal{E}}_{\varphi},{\mathcal{E}}_{\varphi}\big{)}=2k+1$.

The proof of the statement is then complete. ∎

Before starting the proof of Theorem 1.6 we collect below some helpful
remarks. We use the same notation as in Construction 6.2.

######  Remark 6.4.

If $Y\subseteq X$ is any pure $1$–dimensional scheme, then

$\mathrm{Ext}^{1}_{X}\big{(}{\mathcal{O}}_{Y},{\mathcal{E}}_{k,\varphi}\big{)}=\left\\{\begin{array}[]{ll}0&\text{if
$C\neq Y$,}\\\ 1&\text{if $C=Y$.}\end{array}\right.$

Indeed, by applying $\mathrm{Hom}_{X}\big{(}{\mathcal{O}}_{Y},-\big{)}$ to
Sequence (6.1) and taking into account Equality (2.4) we obtain

$\mathrm{Ext}^{1}_{X}\big{(}{\mathcal{O}}_{Y},{\mathcal{E}}_{k,\varphi}\big{)}\cong\mathrm{Hom}_{X}\big{(}{\mathcal{O}}_{Y},{\mathcal{O}}_{C}\big{)}\cong
H^{0}\big{(}\mathcal{H}\kern-1.07639pt{\mathit{o}m}_{X}\big{(}{\mathcal{O}}_{Y},{\mathcal{O}}_{C}\big{)}\big{)}.$

We conclude by noticing that

$\mathcal{H}\kern-1.07639pt{\mathit{o}m}_{X}\big{(}{\mathcal{O}}_{Y},{\mathcal{O}}_{C}\big{)}=\left\\{\begin{array}[]{ll}0&\text{if
$C\neq Y$,}\\\ {\mathcal{O}}_{C}&\text{if $C=Y$.}\end{array}\right.$

######  Remark 6.5.

By applying $\mathrm{Hom}_{X}\big{(}-,{\mathcal{E}}_{k}\big{)}$ to Sequence
(6.1) and taking into account Equality (2.4) we obtain
$\mathrm{Hom}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k}\big{)}\cong\mathbb{C}$.

By applying
$\mathcal{H}\kern-1.07639pt{\mathit{o}m}_{X}\big{(}-,{\mathcal{O}}_{X}\big{)}$
to Sequence (6.1) we deduce
${\mathcal{E}}_{k,\varphi}^{\vee\vee}\cong{\mathcal{E}}_{k}$, because
${\mathcal{O}}_{C}$ is a torsion sheaf on $X$ and
$\mathcal{E}\kern-1.07639pt{\mathit{x}t}^{1}_{X}\big{(}{\mathcal{O}}_{C},{\mathcal{O}}_{X}\big{)}\cong\mathcal{E}\kern-1.07639pt{\mathit{x}t}^{1}_{X}\big{(}{\mathcal{O}}_{C},\omega_{X}\big{)}\otimes{\mathcal{O}}_{X}(2h)=0$.

The above isomorphisms show that the inclusion
${\mathcal{E}}_{k,\varphi}\to{\mathcal{E}}_{k}$ can be uniquely identified
with the canonical monomorphism
${\mathcal{E}}_{k,\varphi}\to{\mathcal{E}}_{k,\varphi}^{\vee\vee}$, i.e.
Sequence (6.1) is canonically isomorphic with

$0\longrightarrow{\mathcal{E}}_{k,\varphi}\longrightarrow{\mathcal{E}}_{k,\varphi}^{\vee\vee}\longrightarrow{\mathcal{O}}_{C}\longrightarrow
0.$

Let ${\mathcal{S}}_{X}\big{(}2;-\varepsilon
h,\zeta,0\big{)}\subseteq{\mathcal{S}}_{X}$ be the locus of simple rank $2$
torsion–free sheaves with fixed Chern classes $-\varepsilon h\in H^{2}(X)$,
$\zeta\in H^{4}(X)$ and $0\in H^{6}(X)$ in what follows In Proposition 6.3 we
proved that ${\mathcal{E}}_{k,\varphi}$ represents a point in
${\mathcal{S}}_{X}\big{(}2;0,(k+2)\ell,0\big{)}$. The core of the proof of
Theorem 1.6 is the following proposition.

######  Proposition 6.6.

Let $X$ be an ordinary prime Fano threefold with very ample
${\mathcal{O}}_{X}(h)$.

For each even $k\geq 2$, a general deformation
${\mathcal{E}}\in{\mathcal{S}}_{X}\big{(}2;0,(k+2)\ell,0\big{)}$ of
${\mathcal{E}}_{k,\varphi}$ is a $\mu$–stable vector bundle such that
$\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}},{\mathcal{E}}\big{)}=0$ and
${\mathcal{E}}\otimes{\mathcal{O}}_{Y}\cong{\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus
2}$ for each general $Y\in\Gamma(X)_{0}$.

###### Proof.

In what follows we will denote by ${\mathcal{S}}(2)$ the locus inside
${\mathcal{S}}_{X}\big{(}2;0,2\ell,0\big{)}$ of bundles obtained via
Construction 5.2 from a general $Z\in\Gamma(X)_{0}$ and by
${\mathcal{S}}(k+2)$ the component of
${\mathcal{S}}_{X}\big{(}2;0,(k+2)\ell,0\big{)}$ containing the point
corresponding to the sheaf ${\mathcal{E}}_{k,\varphi}$ for each even $k\geq
2$. Moreover, ${\mathcal{S}}(k+2)$ is smooth at ${\mathcal{E}}_{k,\varphi}$ of
dimension

$\dim\mathrm{Ext}^{1}_{X}\big{(}{\mathcal{E}}_{k,\varphi},{\mathcal{E}}_{k,\varphi}\big{)}=2k+1.$

Let ${\mathcal{S}}^{bad}(k+2)\subseteq{\mathcal{S}}(k+2)$ be the locus of
sheaves obtained via Construction 6.2. We claim that

$\dim({\mathcal{S}}^{bad}(k+2))\leq 2k.$ (6.4)

If $k=2$, then the points in ${\mathcal{S}}^{bad}(k+2)$ are parameterized by
two points in $\Gamma(X)_{0}$ (the conics $Z$ and $C$) and elements in
$\mathbb{P}\big{(}\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{I}}_{Z|X},{\mathcal{I}}_{C|X}\big{)}\big{)}$
(the Sequence (6.3)). Thanks to the computations in [9, Proof of Step 3 of the
proof of Theorem 4.1], we know that the latter projective space reduces to a
single point, whence we deduce that Inequality (6.4) is fulfilled.

If $k\geq 4$ is even, then the points in ${\mathcal{S}}^{bad}(k+2)$ are
parameterized by the points of ${\mathcal{S}}(k)$ (the choice of
${\mathcal{E}}_{k}$), the points in $\Gamma(X)_{0}$ (the conic $C$) and
elements in
$\mathbb{P}\big{(}\mathrm{Hom}_{{\mathbb{P}^{1}}}\big{(}{\mathcal{O}}_{{\mathbb{P}^{1}}}^{\oplus
2},{\mathcal{O}}_{{\mathbb{P}^{1}}}\big{)}\big{)}$ (the morphism
$\mathrm{Hom}_{X}\big{(}{\mathcal{E}}_{k}\otimes{\mathcal{O}}_{C},{\mathcal{O}}_{C}\big{)}$
up to scalars). Thus Inequality (6.4) is again fulfilled.

It follows the existence of a flat family of torsion–free sheaves in
${\mathcal{S}}(k+2)$ over a smooth connected curve $\mathfrak{E}\to S$ with
$\mathfrak{E}_{s_{0}}\cong{\mathcal{E}}_{k,\varphi}$ and
$\mathfrak{E}_{s}\not\in{\mathcal{S}}^{bad}(k+2)$ for $s\neq s_{0}$. Thanks to
[12, Satz 3] we can assume
$\mathrm{Ext}^{2}_{X}\big{(}\mathfrak{E}_{s},\mathfrak{E}_{s}\big{)}=0$ for
general $s\in S$.

For each $s\in S$ we have a natural exact sequence of sheaves over $X$

$0\longrightarrow\mathfrak{E}_{s}\longrightarrow(\mathfrak{E}_{s})^{\vee\vee}\longrightarrow\mathfrak{T}_{s}\longrightarrow
0.$ (6.5)

where $\mathfrak{T}_{s}$ is a torsion sheaf on $X$ and the dual is taken with
respect to ${\mathcal{O}}_{X}$. Since $\mathfrak{E}_{s}$ is torsion–free, it
follows that the support $T_{s}$ of $\mathfrak{T}_{s}$ has codimension at
least $2$ in $X$ (see [46, Corollary to Theorem II.1.1.8]). Thanks to
Proposition 6.3, by semicontinuity we can assume that $\mathfrak{E}_{s}$ is
simple and $\mu$–semistable,
$h^{0}\big{(}\mathfrak{E}_{s}\big{)}=h^{1}\big{(}\mathfrak{E}_{s}(-h)\big{)}=0$
and
$\mathfrak{E}_{s}\otimes{\mathcal{O}}_{Y}\cong{\mathcal{O}}_{{\mathbb{P}^{1}}}^{\oplus
2}$ for a general $Y\in\Gamma(X)_{0}$. Actually, we have two possible cases,
namely either $T_{s}=\emptyset$ for general $s\in S$ or $T_{s}\neq\emptyset$
for each $s\in S$.

In the former case we can find $s\in S$ such that
$\mathfrak{E}_{s}\cong(\mathfrak{E}_{s})^{\vee\vee}$ is reflexive. We deduce
that $\mathfrak{E}_{s}$ is a vector bundle, because
$c_{3}(\mathfrak{E}_{s})=0$, thanks to [27, Proposition 2.6]. We already know
that $\mathrm{Ext}^{2}_{X}\big{(}\mathfrak{E}_{s},\mathfrak{E}_{s}\big{)}=0$:
an easy computation returns also the other dimensions. Moreover,
${\mathcal{E}}\otimes{\mathcal{O}}_{Y}\cong{\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus
2}$ for each general $Y\in\Gamma(X)_{0}$. Finally,
$h^{0}\big{(}\mathfrak{E}_{s}\big{)}=0$ by semicontinuity and Proposition 6.3,
hence $\mathfrak{E}_{s}$ can be assumed $\mu$–stable by Lemma 2.3 as well.
Thus the statement is proved with ${\mathcal{E}}:=\mathfrak{E}_{s}$.

In what follows we will show that the case $T_{s}\neq\emptyset$ for each $s\in
S$ does not occur. First we notice that if
$h^{0}\big{(}(\mathfrak{E}_{s})^{\vee\vee}(-h)\big{)}\neq 0$, then we can find
a subsheaf $\mathcal{L}\subseteq(\mathfrak{E}_{s})^{\vee\vee}$ with
$\mu(\mathcal{L})=h^{3}>0$. If $\mathcal{K}:=\mathcal{L}\cap\mathfrak{E}_{s}$,
then $\mu(\mathcal{K})=\mu(\mathcal{L})$, because the cokernel of the map
$\mathcal{K}\to\mathcal{L}$ induced by
$\mathfrak{E}_{s}\to(\mathfrak{E}_{s})^{\vee\vee}$ has support contained in
$T_{s}$: this fact contradicts the $\mu$–semistability of $\mathfrak{E}_{s}$,
because $\mathcal{K}\subseteq\mathfrak{E}_{s}$. It follows from the cohomology
of Sequence (6.5) tensored by ${\mathcal{O}}_{X}(-h)$ that

$h^{0}\big{(}\mathfrak{T}_{s}(-h)\big{)}=0,$ (6.6)

hence $T_{s}$ cannot contain embedded points, i.e. it has pure dimension $1$.
Equality (6.6) implies

$\chi(\mathfrak{T}_{s}(th))=-h^{1}\big{(}\mathfrak{T}_{s}(th)\big{)}$ (6.7)

for each $t\leq-1$. Since $(\mathfrak{E}_{s})^{\vee\vee}$ is reflexive, it
follows that $h^{1}\big{(}(\mathfrak{E}_{s})^{\vee\vee}(th)\big{)}=0$ for
$t\ll 0$ thanks to [27, Remark 2.5.1]. Thus the cohomology of Sequence (6.5)
tensored by ${\mathcal{O}}_{X}(th)$ yields

$h^{1}\big{(}\mathfrak{T}_{s}(th)\big{)}\leq
h^{2}\big{(}\mathfrak{E}_{s}(th)\big{)}\leq
h^{2}\big{(}{\mathcal{E}}_{k,\varphi}(th)\big{)}$ (6.8)

when $t\ll 0$ by semicontinuity.

We have $c_{1}((\mathfrak{E}_{s})^{\vee\vee})=c_{1}(\mathfrak{E}_{s})$.
Moreover, we know that $c_{3}((\mathfrak{E}_{s})^{\vee\vee}(th))$,
$c_{3}(\mathfrak{E}_{s}(th))$ and $c_{2}(\mathfrak{T}_{s}(th))$ are
independent of $t$ for each $t\in\mathbb{Z}$, thanks to [27, Lemma 2.1]. An
easy Chern class computation from Sequence (6.5) then yields

$\begin{gathered}c_{1}(\mathfrak{T}_{s}(th))=0,\qquad
hc_{2}(\mathfrak{T}_{s}(th))=hc_{2}((\mathfrak{E}_{s})^{\vee\vee})-k-2,\\\
c_{3}(\mathfrak{T}_{s}(th))=c_{3}((\mathfrak{E}_{s})^{\vee\vee})-2thc_{2}(\mathfrak{T}_{s}),\end{gathered}$

because $c_{i}(\mathfrak{E}_{s})=c_{i}({\mathcal{E}}_{k,\varphi})$. Equality
(2.6) for the sheaf $\mathfrak{T}(th)$ yields

$\chi(\mathfrak{T}_{s}(th))=\frac{c_{3}((\mathfrak{E}_{s})^{\vee\vee})-hc_{2}(\mathfrak{T}_{s})}{2}-thc_{2}(\mathfrak{T}_{s}).$
(6.9)

Since ${\mathcal{E}}_{k}$ is locally free, we know that
$h^{i}\big{(}{\mathcal{E}}_{k}(th)\big{)}=0$ for $t\ll 0$ and $i=1,2$, hence
the cohomology of Sequence (6.1) tensored by ${\mathcal{O}}_{X}(th)$ returns

$h^{2}\big{(}{\mathcal{E}}_{k,\varphi}(th)\big{)}=h^{1}\big{(}{\mathcal{O}}_{C}(th)\big{)}=-\chi({\mathcal{O}}_{C}(th))=-2t-1,$
(6.10)

for $t\ll 0$.

By combining Equalities (6.9) and (6.10) with Inequalities (6.8) and (6.7) we
finally obtain

$\frac{1}{2}\left({c_{3}((\mathfrak{E}_{s})^{\vee\vee})}-hc_{2}(\mathfrak{T}_{s})\right)-thc_{2}(\mathfrak{T}_{s})\geq
2t+1$

for $t\ll 0$. Thus $hc_{2}(\mathfrak{T}_{s})\geq-2$: it follows that either
the irreducible components of $T_{s}$ are up to two lines or $T_{s}$ is an
integral conic.

In what follows we prove that the first configuration cannot occur. Let
$L\subseteq T_{s}$ be a line: if equality holds, then there is an integer $a$
such that $\mathfrak{T}_{s}\cong{\mathcal{O}}_{L}(a)$. Assume that $L\neq
T_{s}$, so that $\deg(T_{s})=2$. Let $M\subseteq X$ be another line such that
$L+M=T_{s}$ in $A^{2}(X)$. The restriction map
${\mathcal{O}}_{T_{s}}\to{\mathcal{O}}_{L}$ induces an exact sequence

$0\longrightarrow\mathcal{B}\longrightarrow\mathfrak{T}_{s}\longrightarrow\mathcal{A}\longrightarrow
0,$ (6.11)

where $\mathcal{A}$ is supported on $L$ and $\mathcal{B}$ on $M$. Since
$\mathfrak{T}_{s}$ is torsion free, then again there are integers $a,b$ such
that $\mathcal{A}\cong{\mathcal{O}}_{L}(a)$ and
$\mathcal{B}\cong{\mathcal{O}}_{M}(b)$.

We claim that for each line $N\subseteq X$ and integer $t$ we have

$\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{O}}_{N}(t),\mathfrak{E}_{s}\big{)}=0.$
(6.12)

Indeed, by semicontinuity it suffices to check that
$\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{O}}_{N}(t),{\mathcal{E}}_{k,\varphi}\big{)}=0$.
To show such a vanishing we apply the functor
$\mathrm{Hom}_{X}\big{(}{\mathcal{O}}_{N}(t),-\big{)}$ to Sequence (6.1)
obtaining the exact sequence

$\mathrm{Hom}_{X}\big{(}{\mathcal{O}}_{N}(t),{\mathcal{O}}_{C}\big{)}\longrightarrow\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{O}}_{N}(t),{\mathcal{E}}_{k,\varphi}\big{)}\longrightarrow\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{O}}_{N}(t),{\mathcal{E}}_{k}\big{)}.$

Notice that the space on the left vanishes because $C\in\Gamma(X)_{0}$ is
general, hence integral, thanks to Remark 6.4. Equality (2.4) implies that

$\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{O}}_{N}(t),{\mathcal{E}}_{k}\big{)}\cong\mathrm{Ext}_{X}^{2}\big{(}{\mathcal{E}}_{k},{\mathcal{O}}_{N}(t)\otimes{\mathcal{O}}_{X}(-h)\big{)}\cong
H^{2}\big{(}{\mathcal{E}}_{k}(-h)\otimes{\mathcal{O}}_{N}(t)\big{)}=0,$

hence the claimed Vanishing (6.12) is completely proved.

By applying $\mathrm{Hom}_{X}\big{(}-,\mathfrak{E}_{s}\big{)}$ to Sequence
(6.11), we deduce that
$\mathrm{Ext}_{X}^{1}\big{(}\mathfrak{T}_{s},\mathfrak{E}_{s}\big{)}=0$,
thanks to Vanishing (6.12), hence Sequence (6.5) should split. Thus
$\mathfrak{T}_{s}$ should be the torsion subsheaf of
$(\mathfrak{E}_{s})^{\vee\vee}$, which is reflexive, hence torsion free, a
contradiction.

Therefore $T_{s}$ is an integral conic and
$\mathfrak{T}_{s}\cong{\mathcal{O}}_{T_{s}}(U)$ where $U\subseteq
T_{s}\cong{\mathbb{P}^{1}}$ is a divisor linearly equivalent to $up$ for some
point $p\in T_{s}$. Equality (6.9) with $t=0$ and the Riemann–Roch theorem on
$T_{s}$ yields $2u=c_{3}((\mathfrak{E})^{\vee\vee})$. On the one hand, the
sheaf $(\mathfrak{E}_{s})^{\vee\vee}$ is reflexive, thus the locus where it is
not locally free has degree $c_{3}((\mathfrak{E}_{s})^{\vee\vee})\geq 0$,
thanks to [27, Proposition 2.6], hence $u\geq 0$. On the other hand, Equality
(6.6) forces $u-2<0$. We conclude that $u\in\\{\ 0,1\ \\}$.

Let $u=1$: we will prove that such a case cannot occur, by showing that
$\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{O}}_{T_{s}}(U),\mathfrak{E}_{s}\big{)}=0$
as above. By semicontinuity it suffices to check that
$\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{O}}_{T_{s}}(U),{\mathcal{E}}_{k,\varphi}\big{)}=0$.
To this purpose we apply the functor
$\mathrm{Hom}_{X}\big{(}{\mathcal{O}}_{T_{s}}(U),-\big{)}$ to Sequence (6.1)
obtaining the exact sequence

$\mathrm{Hom}_{X}\big{(}{\mathcal{O}}_{T_{s}}(U),{\mathcal{O}}_{C}\big{)}\longrightarrow\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{O}}_{T_{s}}(U),{\mathcal{E}}_{k,\varphi}\big{)}\longrightarrow\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{O}}_{T_{s}}(U),{\mathcal{E}}_{k}\big{)}.$

Equality (2.4) implies that

$\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{O}}_{T_{s}}(U),{\mathcal{E}}_{k}\big{)}\cong\mathrm{Ext}_{X}^{2}\big{(}{\mathcal{E}}_{k},{\mathcal{O}}_{T_{s}}(U)\otimes{\mathcal{O}}_{X}(-h)\big{)}\cong
H^{2}\big{(}{\mathcal{E}}_{k}(-h)\otimes{\mathcal{O}}_{T_{s}}(U)\big{)}=0.$

We certainly have
$\mathrm{Hom}_{X}\big{(}{\mathcal{O}}_{T_{s}}(U),{\mathcal{O}}_{C}\big{)}\cong
H^{0}\big{(}\mathcal{H}\kern-1.07639pt{\mathit{o}m}_{X}\big{(}{\mathcal{O}}_{T_{s}}(U),{\mathcal{O}}_{C}\big{)}\big{)}$.
The hypothesis $u=1$ and Remark 6.4 then imply
$\mathrm{Hom}_{X}\big{(}{\mathcal{O}}_{T_{s}}(U),{\mathcal{O}}_{C}\big{)}=0$.

Thus the case $u=1$ is not possible, i.e. $u=0$, hence
$\mathfrak{T}_{s}\cong{\mathcal{O}}_{T_{s}}$ for each $s\in S$. It follows the
existence of a morphism $S\to\Gamma(X)_{0}$, such that $\mathfrak{T}$ is
exactly the pull–back of the universal conic on $\Gamma(X)_{0}$.

Thus $\mathfrak{T}\to S$ is a flat family and the flatness of the families
$\mathfrak{E}\to S$ and $\mathfrak{T}\to S$ yields the flatness of the induced
family $\bigcup_{s\in S}({\mathfrak{E}}_{s})^{\vee\vee}\to S$. In particular
$(\mathfrak{E}_{s_{0}})^{\vee\vee}\cong{\mathcal{E}}_{k}$ thanks to Remark
6.5, hence $(\mathfrak{E}_{s})^{\vee\vee}\in{\mathcal{S}}(k+2)$ and,
consequently, $\mathfrak{E}_{s}\in{\mathcal{S}}^{bad}(k+2)$ for general $s\in
S$, contradicting our initial choice. We conclude that the case
$T_{s}\neq\emptyset$ for each $s\in S$ cannot occur.

It follows that for general $s\in S$ the sheaf
${\mathcal{E}}:=\mathfrak{E}_{s}$ is a vector bundle. Since the locus of
$\mu$–stable sheaves inside ${\mathcal{S}}_{X}(2;0,(k+2)\ell,0)$ is open, it
follows that we can assume that ${\mathcal{E}}$ is $\mu$–stable. By
semicontinuity, thanks to Proposition 6.3 and [12], we know that
$\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}},{\mathcal{E}}\big{)}=0$ and
${\mathcal{E}}\otimes{\mathcal{O}}_{Y}\cong{\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus
2}$ for each general $Y\in\Gamma(X)_{0}$. ∎

We are finally ready to prove Theorem 1.6 stated in the introduction by
induction on even integers $k\geq 4$. The base step is [9, Theorem 4.1].

Proof of Theorem 1.6. If $k=2$, then Proposition 6.6 guarantees that the
general deformation ${\mathcal{E}}\in{\mathcal{S}}_{X}(2;0,4\ell,0)$ of
${\mathcal{E}}_{2,\varphi}$ is a vector bundle such that
$\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}},{\mathcal{E}}\big{)}=0$. Moreover,
by semicontinuity we can also assume $h^{1}\big{(}{\mathcal{E}}(-h)\big{)}=0$
and $h^{0}\big{(}{\mathcal{E}}\big{)}=0$. We deduce that ${\mathcal{E}}$ is
$\mu$–stable by Lemma 2.3, hence it is an instanton bundle with quantum number
$hc_{2}({\mathcal{E}})=hc_{2}({\mathcal{E}}_{2})+hC=4$.

Let $k\geq 4$ be even and assume the existence of a $\mu$–stable instanton
bundle ${\mathcal{E}}_{k}$ with quantum number $k$ such that
$\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}}_{k},{\mathcal{E}}_{k}\big{)}=0$ and
${\mathcal{E}}_{k}\otimes{\mathcal{O}}_{Y}\cong{\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus
2}$ for each general integral $Y\in\Gamma(X)_{0}$. The same argument used in
the base step implies the existence of a $\mu$–stable instanton bundle
${\mathcal{E}}\in{\mathcal{S}}_{X}(2;0,(k+2)\ell,0)$, such that
$\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}},{\mathcal{E}}\big{)}=0$ and
${\mathcal{E}}\otimes{\mathcal{O}}_{Y}\cong{\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus
2}$ for each general $Y\in\Gamma(X)_{0}$. ∎

We conclude this part by showing another possible approach to the construction
of even instanton bundles on a Fano threefold $X$ with $i_{X}=1$.

######  Remark 6.7.

Each Fano threefold $X$ with $i_{X}=\varrho_{X}=1$ and $g_{X}\geq 4$ contains
a canonical curve $Z\subseteq{\mathbb{P}^{g_{X}+1}}$ as pointed out in [13,
Proposition 8]. Indeed, if ${\mathcal{E}}$ is the bundle constructed in [9,
Theorem 4.1], then ${\mathcal{E}}(h)$ is globally generated and its general
section in $H^{0}\big{(}{\mathcal{E}}(h)\big{)}$ vanishes on a canonical curve
$Z$ with $p_{a}(Z)=g_{X}+2$. The same is true if $g_{X}=3$ and we restrict to
general $X$.

More in general, let $X$ be any Fano threefold with $i_{X}=1$ and very ample
${\mathcal{O}}_{X}(h)$. If $X$ contains a canonical curve
$Z\subseteq{\mathbb{P}^{g_{X}+1}}$, then
$\omega_{Z}\cong{\mathcal{O}}_{X}(h)\otimes{\mathcal{O}}_{Z}$,
$p_{a}(Z)=g_{X}+2$ and $\deg(Z)=2g_{X}+2$. Since
$h^{i}\big{(}{\mathcal{O}}_{X}(h)\big{)}=0$ for $i=1,2$, it follows from
Theorem 2.1 the existence of vector bundle ${\mathcal{E}}$ fitting into the
exact sequence

$0\longrightarrow{\mathcal{O}}_{X}(-h)\longrightarrow{\mathcal{E}}\longrightarrow{\mathcal{I}}_{Z|X}(h)\longrightarrow
0.$ (6.13)

Notice that $c_{1}({\mathcal{E}})=0$ and $c_{2}({\mathcal{E}})=Z-h^{2}$. Thus
$hc_{2}({\mathcal{E}})=4$.

If $\varrho_{X}=1$, then ${\mathcal{E}}$ is a minimal $\mu$–stable even
instanton bundle ${\mathcal{E}}$. Indeed, the cohomology of Sequence (6.13)
returns $h^{0}\big{(}{\mathcal{E}}\big{)}=0$, because $Z$, being a canonical
curve, is non–degenerate inside ${\mathbb{P}^{g_{X}+1}}$. Lemma 2.3 then
implies that ${\mathcal{E}}$ is $\mu$–stable.

The cohomology of Sequence (2.2) yields
$h^{1}\big{(}{\mathcal{I}}_{Z|X})\big{)}=0$, because $Z$ is integral. Thus
again the cohomology of Sequence (6.13) tensored by ${\mathcal{O}}_{X}(-h)$
returns $h^{1}\big{(}{\mathcal{E}}(-h)\big{)}=0$.

If $\varrho_{X}\geq 2$, then the $\mu$–(semi)stability of ${\mathcal{E}}$
cannot be deduced as easily as when $\varrho_{X}=1$. Indeed, if
$\mu({\mathcal{O}}_{X}(D))\geq\mu({\mathcal{E}})=0$ for a line bundle
${\mathcal{O}}_{X}(D)\subseteq{\mathcal{E}}$, then Sequence (6.13) implies
$h^{0}\big{(}{\mathcal{I}}_{Z|X}(h-D)\big{)}=h^{0}\big{(}{\mathcal{E}}(-D)\big{)}$.
It follows that ${\mathcal{E}}$ is $\mu$–stable if there are no surfaces
$S\subseteq X$ through $Z$ with $\deg(S)\leq 2g_{X}-2$.

Notice that there always exists an infinite family of surfaces of degree
$2g_{X}+1$ through $Z$, namely the cones projecting $Z$ from its points. For
some results on the computation of the minimal degree of surfaces through a
canonical curve see [21].

##  7\. Some further properties of instanton bundles

In this sections we first study the instanton bundles obtained in the previous
section when restricted to suitable divisors. Then we will inspect their
splitting behaviour when restricted to lines.

The following property has been introduced in [14]

######  Definition 7.1.

Let $X$ be a Fano threefold.

We say that an instanton bundle ${\mathcal{E}}$ on $X$ with
$c_{1}({\mathcal{E}})=-\varepsilon h$ is earnest if
$h^{1}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h-D)\big{)}=0$ for each smooth
integral divisor $D\subseteq X$.

Let us spend few words on earnest instanton bundles. Let $D$ be an integral,
smooth divisor on $X$ and assume that ${\mathcal{O}}_{X}(h)$ is very ample.
When $\varrho_{X}=1$, then ${\mathcal{O}}_{X}(D)$ is very ample, hence the
cohomology of Sequence (2.8) tensored by ${\mathcal{E}}(-q_{X}^{\varepsilon})$
implies
$h^{1}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h-D)\big{)}=h^{0}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h)\otimes{\mathcal{O}}_{D}\big{)}$.
When $q_{X}^{\varepsilon}\geq 1$ the dimension on the right vanishes by [42,
Theorem 3.1]. When $q_{X}^{\varepsilon}=0$, then $i_{X}=\varepsilon=1$: in
this case $\mu({\mathcal{E}}\otimes{\mathcal{O}}_{D})<0$ and we can argue as
in the previous case.

When $\varrho_{X}\geq 2$ such a deduction does not hold. Indeed there could be
integral smooth divisors $D$ such that ${\mathcal{O}}_{X}(D)$ is even not
ample. E.g. such an issue always occurs for ordinary instanton bundles on the
blow up of ${\mathbb{P}^{3}}$ at a point or along a line: see respectively
[14, 17, 3]. Nevertheless, in these cases ordinary earnest instanton bundles
${\mathcal{E}}$ are characterized by a finite number of further exotic
instantonic conditions, i.e. vanishings
$h^{1}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h-D)\big{)}=0$ for a suitable
divisor $D\subseteq X$.

In the case of the non–ordinary instanton bundles obtained in the previous
sections we can prove the following result.

######  Proposition 7.2.

Let $X$ be a Fano threefold with very ample ${\mathcal{O}}_{X}(h)$ and let
${\mathcal{E}}$ be any bundle whose existence is guaranteed by Theorems 1.5
and 1.6.

  1. (1)

If $X\not\cong F_{7}$, then ${\mathcal{E}}$ is earnest.

  2. (2)

If $X\cong F_{7}$, then ${\mathcal{E}}$ is earnest if and only if $\dim(Z\cap
E)=0$. Moreover, if ${\mathcal{E}}$ is not earnest, then
$h^{1}\big{(}{\mathcal{E}}(-h-D)\big{)}=0$ for each smooth integral divisor
$D\neq E$ and $h^{1}\big{(}{\mathcal{E}}(-h-E)\big{)}=1$.

###### Proof.

As pointed out above ${\mathcal{E}}$ is automatically earnest if
$\varrho_{X}=1$. Thus we have to prove the statement only when $X$ is one of
the following threefolds: $F_{6,2}$, $F_{6,3}$, $F_{7}$.

The proof that ${\mathcal{E}}$ is earnest when $X$ is either $F_{6,2}$ or
$F_{6,3}$ is similar, hence we will deal with the first case in what follows.
If $D\in|\alpha_{1}h_{1}+\alpha_{2}h_{2}|\neq\emptyset$, then
$\alpha_{1}=Dh_{2}^{2}\geq 0$ and $\alpha_{2}=Dh_{1}^{2}\geq 0$. In particular
${\mathcal{O}}_{F_{6,2}}(D)$ is globally generated, hence
${\mathcal{O}}_{F_{6,2}}(h+D)$ is ample. It follows from [42, Theorem 3.1]
that $h^{1}\big{(}{\mathcal{E}}(-h-D)\big{)}=0$ because ${\mathcal{E}}$ is
$\mu$–semistable (see the argument relating the earnest property with the
$\mu$–semistability of the restriction of ${\mathcal{E}}$ to any ample
divisor).

If $X\cong F_{7}$, then $|\alpha_{1}\xi+\alpha_{2}f|$ contains a smooth $D$ if
and only if either $\alpha_{1}=-\alpha_{2}=1$ or $\alpha_{1},\alpha_{2}\geq 0$
(see [14, Remark 5.7]). In the latter case $h+D$ is actually ample, hence
$h^{1}\big{(}{\mathcal{E}}(-h-D)\big{)}=0$ thanks to [42, Theorem 3.1]. We
deduce that ${\mathcal{E}}$ is earnest if and only if
$h^{1}\big{(}{\mathcal{E}}(-h-E)\big{)}=0$. Since $h+E=2\xi$ and
$2h+E=3\xi+f$, it follows from the cohomology of Sequences (5.2) and (2.2)
tensored by ${\mathcal{O}}_{F_{7}}(-h-E)$ and [14, Proposition 5.4] that

$h^{1}\big{(}{\mathcal{E}}(-h-E)\big{)}=h^{1}\big{(}{\mathcal{I}}_{Z|F_{7}}(-h-E)\big{)}=h^{0}\big{(}{\mathcal{O}}_{Z}(-h-E)\big{)}.$

Notice that the dimension on the right is the sum of
$h^{0}\big{(}{\mathcal{O}}_{C}(-h-E)\big{)}$ as $C\cong{\mathbb{P}^{1}}$ runs
over the conics contained $Z$. If the class of $C$ in $A^{2}(F_{7})$ is $\xi
f$, then $(h+E)C=2$, hence $h^{0}\big{(}{\mathcal{O}}_{C}(-h-E)\big{)}=0$. If
the class of $C$ is $2(\xi-f)f$, then $(h+E)C=0$, hence
$h^{0}\big{(}{\mathcal{O}}_{C}(-h-E)\big{)}=1$. Since the components of $Z$
are pairwise disjoint, then at most one of them can be contained in
$E\cong{\mathbb{P}^{2}}$. We deduce that if $\dim(Z\cap E)=0$, then
$h^{0}\big{(}{\mathcal{O}}_{Z}(-h-E)\big{)}=0$, while if $\dim(Z\cap E)=1$,
then $h^{0}\big{(}{\mathcal{O}}_{Z}(-h-E)\big{)}=1$. ∎

We are interested in the splitting behaviour of instanton bundles when
restricted to general lines. Indeed we finally prove also Theorem 1.8 stated
in the Introduction.

Proof of Theorem 1.8. Assume that ${\mathcal{E}}$ is obtained via Construction
5.2.

Let us first consider the case $\varrho_{X}=1$. Since ${\mathcal{E}}$ is
$\mu$–stable, it follows that when $X\cong{\mathbb{P}^{3}}$, i.e. $i_{X}=4$,
then ${\mathcal{E}}$ is generically trivial thanks to [46, Corollary 1 of
Theorem II.2.1.4]. The same is true when $i_{X}=3$, thanks to [20, Proposition
5.3].

Now let $i_{X}=2$, $\mathscr{L}\subseteq\Lambda(X)\times X$ the universal line
and $\lambda\colon\mathscr{L}\to\Lambda(X)$ the projection to the first
factor: the natural projection map $\chi\colon\mathscr{L}\to X$ is finite if
$h^{3}\geq 4$ (e.g. see [39, Remark 2.2.7]). If $h^{3}=3$ then it is
generically finite and its fibres have positive dimension at most in $30$
points (e.g. see [19, p. 315 and Section 10]). It follows that for each conic
$C\subseteq X$, then $\lambda(\chi^{-1}(C))\subseteq\Lambda(X)$ is a union of
a finite number of curves. Since $\Lambda(X)$ is a surface (e.g. see [34,
Propositions 3.5.6 and 3.5.8]), it follows that the general line on $X$ does
not intersect $C$.

Thus the definition of $Z$ implies the existence of a line
$L\cong{\mathbb{P}^{1}}$ which does not intersect $Z$. It follows that the
restriction of Sequence (5.2) to $L$ induces a surjection
${\mathcal{E}}\otimes{\mathcal{O}}_{L}\twoheadrightarrow{\mathcal{O}}_{{\mathbb{P}^{1}}}$,
hence
${\mathcal{E}}\otimes{\mathcal{O}}_{L}\cong{\mathcal{O}}_{{\mathbb{P}^{1}}}\oplus{\mathcal{O}}_{{\mathbb{P}^{1}}}(-1)$.

If $\varrho_{X}\geq 2$ the argument is analogous and easier. Indeed, in this
cases we know exactly the structure of $\Lambda(X)$ and $\Gamma(X)$ (see
Remarks 4.2, 4.3, 4.4), hence it is immediate to check that the general line
does not intersect $Z$ in these cases.

Finally consider a bundle ${\mathcal{E}}$ obtained as in Theorem 1.6. Thanks
to Lemma 6.1, we know that given general $Z,C\in\Gamma(X)_{0}$ the general
line $L$ in each component of $\Lambda(X)$ does not intersect them.
Restricting Sequence (5.2) to $L$, we then deduce
${\mathcal{E}}_{2}\otimes{\mathcal{O}}_{L}\cong{\mathcal{O}}_{{\mathbb{P}^{1}}}^{\oplus
2}$. Thus the restriction of Sequence (6.1) to the same line yields
${\mathcal{E}}_{2,\varphi}\otimes{\mathcal{O}}_{L}\cong{\mathcal{O}}_{{\mathbb{P}^{1}}}^{\oplus
2}$ too. In particular every general deformation
${\mathcal{E}}\in{\mathcal{S}}_{X}\big{(}2;0,(k+2)\ell,0\big{)}$ of
${\mathcal{E}}_{2,\varphi}$ enjoys the same property by semicontinuity.

If we now assume the existence of a generically trivial $\mu$–stable even
instanton bundle ${\mathcal{E}}_{k}$ with quantum number $k\geq 4$, then the
same argument as above yields the existence of a generically trivial
$\mu$–stable even instanton bundle ${\mathcal{E}}$ with quantum number $k+2$,
hence the statement is proved by induction on $k$. ∎

##  8\. Moduli spaces of non–ordinary instanton bundles

In this section we deal with the component of the moduli spaces of instanton
bundles defined in the previous sections.

Each instanton bundle ${\mathcal{E}}$ obtained via Construction 5.2 is
$\mu$–stable, hence simple. Thus it represents a point in an open subset
${\mathcal{S}}{\mathcal{I}}_{X}\big{(}\varepsilon,\zeta)$ of
${\mathcal{S}}_{X}\big{(}2;-\varepsilon h,\zeta,0\big{)}$ where $\zeta\in
H^{4}(X)$ is such that $\zeta h=k$.

######  Proposition 8.1.

Let $X$ be a Fano threefold with $i_{X}\geq 2$ and very ample
${\mathcal{O}}_{X}(h)$.

For each even integer $k$ satisfying Inequality (1.2) all the bundles obtained
via Construction 5.2 such that the class $\zeta\in H^{4}(X)$ of Z satisfies
$\zeta h=k$ represent smooth points in one and the same irreducible component
${\mathcal{S}}{\mathcal{I}}^{0}_{X}\big{(}\varepsilon,\zeta)\subseteq{\mathcal{S}}{\mathcal{I}}_{X}\big{(}\varepsilon,\zeta\big{)}$
with dimension

$\dim({\mathcal{S}}{\mathcal{I}}^{0}_{X}\big{(}\varepsilon,\zeta))=\left\\{\begin{array}[]{ll}8k-5&\text{if
$i_{X}=4$,}\\\ 6k-3&\text{if $i_{X}=3$,}\\\ 4k-h^{3}-3&\text{if
$i_{X}=2$.}\end{array}\right.$

###### Proof.

In order to prove the statement, it suffices to check that 5.2

$h^{0}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=1,\qquad
h^{2}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=h^{3}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=0,\\\
$ (8.1)

thanks to Equality (3.8).

To this purpose, since ${\mathcal{E}}$ is simple, it follows that
$h^{0}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=1$ and, by Lemma
2.2, also $h^{3}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=0$. It
remains to check that
$h^{2}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=0$. We have

$h^{i}\big{(}{\mathcal{E}}((1-q_{X}^{\varepsilon})h)\big{)}=h^{3-i}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h)\big{)}.$

thanks to Equality (2.4). In particular

$\displaystyle
h^{3}\big{(}{\mathcal{E}}((1-q_{X}^{\varepsilon})h)\big{)}=h^{0}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h)\big{)}\leq
h^{0}\big{(}{\mathcal{E}}\big{)}=0,$ $\displaystyle
h^{2}\big{(}{\mathcal{E}}((1-q_{X}^{\varepsilon})h)\big{)}=h^{1}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h)\big{)}=0.$

Thus the cohomology of Sequence (5.2) tensored by
${\mathcal{E}}^{\vee}\cong{\mathcal{E}}(\varepsilon h)$ yields

$h^{2}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=h^{2}\big{(}{\mathcal{I}}_{Z|X}\otimes{\mathcal{E}}((q_{X}^{\varepsilon}-1+\varepsilon)h)\big{)}.$

The cohomology of Sequence (2.2) tensored by
${\mathcal{E}}((q_{X}^{\varepsilon}-1+\varepsilon)h)$ implies

$\displaystyle
h^{2}\big{(}{\mathcal{I}}_{Z|X}\otimes{\mathcal{E}}((q_{X}^{\varepsilon}-1+\varepsilon)h)\big{)}$
$\displaystyle\leq
h^{2}\big{(}{\mathcal{E}}((q_{X}^{\varepsilon}-1+\varepsilon)h)\big{)}$ (8.2)
$\displaystyle+h^{1}\big{(}{\mathcal{O}}_{Z}\otimes{\mathcal{E}}((q_{X}^{\varepsilon}-1+\varepsilon)h)\big{)}.$

On the one hand, again the cohomologies of Sequences (5.2) and (2.2) tensored
by ${\mathcal{E}}((q_{X}^{\varepsilon}-1+\varepsilon)h)$ and
${\mathcal{O}}_{X}((i_{X}-1)h)$ respectively imply

$h^{2}\big{(}{\mathcal{E}}((q_{X}^{\varepsilon}-1+\varepsilon)h)\big{)}=h^{2}\big{(}{\mathcal{I}}_{Z|X}((i_{X}-1)h)\big{)}\leq
h^{1}\big{(}{\mathcal{O}}_{X}((i_{X}-1)h)\otimes{\mathcal{O}}_{Z}\big{)}.$

The latter dimension is zero because $Z$ is the disjoint union of rational
curves. Thus the first righthand summand in Inequality (8.2) vanishes.

On the other hand, for each integral conic $C\subseteq Z$, Equality (2.3)
implies

${\mathcal{O}}_{C}\otimes{\mathcal{E}}((q_{X}^{\varepsilon}-1+\varepsilon)h)\cong{\mathcal{N}}_{C|X}\cong{\mathcal{O}}_{{\mathbb{P}^{1}}}(a_{1})\oplus{\mathcal{O}}_{{\mathbb{P}^{1}}}(a_{2})$

where $a_{1}\leq a_{2}$ and $a_{1}+a_{2}=2i_{X}-2$ by adjunction on $X$. If
$i_{X}\geq 3$, then $C$ is the complete intersection of hypersurfaces of
degrees $1$ and $i_{X}-2$, hence $(a_{1},a_{2})=(2,2i_{X}-4)$. Let $i_{X}=2$
and choose a general hyperplane section $H\subseteq X$ containing $C$. Thus
${\mathcal{N}}_{H|X}\cong{\mathcal{O}}_{X}(h)\otimes{\mathcal{O}}_{H}$ and
there exists a surjective morphism
${\mathcal{N}}_{C|X}\to{\mathcal{N}}_{H|X}$. It follows that $(a_{1},a_{2})$
is either $(0,2)$ or $(1,1)$. Since $Z$ is the disjoint union of conics in
$X$, it follows that

$h^{1}\big{(}{\mathcal{O}}_{Z}\otimes{\mathcal{E}}((q_{X}^{\varepsilon}-1+\varepsilon)h)\big{)}=h^{1}\big{(}{\mathcal{N}}_{Z|X}\big{)}=0$

regardless of $i_{X}$. Thus the second righthand summand in Inequality (8.2)
vanishes too and the proof of the Equalities (8.1) is complete.

The vanishing $h^{2}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=0$
implies that ${\mathcal{E}}$ in ${\mathcal{S}}_{X}\big{(}2;-\varepsilon
h,\zeta,0\big{)}$ is smooth and lies in a component of dimension
$\dim\mathrm{Ext}_{X}^{1}\big{(}{\mathcal{E}},{\mathcal{E}}\big{)}\cong
h^{1}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}$. Moreover,
${\mathcal{E}}$ is in the image of the natural rational map $\Gamma(X)^{\times
s}\dashrightarrow{\mathcal{S}}_{X}\big{(}2;-\varepsilon h,\zeta,0\big{)}$.
Thus all the bundles obtained via Construction 5.2 lie in the same component
of ${\mathcal{S}}{\mathcal{I}}_{X}\big{(}\varepsilon,k\big{)}$. ∎

When $\varrho_{X}=1$, each instanton bundle ${\mathcal{E}}$ is automatically
$\mu$–stable because $h^{0}\big{(}{\mathcal{E}}\big{)}=0$. In [41, 4] the
authors proved the existence of ordinary instanton bundles which are not
$\mu$–stable on $F_{6,2}$ and $F_{6,3}$. When $X\cong F_{7}$, each instanton
bundle with quantum number $k\leq 14$ is $\mu$–stable (see [14]).

The picture for non–ordinary instanton bundles is clearer.

######  Proposition 8.2.

Let $X$ be a Fano threefold with $i_{X}\geq 2$ and very ample
${\mathcal{O}}_{X}(h)$.

Each non–ordinary instanton bundle on $X$ is $\mu$–stable if and only if
$X\not\cong F_{6,2}$.

###### Proof.

If $\varrho_{X}=1$ each instanton bundle is $\mu$–stable, hence we have to
deal only with the case $\varrho_{X}\geq 2$. To this purpose, thanks to Lemma
2.3, it suffices to look at the divisors $D$ such that
$\mu({\mathcal{O}}_{X}(D))=\mu({\mathcal{E}})$, when $X$ is $F_{6,2}$,
$F_{6,3}$, $F_{7}$.

Since $\mu({\mathcal{E}})$ is either $-3$ or $-7/2$ according $X$ is either
$F_{6,2}$ and $F_{6,3}$ or $F_{7}$ respectively, it follows that we can
restrict to the two cases $F_{6,2}$ and $F_{6,3}$. In the latter case, using
the notation of Remark 4.3, we know that
$h^{2}=2(h_{2}h_{3}+h_{1}h_{3}+h_{1}h_{2})$, hence $\mu({\mathcal{O}}_{X}(D))$
is certainly even.

Thus, the only case we have to deal with is $X:=F_{6,2}$. In this case, we
claim the existence of a strictly $\mu$–semistable non–ordinary instanton
${\mathcal{E}}$. To this purpose, thanks to Lemma 2.3, it suffices to show
that $h^{0}\big{(}{\mathcal{E}}(-\alpha_{1}h_{1}-\alpha_{2}h_{2})\big{)}=0$
for each
${\mathcal{O}}_{F_{6,2}}(\alpha_{1}h_{1}+\alpha_{2}h_{2})\subseteq{\mathcal{E}}$
such that
$3(\alpha_{1}+\alpha_{2})=\mu({\mathcal{O}}_{F_{6,2}}(\alpha_{1}h_{1}+\alpha_{2}h_{2}))>\mu({\mathcal{E}})=-3$,
i.e. $\alpha_{1}+\alpha_{2}\geq 0$, and that
$h^{0}\big{(}{\mathcal{E}}(h_{1})\big{)}>0$.

In order to prove the claim, recall that $F_{6,2}$ can be also viewed as the
incidence variety $\\{\ (p,L)\ |\ p\in L\
\\}\subseteq{\mathbb{P}^{2}}\times({\mathbb{P}^{2}})^{\vee}$. If we fix
coordinates $u_{0},u_{1},u_{2}$ on ${\mathbb{P}^{2}}$ and we denote by
$v_{0},v_{1},v_{2}$ the dual coordinates in $({\mathbb{P}^{2}})^{\vee}$, then
the $F_{6,2}$ coincides with intersection of the image of the Segre map
${\mathbb{P}^{2}}\times({\mathbb{P}^{2}})^{\vee}\to{\mathbb{P}^{8}}$ defined
by

$\displaystyle(x_{0},\dots,x_{8})=(u_{0}v_{0},u_{0}v_{1},u_{0}v_{2},u_{1}v_{0},u_{1}v_{1},u_{1}v_{2},u_{2}v_{0},u_{2}v_{1},u_{2}v_{2})$

with the hyperplane $H:=\\{\ x_{0}+x_{4}+x_{8}=0\ \\}$ corresponding the
incidence relation $p\in L$ translated analytically as
$u_{0}v_{0}+u_{1}v_{1}+u_{2}v_{2}=0$. Thus the ideal of $F_{6,2}$ inside
$H\cong{\mathbb{P}^{7}}$ with coordinates $x_{0},\dots,x_{7}$ is generated by
the $2\times 2$–minors of the matrix

$\left(\begin{array}[]{ccc}x_{0}&x_{1}&x_{2}\\\ x_{3}&x_{4}&x_{5}\\\
x_{6}&x_{7}&-x_{0}-x_{4}\end{array}\right).$

If $L\subseteq F_{6,2}$ is a line with class $h_{1}^{2}$, then it is the fibre
of the projection map on the $u$–plane ${\mathbb{P}^{2}}$. It follows that up
to an automorphism of $F_{6,2}$ we can assume that $L:=\\{\
x_{0}=\dots=x_{5}=0\ \\}$. Thanks to [45, Theorem 2.4] (or using any software
for symbolic computation) one checks that the ideal

$I_{C}:=(x_{0},\dots,x_{5})^{2}+(x_{2},x_{5},x_{0}-x_{4},x_{1}x_{6}-x_{0}x_{7},x_{4}x_{6}-x_{3}x_{7})$

defines a scheme $C\subseteq F_{6,2}$ with $\deg(C)=2$ and $p_{a}(C)=-2$. The
curve $C$ is obtained by doubling the line $L$ inside the surface $S$ with
ideal

$I_{S}:=(x_{2},x_{5},x_{0}-x_{4},x_{1}x_{6}-x_{0}x_{7},x_{4}x_{6}-x_{3}x_{7}).$

It is easy to check that
$I_{C}+I_{F_{6,2}}=(x_{1}^{2},x_{3}^{2})+I_{S}+I_{F_{6,2}}$ and $x_{4}^{2}\in
I_{S}+I_{F_{6,2}}$. It follows that the localization of $I_{C}+I_{F_{6,2}}$ at
the prime ideals $(x_{7})$ and $(x_{6})$ coincides with the localizations of
the ideals $(x_{1}^{2})+I_{S}+I_{F_{6,2}}$ and $(x_{3}^{2})+I_{S}+I_{F_{6,2}}$
respectively. Thus $C$ is locally complete intersection inside $F_{6,2}$
because $S$ is smooth at the points of $L$.

Moreover, each Cartier divisor on $C$ is completely identified by its degree
which is twice the degree of the same divisor restricted to $L$: see [8,
Propositions 4.1 and 4.2]. Since $\deg(\omega_{C})=-6$ and
${\mathcal{O}}_{F_{6,2}}(-h_{1}-3h_{2})\otimes{\mathcal{O}}_{L}\cong{\mathcal{O}}_{{\mathbb{P}^{1}}}(-3)$,
it follows that
$\omega_{C}\cong{\mathcal{O}}_{F_{6,2}}(-h_{1}-3h_{2})\otimes{\mathcal{O}}_{C}$.
Thus

$\det({\mathcal{N}}_{C|X})\cong\omega_{C}\otimes{\mathcal{O}}_{F_{6,2}}(2h)\cong{\mathcal{O}}_{F_{6,2}}(h_{1}-h_{2})\otimes{\mathcal{O}}_{C},$

by adjunction. Since ${\mathcal{O}}_{F_{6,2}}(h_{1})$ is globally generated
and $h_{1}^{3}=0$, it follows that

${\mathcal{O}}_{F_{6,2}}(-h_{1}-2h_{2})\otimes{\mathcal{O}}_{C}\cong{\mathcal{O}}_{F_{6,2}}(-2h)\otimes{\mathcal{O}}_{C}.$

Thus [18, Theorem 2.1] yields
$h^{0}\big{(}{\mathcal{O}}_{F_{6,2}}(-2h)\otimes{\mathcal{O}}_{C}\big{)}=h^{1}\big{(}{\mathcal{I}}_{C|{\mathbb{P}^{7}}}(-2h)\big{)}=0$.

Let $Z$ be the union of $s:=(k-2)/2$ pairwise disjoint schemes of degree $2$
and genus $-2$ in $F_{6,2}$ supported on lines in class $h_{1}^{2}$ as above.
The above discussion and Theorem 2.1 yield the existence of a vector bundle
${\mathcal{E}}$ fitting into the exact sequence

$0\longrightarrow{\mathcal{O}}_{F_{6,2}}(-h_{1})\longrightarrow{\mathcal{E}}\longrightarrow{\mathcal{I}}_{Z|F_{6,2}}(-h_{2})\longrightarrow
0.$

It is immediate to check that $c_{1}({\mathcal{E}})=-h$,
$hc_{2}({\mathcal{E}})=k$, $h^{0}\big{(}{\mathcal{E}}\big{)}=0$ and
$h^{1}\big{(}{\mathcal{E}}(-h)\big{)}=h^{1}\big{(}{\mathcal{I}}_{Z|F_{6,2}}(-h_{1}-2h_{2})\big{)}$.
Thus

$h^{1}\big{(}{\mathcal{I}}_{Z|F_{6,2}}(-h_{1}-2h_{2})\big{)}=h^{0}\big{(}{\mathcal{O}}_{F_{6,2}}(-h_{1}-2h_{2})\otimes{\mathcal{O}}_{Z}\big{)}=sh^{0}\big{(}{\mathcal{O}}_{F_{6,2}}(-2h)\otimes{\mathcal{O}}_{C}\big{)}=0,$

whence we finally deduce $h^{1}\big{(}{\mathcal{E}}(-h)\big{)}=0$. Moreover,
$h^{0}\big{(}{\mathcal{E}}(-\alpha_{1}h_{1}-\alpha_{2}h_{2})\big{)}=0$ if
$\alpha_{1}+\alpha_{2}\geq 0$ and $h^{0}\big{(}{\mathcal{E}}(h_{1})\big{)}=1$.
Thus the claim is proved and the proof of the proposition is complete. ∎

If $k=2$, then the bundle constructed in the proof of Proposition 8.2 splits
as ${\mathcal{O}}_{F_{6,2}}(-h_{1})\oplus{\mathcal{O}}_{F_{6,2}}(-h_{2})$:
hence the above construction extends Remark 4.5 to higher quantum numbers.
Moreover, it is possible to show that
$h^{2}\big{(}{\mathcal{E}}\otimes{\mathcal{E}}^{\vee}\big{)}=0$ regardless of
the value of $k$.

In order to conclude the analysis in the case $i_{X}\geq 2$, we deal below
with non–ordinary instanton bundles with low quantum numbers and their moduli
spaces. Recall that the minimal non–ordinary instanton bundles are the ones
with quantum number $k=2$ when $\varrho_{X}=1$ and $k=4$ when $\varrho_{X}\geq
2$ (see Theorems 1.4 and 1.5).

######  Remark 8.3.

If $\varrho_{X}=1$, then each instanton bundle is automatically $\mu$–stable.
The moduli space
${\mathcal{S}}{\mathcal{I}}_{X}\big{(}\varepsilon,k\ell\big{)}$ is the open
non–empty subset of bundles ${\mathcal{E}}$ such that
$h^{1}\big{(}{\mathcal{E}}(-q_{X}^{\varepsilon}h)\big{)}=0$ inside the
Maruyama moduli space ${\mathcal{M}}_{X}(2;-\varepsilon h,k\ell)$ of
$\mu$–stable rank $2$ bundles (see [43] and [44]).

If $i_{X}=4,3,2$ and $\varepsilon=1,0,1$ respectively, then the space
${\mathcal{M}}_{X}(2;-\varepsilon h,2\ell)$, hence
${\mathcal{S}}{\mathcal{I}}_{X}\big{(}\varepsilon,2\big{)}$, is irreducible
and smooth: moreover it is also rational (see [31, Theorem 3.1], [48, Theorem
2.1] and [49] for the proof of this facts in the cases $i_{X}=4,3,2$
respectively). In particular
${\mathcal{S}}{\mathcal{I}}_{X}(\varepsilon,2\ell\big{)}={\mathcal{S}}{\mathcal{I}}_{X}^{0}(\varepsilon,2\ell\big{)}$
in these cases and it actually coincides with ${\mathcal{M}}_{X}(2;-h,2\ell)$
when $i_{X}\neq 3$ (see [31, Proposition 2.3] and [49, Proposition 2.1]).

If $X={\mathbb{P}^{3}}$, then the moduli space
${\mathcal{M}}_{{\mathbb{P}^{3}}}(2;-h,4\ell)$ has been described in [11,
Theorems 1 and 2]. More precisely (see [11, Section 1.1]) such a moduli space
has two components, both rational. One of them is smooth of dimension $28$ and
its points represent bundles ${\mathcal{E}}$ such that
$h^{1}\big{(}{\mathcal{E}}(-2h)\big{)}=1$, thus it does not contain any point
corresponding to an instanton bundle. The second component has dimension $27$
and its points represent bundles ${\mathcal{E}}$ such that
$h^{1}\big{(}{\mathcal{E}}(-2h)\big{)}=0$: it follows that it coincides with
${\mathcal{S}}{\mathcal{I}}_{{\mathbb{P}^{3}}}(1,4\ell)={\mathcal{S}}{\mathcal{I}}_{{\mathbb{P}^{3}}}^{0}(1,4\ell\big{)}$.
More generally in [52] the author defines a component
${\mathcal{M}}\subseteq{\mathcal{M}}_{{\mathbb{P}^{3}}}(2;-h,k\ell)$ for each
even $k$, shows that it is different from
${\mathcal{S}}{\mathcal{I}}_{{\mathbb{P}^{3}}}^{0}(1,k\ell\big{)}$ for $k=6$
and conjectures that this is true for all $k\geq 6$.

If $X=Q$, then ${\mathcal{M}}_{Q}(2;0,4\ell)$, hence
${\mathcal{S}}{\mathcal{I}}_{Q}\big{(}0,4\ell\big{)}$, is irreducible, reduced
and unirational (see [48, Theorem 3.4]). It follows that
${\mathcal{S}}{\mathcal{I}}_{Q}(0,4\ell\big{)}={\mathcal{S}}{\mathcal{I}}_{Q}^{0}(0,4\ell\big{)}$.

When $\varrho_{X}\geq 2$, then $h^{3}\geq 6$ and minimal non–ordinary
instanton bundle must have quantum number $4$. We showed in Theorem 1.5 that
such bundles exist.

We have $h^{2}\big{(}{\mathcal{E}}(h)\big{)}=0$ for such an ${\mathcal{E}}$ by
Lemma 2.4. Thus Equality (2.6) implies
$h^{0}\big{(}{\mathcal{E}}(h)\big{)}\geq h^{3}-3\geq 3$. If $s\in
H^{0}\big{(}{\mathcal{E}}(h)\big{)}$ is a non–zero section whose zero–locus is
a curve $Z$, then $\deg(Z)=hc_{2}({\mathcal{E}}(h))=hc_{2}({\mathcal{E}})=4$.
Equality (2.3) implies

$\det({\mathcal{N}}_{Z|X})\cong\det({\mathcal{E}}(h))\otimes{\mathcal{O}}_{Z}\cong{\mathcal{O}}_{X}(h)\otimes{\mathcal{O}}_{Z},$

hence the adjunction formula yields

$\omega_{Z}\cong\omega_{X}\otimes{\mathcal{O}}_{X}(h)\otimes{\mathcal{O}}_{Z}\cong{\mathcal{O}}_{X}(-h)\otimes{\mathcal{O}}_{Z}.$

Thus $p_{a}(Z)=-1$.

We conclude the analysis of the case $i_{X}\geq 2$ by characterizing minimal
instanton bundles on $F_{6,2}$ which are not $\mu$–semistable as the ones
associated to double structures $Z$ with $p_{a}(Z)=-2$ on lines inside
$F_{6,2}$.

######  Remark 8.4.

In what follows we will use the notation of Remark 4.2. It is easy to see that
the condition $\mu({\mathcal{O}}_{F_{6,2}}(D))=\mu({\mathcal{E}})$ forces the
existence of an integer $\alpha$ such that $D=\alpha h_{1}-(1+\alpha)h_{2}$.
Let $s\in H^{0}\big{(}{\mathcal{O}}_{F_{6,2}}(-D)\big{)}$ be a non–zero
section. If the zero–locus of $s$ contains a surface $S$, then

$\deg(S)=Sh^{2}=\mu({\mathcal{O}}_{F_{6,2}}(S))\leq\mu({\mathcal{E}}(-D))=\mu({\mathcal{E}})-\mu({\mathcal{O}}_{F_{6,2}}(D))=0.$

We deduce that $(s)_{0}=Z$ is a curve with degree $c_{2}({\mathcal{E}}(-D))h$.
One easily checks that $c_{2}({\mathcal{E}}(-D))h=2(1-\alpha-\alpha^{2})$
which is negative, unless $\alpha\in\\{\ -1,0\ \\}$, i.e. $D=h_{i}$. Thus we
deduce $\deg(Z)=2$.

Let $\beta_{1}h_{2}^{2}+\beta_{2}h_{1}^{2}$ be the class of $Z$. Since
$\beta_{i}=Zh_{i}\geq 0$ and $\beta_{1}+\beta_{2}=\deg(Z)=2$, it follows that
the possible values of $(\beta_{1},\beta_{2})$ are $(2,0)$, $(1,1)$ and
$(0,2)$. Let $D=h_{1}$: Equality (2.3) and the adjunction formula imply
$2p_{a}(Z)-2=(-h_{1}-3h_{2})Z$. Thus $p_{a}(Z)$ is $-2$, $-1$ and $0$
respectively in the three cases.

If the class of $Z$ is $2h_{2}^{2}$, then $p_{a}(Z)=0$, hence $Z$ would be a
conic supported on a line. We can exclude this case as in Remark 4.2.

If the class of $Z$ is $h_{1}^{2}+h_{2}^{2}$, then $p_{a}(Z)=-1$. In this case
$Z$ is either the union of two disjoint lines, or a double structure on a
line. The latter case cannot occur because $h_{1}^{2}+h_{2}^{2}$ is not
divisible in $A^{2}(F_{6,2})$, hence $Z$ is the union of lines $L_{i}$ with
class is $h_{i}^{2}\in A^{2}(F_{6,2})$. Equality (2.3) implies that

$\det({\mathcal{N}}_{L_{i}|F_{6,2}})\cong\det({\mathcal{E}}(h_{1}))\otimes{\mathcal{O}}_{L_{i}}\cong\det({\mathcal{O}}_{F_{6,2}}(h_{1}-h_{2}))\otimes{\mathcal{O}}_{L_{i}}$

because $L_{1}\cap L_{2}=\emptyset$. Thus this case cannot occur too, because
$(h_{1}-h_{2})h_{1}^{2}=-1$ and
${\mathcal{N}}_{L_{1}|F_{6,2}}\cong{\mathcal{O}}_{{\mathbb{P}^{1}}}^{\oplus
2}$ as shown above.

It follows that the class of $Z$ is $2h_{1}^{2}$ and $p_{a}(Z)=-2$. Thus $Z$
is necessarily a double structure on a line $L$ with class $h_{1}^{2}$ and
Sequence (2.1) becomes

$0\longrightarrow{\mathcal{O}}_{F_{6,2}}\longrightarrow{\mathcal{E}}(h_{1})\longrightarrow{\mathcal{I}}_{Z|F_{6,2}}(h_{1}-h_{2})\longrightarrow
0,$

hence $h^{0}\big{(}{\mathcal{E}}(h_{1})\big{)}=1$, i.e. the curve $Z$
completely identifies ${\mathcal{E}}$.

Tensoring the above sequence by ${\mathcal{O}}_{F_{6,2}}(h_{2}-h_{1})$, one
deduces $h^{0}\big{(}{\mathcal{E}}(h_{2})\big{)}=0$. Thus if we make a similar
construction with $h_{2}$ instead of $h_{1}$, we obtain a different family of
$\mu$–semistable minimal instanton bundles.

The instanton bundles obtained in Theorem 1.6 are $\mu$–stable, hence simple.
Thus they represent points in an open subset
${\mathcal{S}}{\mathcal{I}}_{X}\big{(}\varepsilon,k\ell)\subseteq{\mathcal{S}}_{X}\big{(}2;-\varepsilon
h,k\ell,0\big{)}$.

Theorem 1.6 is less explicit than Construction 5.2. Thus we have no way to
confront the bundles constructed in Section 6 in order to show that they lie
in the same component. Nevertheless, at least the following statement holds.

######  Proposition 8.5.

Let $X$ be an ordinary prime Fano threefold with very ample
${\mathcal{O}}_{X}(h)$.

For each even integer $k$ satisfying Inequality (1.2) there exists an earnest
instanton bundle ${\mathcal{E}}$ with quantum number $k$ representing a smooth
point in an irreducible component
${\mathcal{S}}{\mathcal{I}}^{0}_{X}\big{(}\varepsilon,k\ell)\subseteq{\mathcal{S}}{\mathcal{I}}_{X}\big{(}\varepsilon,k\ell\big{)}$
with dimension

$\dim({\mathcal{S}}{\mathcal{I}}^{0}_{X}\big{(}\varepsilon,k\ell))=2k-3.$

###### Proof.

As pointed out in the proof of Theorem 1.6 we can construct a $\mu$–stable
instanton bundle ${\mathcal{E}}$ with
$\mathrm{Ext}^{2}_{X}\big{(}{\mathcal{E}},{\mathcal{E}}\big{)}=0$ and quantum
number $k$. Thanks to Equality (3.8) we obtain
$\dim\mathrm{Ext}^{1}_{X}\big{(}{\mathcal{E}},{\mathcal{E}}\big{)}=2k-3$. ∎

##  9\. Monads for non–ordinary instanton bundles when $i_{X}\geq 3$

In this section we prove the existence of monads for the bundles we are
interested in when $X$ is either ${\mathbb{P}^{3}}$ or $Q$.

######  Proposition 9.1.