We now want to understand how non locally free instantons behave in families.
We consider therefore the set
${\mathcal{D}}(1,1):={\mathcal{L}}(2)\setminus{\mathcal{I}}(2)$ that
parameterizes non locally free instantons.

######  Proposition 48.

${\mathcal{D}}(1,1)$ is a locally closed subscheme of ${\mathcal{M}}$; it is
smooth, irreducible and of dimension 5.

###### Proof.

By semicontinuity, the instanton locus ${\mathcal{L}}(2)$ is open in
${\mathcal{M}}$; since being non locally free is a closed condition, we have
that ${\mathcal{D}}(1,1)$ is locally closed in ${\mathcal{M}}$. To prove the
rest of the proposition we mimic the proof of Theorem 43. From Proposition 46
we know that for $[E]\in{\mathcal{D}}(1,1)$, the curves in the pencil
${\mathbb{P}}(H^{0}(E(1)))\subset\operatorname{Hilb}_{2t+2}(X)$ are of the
form $l\cup l^{\prime}$ with $l=\operatorname{Sing}(E)$ and with the
$l^{\prime}$s varying in a ruling of a smooth hyperplane section of $X$.

As it was the case for ${\mathcal{I}}(2),$ ${\mathcal{D}}(1,1)$ is endowed as
well with a surjective map
${\mathcal{D}}(1,1)\xrightarrow{\phi^{(1,1)}}{\mathbb{P}^{4}}^{*}_{sm}$
fitting in a commutative diagram

---
$\textstyle{{\mathcal{D}}(1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{(1,1)}}$$\scriptstyle{\phi^{(1,1)}_{{\mathcal{U}}}}$$\textstyle{{\mathbb{P}^{4}}^{*}_{sm}}$$\textstyle{\mathcal{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

but this time the fibre over a ruling
$|{\mathcal{O}}_{Q}(1,0)|\simeq{\mathbb{P}^{1}}\in{\mathcal{U}},\
[Q]\in({\mathbb{P}^{4}})^{*}_{sm}$ consists of pencils of divisors of type
$(2,0)$ with a base point. Each fibre of $\phi^{(1,1)}_{{\mathcal{U}}}$ is
therefore isomorphic to $C_{2}\subset\mathbb{G}(1,|{\mathcal{O}}_{Q}(2,0)|)$;
the smooth conic parameterizing the tangents to the locus of singular divisors
in $|{\mathcal{O}}_{Q}(2,0)|$. This proves that ${\mathcal{D}}(1,1)$ is smooth
irreducible and of dimension equal to five. ∎

######  Remark 49.

Applying arguments equivalent to [13] Lemma 9.3., we can make the following
considerations. Let $[E]$ be a point corresponding to an instanton and let us
consider the short exact sequence

$0\rightarrow{\mathcal{O}}_{X}\xrightarrow{s}E(1)\rightarrow{\mathcal{I}}_{Y}(1)\rightarrow
0$

induced by $s\in H^{0}(E(1))$. From this short exact sequence we deduce that
the image of $t\in H^{0}(E(1))$ in $H^{0}({\mathcal{I}}_{Y}(1))$ gives an
equation for the hyperplane $\langle Y\rangle$ and that moreover for any $t\in
H^{0}(E(1))$ independent from $s$, $E(1)/(s,t)\simeq{\mathcal{I}}_{Y,Q}(1)$,
for $Q:=X\cap\langle Y\rangle$. We have therefore a well defined linear map
$H^{0}(E(1))\rightarrow H^{0}({\mathcal{O}}_{Q}(2,0))$ that maps each $s\in
H^{0}(E(1))$ to the form defining $V(s)$ on $Q$.

###  5.3. A description of ${\mathcal{L}}(2)$ via Serre correspondence.

For the moment we just know that ${\mathcal{I}}(2)$ and ${\mathcal{D}}(1,1)$
are locally trivial fibrations over ${\mathcal{U}}$. Using Serre
correspondence we are now going to show that actually, the entire
${\mathcal{L}}(2)$ identifies with a ${\mathbb{P}}^{2}$ bundle over
${\mathcal{U}}$ and that ${\mathcal{D}}(1,1)$ and ${\mathcal{I}}(2)$ are,
respectively, a divisor and an open subset of ${\mathcal{L}}(2)$. The key
ingredient to prove this is the Serre correspondence which enables us to
collect information about the geometry of ${\mathcal{L}}(2)$ studying the
geometric properties of the families of the corresponding curves. Our starting
point is therefore the inspection of the open
${\mathcal{H}}\subset\operatorname{Hilb}_{2t+2}(X)$ that parameterises locally
Cohen Macaulay curves. Note that any locally Cohen Macaulay curve with
Hilbert’s polynomial $2t+2$ is indeed either the union of two disjoint lines
or a double structure on a line of arithmetic genus -1.

######  Lemma 50.

$\forall\>[Y]\in{\mathcal{H}}$, $h^{0}({\mathcal{N}}_{Y/X})=6$ and
$h^{1}({\mathcal{N}}_{Y/X})=0$.

###### Proof.

We first show that for $[Y]\in{\mathcal{H}}$, $Y$ lies in a unique smooth
hyperplane section $Q\subset X$.

If $Y=l_{1}\cup l_{2},\ l_{1}\cap l_{2}=\emptyset$ then the only hyperplane
containing $Y$ is $\langle Y\rangle=\langle l_{1},l_{2}\rangle$. If otherwise
$Y$ is a double line supported on $l$, we have that ${\mathcal{I}}_{Y}$ fits
in the exact sequence

$0\rightarrow{\mathcal{I}}_{Y}\rightarrow{\mathcal{I}}_{l}\rightarrow{\mathcal{O}}_{l}\rightarrow
0$

from which we compute that $h^{0}({\mathcal{I}}_{Y}(1))\neq 0$ (since
$H^{0}({\mathcal{I}}_{l}(1))$ can not inject in $H^{0}({\mathcal{O}}_{l}(1))$)
and $h^{0}({\mathcal{I}}_{Y}(1))<2$ (since no planar l.c.m. curve has negative
arithmetic genus). Thus $h^{0}({\mathcal{I}}_{Y}(1))=1$ and $Y$ is contained
in a unique hyperplane section $Q$ of $X$. $Q$ must be smooth since a degree 2
l.c.m. curves on a singular hyperplane section of $X$ is planar. The only
degree 2 l.c.m. curve on a cone of vertex $p$ over a smooth conic $C$ indeed
are $C$ itself or cones degree 2 divisors $D\subset C$ of $C$. These latter
are always contained in a plane $\langle p,L_{D}\rangle$ with
$L_{D}\subset\langle C\rangle$ the unique line spanned by $D$ for $D$ reduced
whilst $L_{D}={\mathbb{T}}_{q}C$ for $D_{\rm red}=q$.

Let us now compute $h^{i}({\mathcal{N}}_{Y/X}),\ i=0,1$. Consider the smooth
quadric surface $Q:=\langle Y\rangle\cap X$ and denote by $L_{A},\ L_{B}$ the
two generators of $\operatorname{{Pic}}(Q)$. The only degree 2 effective
divisors in $Q$ having arithmetic genus -1 belong either to the class $2L_{A}$
or to $2L_{B}$.

Without loss of generality, we suppose then $Y\sim 2L_{A}$ and we consider

$0\rightarrow{\mathcal{O}}_{Q}\rightarrow{\mathcal{O}}_{Q}(2L_{A})\rightarrow{\mathcal{O}}_{Y}(2L_{A})\rightarrow
0.$

From this short exact sequence, since
$H^{i}({\mathcal{O}}_{Q})=H^{i}({\mathcal{O}}_{Q}(L_{A}))=0,\ \forall\>i\geq
1$, we compute $h^{0}({\mathcal{O}}_{Y}(2L_{A}))=2$ and
$h^{1}({\mathcal{O}}_{Y}(2L_{A}))=h^{1}(\mathcal{N}_{Y/Q})=0.$

We finally consider:

$0\rightarrow\mathcal{N}_{Y/Q}\rightarrow\mathcal{N}_{Y/X}\rightarrow{\mathcal{N}_{Q/X}|}_{Y}\rightarrow
0.$

${\mathcal{N}_{Q/X}|}_{Y}$ is isomorphic to ${\mathcal{O}}_{Y}(1)$ and from

$0\rightarrow{\mathcal{O}}_{l}(1)\rightarrow{\mathcal{O}}_{Y}(1)\rightarrow{\mathcal{O}}_{l}(1)\rightarrow
0$

we obtain $h^{0}({\mathcal{O}}_{Y}(1))=4$ and $h^{1}({\mathcal{O}}_{Y}(1))=0.$

From these arguments we deduce the vanishing of $H^{1}(\mathcal{N}_{Y/X})$,
which implies the smoothness of $\operatorname{Hilb}_{2t+2}(X)$, and we
compute that $\operatorname{Hilb}_{2t+2}(X)$ has dimension
$6=h^{0}({\mathcal{N}}_{Y/X})=h^{0}({\mathcal{O}}_{Y}(1))+h^{0}(O_{Y}(2L_{A}))$
at $[Y]$. ∎

######  Lemma 51.

${\mathcal{H}}$ is a ${\mathbb{P}}^{2}$ bundle over
${\mathcal{U}}\subset\mathbb{G}(1,F(X))$.

###### Proof.

Denote by ${\mathcal{T}}_{{\mathcal{U}}}$ the restriction of the tautological
bundle over $\mathbb{G}(1,F(X))$ to ${\mathcal{U}}$. Take then the rank 3
vector bundle $\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee})$
and the projective bundle

${\mathbb{P}}(\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee}))\rightarrow{\mathcal{U}}.$

For a point
$h\in{\mathbb{P}}(\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee}))$
we denote by $l_{1,h},\ l_{2,h}$ the corresponding (possibly coincident)
lines. The incidence correspondence
$\Sigma\subset{\mathbb{P}}(\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee}))\times
X$

$\Sigma:=\\{(h,p)\in{\mathbb{P}}(\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee}))\times
X\mid p\in l_{1,h}\cup l_{2,h}\\}$

induces a bijective morphism
${\mathbb{P}}(\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee}))\to{\mathcal{H}}$
that, since ${\mathcal{H}}$ and
${\mathbb{P}}(\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee}))$
are smooth, is therefore an isomorphism (this is due to Zariski main theorem).
∎

From now on we denote by $\pi_{{\mathcal{H}}}:{\mathcal{H}}\to{\mathcal{U}}$
the standard projection.

Let us now pass to the study of ${\mathcal{L}}(2)$. To begin with we show how,
from the smoothness of ${\mathcal{H}}$ we can deduce the smoothness of
${\mathcal{M}}$ along ${\mathcal{L}}(2)$.

######  Lemma 52.

For any $[E]\in{\mathcal{I}}(2),\ \operatorname{ext}^{1}(E,E)=6$ and
$\operatorname{ext}^{2}(E,E)=0$.

###### Proof.

Consider the short exact sequence:

(36) $0\rightarrow{\mathcal{O}}_{X}(-1)\rightarrow
E\rightarrow{\mathcal{I}}_{Y}\rightarrow 0.$

Applying $\operatorname{Hom}(E,\ \cdot\ )$ we obtain an exact sequence of
vector spaces

$\operatorname{Ext}^{1}(E,{\mathcal{O}}_{X}(-1))\rightarrow\operatorname{Ext}^{2}(E,E)\rightarrow\operatorname{Ext}^{2}(E,{\mathcal{I}}_{Y}).$

The left side term is zero since it is dual to $H^{1}(E(-2))\simeq
H^{1}({\mathcal{I}}_{Y}(-2))=0$. Let us now prove that the right side term
vanishes as well. By stability and by Lemma 30, we have
$h^{i}(E(1))=h^{3-i}(E(-3))=0$ for $i=2,3$; as moreover $P_{E}(1)=2$ and
$h^{0}(E(1))=2$, we conclude that $h^{1}(E(1))=0$ as well. This implies the
vanishing of
$\operatorname{Ext}^{i}(E,{\mathcal{O}}_{X})\simeq\operatorname{Ext}^{i}({\mathcal{O}}_{X},E(1))$
for $i=1,2$ which leads to an isomorphism
$\operatorname{Ext}^{2}(E,{\mathcal{I}}_{Y})\simeq\operatorname{Ext}^{1}(E,{\mathcal{O}}_{Y})$.
But $E$ is locally free therefore:

$\operatorname{Ext}^{1}(E,{\mathcal{O}}_{Y})\simeq H^{1}({\mathcal{H}}{\it
om}(E,{\mathcal{O}}_{Y}))\simeq H^{1}(E|_{Y}^{\vee})\simeq
H^{1}({\mathcal{N}}_{Y/X})=0$

(the isomorphism $E|_{Y}^{\vee}\simeq{\mathcal{N}}_{Y/X}$ is obtained
tensoring (36) for ${\mathcal{O}}_{Y}$ and the vanishing of
$H^{1}({\mathcal{N}}_{Y/X})$ is due to Lemma (50)). Therefore
$\operatorname{Ext}^{2}(E,{\mathcal{I}}_{Y})=0$ which implies
$\operatorname{Ext}^{2}(E,E)=0$. Now, the stability of $E$ leads to
$\operatorname{Hom}(E,E)\simeq{\mathbb{C}}$ and
$\operatorname{Ext}^{3}(E,E)\simeq\operatorname{Hom}(E,E(-3))^{*}=0$. Since
$E$ has homological dimension one, we can apply an argument equivalent to [14,
Proposition 3.4], obtaining:

(37) $\chi(E,E)=\frac{3}{2}c_{1}(E)^{2}-6c_{2}(E)+4=-5.$

which allows to conclude that $\operatorname{ext}^{1}(E,E)=6$. This ensures
that the moduli space ${\mathcal{M}}$ is smooth along ${\mathcal{I}}(2)$ and
that $\overline{{\mathcal{I}}(2)}$ is the only component passing trough any
point in ${\mathcal{I}}(2)$. ∎

We pass now to the case of non locally free instantons.

######  Proposition 53.

${\mathcal{M}}$ is smooth of dimension 6 at any point
$[E]\in{\mathcal{D}}(1,1)$.

###### Proof.

We know that $E$ fits in a short exact sequence:

(38) $0\rightarrow
E\rightarrow{\mathcal{S}}\rightarrow{\mathcal{O}}_{l}\rightarrow 0$

where $E^{\vee\vee}\simeq\mathcal{S}$ and $l=Sing(E).$ Applying
$\operatorname{Hom}(\>\cdot\>,E)$ we end up with a sequence of vector spaces:

$\operatorname{Ext}^{2}({\mathcal{S}},E)\rightarrow\operatorname{Ext}^{2}(E,E)\rightarrow\operatorname{Ext}^{3}({\mathcal{O}}_{l},E);$

$\operatorname{Ext}^{3}({\mathcal{O}}_{l},E)\simeq\operatorname{Hom}(E,{\mathcal{O}}_{l}(-3))\simeq\operatorname{Hom}(E\restriction_{l},{\mathcal{O}}_{l}(-3))$.
Tensoring (38) for $\otimes\>{\mathcal{O}}_{l}$ we obtain:

$0\rightarrow{\mathcal{T}\\!}{\it
or}_{1}({\mathcal{O}}_{l},{\mathcal{O}}_{l})\xrightarrow{\alpha}E\restriction_{l}\xrightarrow{\beta}{\mathcal{S}}\restriction_{l}\xrightarrow{\gamma}{\mathcal{O}}_{l}\rightarrow
0$

and consequently:

(39) $0\rightarrow{\mathcal{T}\\!}{\it
or}_{1}({\mathcal{O}}_{l},{\mathcal{O}}_{l})\xrightarrow{\alpha}E\restriction_{l}\xrightarrow{\beta}Im(\beta)\rightarrow
0,$ (40) $0\rightarrow
Im(\beta)\rightarrow{\mathcal{S}}\restriction_{l}\xrightarrow{\gamma}{\mathcal{O}}_{l}\rightarrow
0.$

${\mathcal{T}\\!}{\it
or}_{1}({\mathcal{O}}_{l},{\mathcal{O}}_{l})\simeq\mathcal{N}_{l/X}^{\vee}\simeq{\mathcal{O}}_{l}\oplus{\mathcal{O}}_{l}(-1)$,
thus $\operatorname{Hom}({\mathcal{T}\\!}{\it
or}_{1}({\mathcal{O}}_{l},{\mathcal{O}}_{l}),{\mathcal{O}}_{l}(-3))=0$ so that
$\operatorname{Hom}(E\restriction_{l},{\mathcal{O}}_{l}(-3))\simeq\operatorname{Hom}(Im(\beta),{\mathcal{O}}_{l}(-3)).$
From (40) $Im(\beta)$ is a rank one torsion free sheaf of degree -1; but $l$
is a line, therefore $Im(\beta)$ is a line bundle of degree -1, so that
$Im(\beta)\simeq{\mathcal{O}}_{l}(-1)$. From this we deduce
$\operatorname{Hom}(Im(\beta),{\mathcal{O}}_{l}(-3))=0$ and consequently that
$\operatorname{Hom}(E\restriction_{l},{\mathcal{O}}_{l}(-3))\simeq\operatorname{Ext}^{3}({\mathcal{O}}_{l},E)=0$.

To prove the vanishing of $\operatorname{Ext}^{2}({\mathcal{S}},E)$, we apply
$\operatorname{Hom}({\mathcal{S}},\cdot)$ to (38), getting:

$\operatorname{Ext}^{1}({\mathcal{S}},{\mathcal{O}}_{l})\rightarrow\operatorname{Ext}^{2}({\mathcal{S}},E)\rightarrow\operatorname{Ext}^{2}({\mathcal{S}},{\mathcal{S}})=0.$

Since ${\mathcal{S}}$ is locally free,
$\operatorname{Ext}^{1}({\mathcal{S}},{\mathcal{O}}_{l})\simeq
H^{1}({\mathcal{H}}{\it om}({\mathcal{S}},{\mathcal{O}}_{l}))$ and this latter
vanishes again due to
${\mathcal{S}}\restriction_{l}\simeq{\mathcal{O}}_{l}\oplus{\mathcal{O}}_{l}(-1)$.
These computations yield to $\operatorname{Ext}^{2}(E,E)=0$ implying the
smoothness of $\mathcal{M}$ at $E$. Also this time the stability of $E$
ensures that $\hom(E,E)=1$ and $\operatorname{ext}^{3}(E,E)=0$, and since the
homological dimension of $E$ is one, we can again argue as in [14, Proposition
3.4], which leads to $\chi(E,E)=\frac{3}{2}c_{1}(E)^{2}-6c_{2}(E)+4=-5$. This
implies that $\operatorname{ext}^{1}(E,E)=6$ ending our proof. ∎

We consider now the scheme ${\mathcal{B}}$ parameterizing the pencils of
curves ${\mathbb{P}}(H^{0}(E(1))$ for $[E]\in{\mathcal{L}}(2)$.
${\mathcal{B}}$ identifies with the Grassmann bundle:

(41)
${\mathcal{B}}:=G_{2}(\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee}))\simeq{\mathbb{P}}(\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee})^{\vee})\xrightarrow{\pi_{{\mathcal{B}}}}{\mathcal{U}}.$

By construction ${\mathcal{B}}$ is a smooth and irreducible 6-dimensional
variety. Our next goal is to show that ${\mathcal{B}}$ is isomorphic to
${\mathcal{L}}(2)$. To prove this we will construct a projective bundle
${\mathbb{P}}({\mathcal{E}})\to{\mathcal{B}}$ that carries a family of
instantons and such that the induced morphism
${\mathbb{P}}({\mathcal{E}})\to{\mathcal{L}}(2)$ factors trough an isomorphism
${\mathcal{B}}\to{\mathcal{L}}(2)$.

We start by considering the universal curve
$\mathbf{Y}\subset{\mathcal{H}}\times X$ and the relative ext sheaf
${\mathcal{E}}:={\mathcal{E}}{\it
xt}^{1}_{p_{1}}({\mathcal{I}}_{\mathbf{Y}}(1),{\mathcal{O}}_{{\mathcal{H}}\times
X})\in Coh({\mathcal{H}})$, where $p_{1}$ is the projection onto the first
factor (here for ${\mathcal{F}}\in Coh({\mathcal{H}}\times X)$ we define
${\mathcal{F}}(n):={\mathcal{F}}\otimes{p_{2}}^{*}{\mathcal{O}}_{X}(n)$.)

######  Proposition 54.

${\mathcal{E}}$ is a rank 2 vector bundle on ${\mathcal{H}}$ and the
projective bundle ${\mathbb{P}}({\mathcal{E}})$ admits the structure of a
${\mathbb{P}^{1}}$ bundle over ${\mathcal{B}}$.

###### Proof.

Recall that ${\mathcal{E}}:=R^{i}({p_{1}}_{*}{\mathcal{H}}{\it
om}({\mathcal{I}}_{\mathbf{Y}}(1),\ \cdot\
))({\mathcal{O}}_{{\mathcal{H}}\times X})$ hence, from the spectral sequence
$R^{p}{p_{1}}_{*}({\mathcal{E}}{\it
xt}^{q}({\mathcal{I}}_{\mathbf{Y}}(1),{\mathcal{O}}_{{\mathcal{H}}\times
X}))\Rightarrow{\mathcal{E}}{\it
xt}^{p+q}_{p_{1}}({\mathcal{I}}_{\mathbf{Y}}(1),{\mathcal{O}}_{{\mathcal{H}}\times
X})$ we obtain an exact sequence:

$\displaystyle 0\rightarrow R^{1}{p_{1}}_{*}({\mathcal{H}}{\it
om}({\mathcal{I}}_{\mathbf{Y}}(1),{\mathcal{O}}_{{\mathcal{H}}\times
X}))\rightarrow{\mathcal{E}}$ $\displaystyle\rightarrow$
$\displaystyle\rightarrow{p_{1}}_{*}({\mathcal{E}}{\it
xt}^{1}({\mathcal{I}}_{\mathbf{Y}}(1),{\mathcal{O}}_{{\mathcal{H}}\times
X}))\rightarrow R^{2}{p_{1}}_{*}({\mathcal{H}}{\it
om}({\mathcal{I}}_{\mathbf{Y}}(1)$
$\displaystyle,{\mathcal{O}}_{{\mathcal{H}}\times X}))\rightarrow 0.$

As $R^{i}{p_{1}}_{*}({\mathcal{H}}{\it
om}({\mathcal{I}}_{\mathbf{Y}}(1),{\mathcal{O}}_{{\mathcal{H}}\times X}))=0,\
i=1,2$, we get

${\mathcal{E}}\simeq{p_{1}}_{*}({\mathcal{E}}{\it
xt}^{1}({\mathcal{I}}_{\mathbf{Y}}(1),{\mathcal{O}}_{{\mathcal{H}}\times
X}))\simeq{p_{1}}_{*}(\tilde{\omega}_{\mathbf{Y}}(2)),$

where $\tilde{\omega}_{\mathbf{Y}}$ is the relative dualizing sheaf.
$\forall\>[Y]\in{\mathcal{H}},\>h^{0}(\omega_{Y}(2))=2$ and since
${\mathcal{H}}$ is integral, we can conclude that ${\mathcal{E}}$ is a rank 2
vector bundle.

The isomorphism
${\mathcal{E}}\simeq{p_{1}}_{*}(\tilde{\omega}_{\mathbf{Y}}(2))$ also implies
that ${\mathcal{E}}$ commutes with base change, and since
${p_{1}}_{*}{\mathcal{H}}{\it
om}({\mathcal{I}}_{\mathbf{Y}}(1),{\mathcal{O}}_{{\mathcal{H}}\times X})=0$,
from [25, Corollary 4.5] we get the existence of a universal extension on
${\mathbb{P}}({\mathcal{E}})\times X$:

(42) $0\rightarrow{\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}})\times
X}\otimes{p_{1}}^{*}{\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}})}(1)\rightarrow\hat{\mathbf{E}}\rightarrow{\mathcal{I}}_{\hat{\mathbf{Y}}}\rightarrow
0$

where $\hat{\mathbf{Y}}\subset{\mathbb{P}}({\mathcal{E}})\times X$ is the
pullback of the universal curve $\mathbf{Y}$. Twisting and applying the
functor ${p_{1}}_{*}$ we obtain a short exact sequence of vector bundles on
${\mathbb{P}}({\mathcal{E}})$:

(43)
$0\rightarrow{\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}})}(1)\rightarrow{p_{1}}_{*}(\hat{\mathbf{E}}(1))\rightarrow\pi_{{\mathcal{E}}}^{*}({p_{1}}_{*}({\mathcal{I}}_{\mathbf{Y}}(1)))\rightarrow
0$

($\pi_{{\mathcal{E}}}:{\mathbb{P}}({\mathcal{E}})\to{\mathcal{H}}$ is the
standard projection). We claim that the rank 2 vector bundle
${p_{1}}_{*}(\hat{\mathbf{E}}(1))$ on ${\mathbb{P}}({\mathcal{E}})$ induces a
morphism ${\mathbb{P}}({\mathcal{E}})\to{\mathcal{B}}$. This is obtained via a
relative version of the argument presented in (49).

Take an affine cover $V_{i}=Spec(A_{i})$ of ${\mathbb{P}}({\mathcal{E}})$ that
is trivialising for ${p_{1}}_{*}(\hat{\mathbf{E}}(1))$; on each $V_{i}$ we
have

${p_{1}}_{*}(\hat{\mathbf{E}}(1)))|_{V_{i}}\simeq
H^{0}(\hat{\mathbf{E}}(1)|_{V_{i}\times X})^{\sim}\simeq(A_{i}^{2})^{\sim}.$

For any non zero $s_{i}\in H^{0}(\hat{\mathbf{E}}(1)|_{V_{i}\times X})$, its
image in $H^{0}({\mathcal{I}}_{\hat{\mathbf{Y}}}(1)|_{V_{i}\times X})$
determines a family $\hat{\mathbf{Q}}_{i}$ of hyperplane sections whose fibre
over $v\in V_{i}$ is $\langle\hat{\mathbf{Y}}_{v}\rangle\cap X$. Moreover for
any pair $s_{i},t_{i}$ of generators of
$H^{0}(\hat{\mathbf{E}}(1)|_{V_{i}\times X}),$ we have isomorphisms
$(\hat{\mathbf{E}}(1)|_{V_{i}\times
X})/(s_{i},t_{i})\simeq{\mathcal{I}}_{\hat{\mathbf{Y}}_{i},\hat{\mathbf{Q}}_{i}}(1)$
($\hat{\mathbf{Y}}_{i}$ being the restriction of $\hat{\mathbf{Y}}$ to
$V_{i}\times X$). We get therefore injective $A_{i}$-linear maps
${p_{1}}_{*}(\hat{\mathbf{E}}(1))|_{V_{i}}\hookrightarrow(\pi_{{\mathcal{H}}}\circ\pi_{{\mathcal{E}}})^{*}(\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee})|_{V_{i}})$
that glue defining an injective morphism
${p_{1}}_{*}(\hat{\mathbf{E}}(1))\hookrightarrow(\pi_{{\mathcal{H}}}\circ\pi_{{\mathcal{E}}})^{*}\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee})$.
By the universal property of ${\mathcal{B}}$,
$\pi_{{\mathcal{H}}}\circ\pi_{{\mathcal{E}}}$ factors therefore trough a
morphism $\rho:{\mathbb{P}}({\mathcal{E}})\to{\mathcal{B}}$.

We finally show that
${\mathbb{P}}({\mathcal{E}})\xrightarrow{\rho}{\mathcal{B}}$ is a projective
bundle. Denote by
${\mathcal{T}}_{{\mathcal{B}}}\subset\pi_{{\mathcal{B}}}^{*}(\operatorname{Sym}^{2}({\mathcal{T}}_{{\mathcal{U}}}^{\vee}))$
the tautological rank 2 sub-bundle. By construction
$\rho^{*}({\mathcal{T}}_{{\mathcal{B}}})\simeq{p_{1}}_{*}(\hat{\mathbf{E}}(1))$
and since
${\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}})}(1)\hookrightarrow{p_{1}}_{*}(\hat{\mathbf{E}}(1))$,
we get that $\rho$ factors through a morphism
${\mathbb{P}}({\mathcal{E}})\xrightarrow{\rho^{\prime}}{\mathbb{P}}({\mathcal{T}}_{{\mathcal{B}}})$
such that
${\rho^{\prime}}^{*}({\mathcal{O}}_{{\mathbb{P}}({\mathcal{T}}_{{\mathcal{B}}})}(-1))\simeq{\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}})}(1)$.
$\rho^{\prime}$ is the morphism mapping a point $(Y,e),\ [Y]\in{\mathcal{H}},\
e\in\operatorname{Ext}^{1}({\mathcal{I}}_{Y}(1),{\mathcal{O}}_{X})$ to
$([{\mathbb{P}}(H^{0}(E(1)))],Y)\in{\mathbb{P}}({\mathcal{T}}_{\mathcal{B}})$,
$E=\hat{\mathbf{E}}_{(Y,e)}$. $\rho^{\prime}$ is a bijective morphism between
smooth varieties, therefore it is an isomorphism. ∎

######  Remark 55.

The variety
${\mathbb{P}}({\mathcal{E}})\simeq{\mathbb{P}}({\mathcal{T}}_{{\mathcal{B}}})$
identifies with the following incidence variety

${\mathbb{P}}({\mathcal{E}})\simeq{\mathbb{P}}({\mathcal{T}}_{{\mathcal{B}}})\simeq\\{(Y,{\mathbb{P}}^{1})\in{\mathcal{H}}\times{\mathcal{B}}\mid[Y]\in{\mathbb{P}}^{1}\\}$

From the proof of Proposition 54, we learn in particular that
${\mathbb{P}}({\mathcal{E}})$ carries a family of instantons
$\hat{\mathbf{E}}$. Accordingly, we have the following:

######  Corollary 56.

There exists a morphism $\psi:{\mathbb{P}}({\mathcal{E}})\to{\mathcal{L}}(2)$
that locally, in the étale topology, has the structure of a
${\mathbb{P}}^{1}$-bundle.

###### Proof.

The family $\hat{\mathbf{E}}$ induces a morphism
$\psi:{\mathbb{P}}({\mathcal{E}})\to{\mathcal{M}}$ that, by Propositions 44
and 46, surjects onto ${\mathcal{L}}(2)$. To prove the rest of the current
proposition, we argue as in [27, Lemma 5.3]. We start considering an étale
cover $\mathbf{W}_{i}\to{\mathcal{L}}(2)$ of ${\mathcal{L}}(2)$ such that each
$\mathbf{W}_{i}\times X$ carries a universal sheaf $\mathbf{E}_{i}$. Define
$\mathbf{G}_{i}:={p_{1}}_{*}(\mathbf{E}_{i}(1))$. This is a rank 2 vector
bundle on $\mathbf{W}_{i}$. Denote by ${\mathcal{E}}_{\mathbf{W}_{i}}$ the
pullback of ${\mathcal{E}}$ to
${\mathbb{P}}({\mathcal{E}})\times_{{\mathcal{L}}(2)}\mathbf{W}_{i}$, so that
${\mathbb{P}}({\mathcal{E}})\times_{{\mathcal{L}}(2)}\mathbf{W}_{i}\simeq{\mathbb{P}}({\mathcal{E}}_{\mathbf{W}_{i}})$.
Define $\psi_{i}$ as the induced morphism
${\mathbb{P}}({\mathcal{E}}_{\mathbf{W}_{i}})\to\mathbf{W}_{i}$, and let
$\hat{\mathbf{E}}_{\mathbf{W}_{i}}$ denote the pullback of $\hat{\mathbf{E}}$
to ${\mathbb{P}}({\mathcal{E}}_{\mathbf{W}_{i}})$; by the universal property
of $\mathbf{E}_{i}$,
$\psi_{i}^{*}(\mathbf{E}_{i})\simeq\hat{\mathbf{E}}_{\mathbf{W}_{i}}\times\mathbf{L}$,
for some line bundle $\mathbf{L}$ on
${\mathbb{P}}({\mathcal{E}}_{\mathbf{W}_{i}})$. Observe now that pulling back
(43) to ${\mathbb{P}}({\mathcal{E}}_{\mathbf{W}_{i}})$, we obtain an injection
${\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}}_{\mathbf{W}_{i}})}(1)\otimes\mathbf{L}^{*}\hookrightarrow\psi_{i}^{*}(\mathbf{G}_{i})$;
this induces a morphism
${\mathbb{P}}({\mathcal{E}}_{\mathbf{W}_{i}})\to{\mathbb{P}}(\mathbf{G}_{i})$
and once again, since this is a bijective morphism between smooth varieties,
we conclude that it is an isomorphism. ∎

From the irreducibility of ${\mathbb{P}}(E)$ we deduce the irreducibility of
${\mathcal{L}}(2)$; this observation together with Lemma 52 and Proposition 53
lead to the following claim.

######  Corollary 57.

${\mathcal{L}}(2)$ is a smooth and irreducible scheme of dimension 6.

Next, we argue that ${\mathcal{L}}(2)$ is not just locally a fibration over
the open subet ${\mathcal{U}}\subset\mathbb{G}(1,F(X))$ (see Remark 45), but
that actually, it is a projective bundle isomorphic to the scheme
${\mathcal{B}}$ defined in display (41).

######  Proposition 58.

There exists an isomorphism ${\mathcal{L}}(2)\xrightarrow{\zeta}{\mathcal{B}}$
such that $\rho=\zeta\circ\psi$.

###### Proof.

Set theoretically, $\zeta$ is the map sending $[E]\in{\mathcal{L}}(2)$ to the
pencil of curves defined by ${\mathbb{P}}(H^{0}(E(1)))$. Let us check that it
is actually a morphism of schemes. For each open $V\subset{\mathcal{B}}$,
$\zeta^{-1}(V)=\psi(\rho^{-1}(V))$ is open in ${\mathcal{L}}(2)$ since $\psi$
is open (this is a consequence of Proposition 56). We have then a morphism
${\mathcal{O}}_{{\mathcal{B}}}(V)\to{\mathcal{O}}_{{\mathcal{L}}(2)}(\zeta^{-1}(V))$
induced by
${\mathcal{O}}_{{\mathcal{B}}}(V)\to{\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}})}(\rho^{-1}(V))$:
indeed from Corollary 56,
$\psi_{*}{\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}})}\simeq{\mathcal{O}}_{{\mathcal{L}}(2)}$,
thus
${\mathcal{O}}_{{\mathcal{L}}(2)}(\zeta^{-1}(V))\simeq{\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}})}(\psi^{-1}(\zeta^{-1}(V)))\simeq{\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}})}(\rho^{-1}(V))$.
$\zeta$ is bijective by construction thus, by the smoothness of
${\mathcal{L}}(2)$ and ${\mathcal{B}}$, is an isomorphism. ∎

We now denote by $\overline{{\mathcal{I}}(2)}$ the closure of
${\mathcal{I}}(2)$ in ${\mathcal{M}}$ and by
$\overline{{\mathcal{I}}(2)}^{inst}$ its open subscheme parameterizing
instantons.

######  Corollary 59.

${\mathcal{L}}(2)\simeq\overline{{\mathcal{I}}(2)}^{inst}$.

###### Proof.

${\mathcal{L}}(2)$ is a smooth irreducible 6-dimensional variety that contains
the moduli ${\mathcal{I}}(2)={\mathcal{L}}(2)\setminus{\mathcal{D}}(1,1)$ as
an open dense subset. Therefore we have equalities
${\mathcal{L}}(2)=\overline{{\mathcal{L}}(2)}\cap{\mathcal{L}}(2)=\overline{{\mathcal{I}}(2)}\cap{\mathcal{L}}(2):=\overline{{\mathcal{I}}(2)}^{inst}$.
Note that ${\mathcal{I}}(2)$ identifies with the following open subset of
${\mathcal{L}}(2)$: for $\mathring{{\mathcal{B}}}$, the open subset of base
point free pencils, we have
${\mathcal{I}}(2)=\zeta^{-1}(\mathring{{\mathcal{B}}})$. ∎

######  Corollary 60.

${\mathcal{D}}(1,1)$ is contained in $\overline{{\mathcal{I}}(2)}$; in
particular a general deformation of a non locally free instanton $[E]$ in
${\mathcal{L}}(2)$ is an instanton bundle.

###### Proof.

${\mathcal{D}}(1,1)\subset\overline{{\mathcal{I}}(2)}$ is an immediate
consequence of Corollary 59.

The possibility to deform $[E]\in{\mathcal{D}}(1,1)$ to an instanton bundle is
due to the smoothness of ${\mathcal{L}}(2)$. Note in particular that we have
the following. The locus of pencils of curves with a base point is a smooth
and irreducible divisor
${\mathcal{Z}}\subset{\mathcal{B}}={\mathcal{B}}\setminus\mathring{{\mathcal{B}}}$
and it is the image of ${\mathcal{D}}(1,1)$ trough $\zeta$. A deformation of
$E$ to an instanton bundle, for $[E]\in{\mathcal{D}}(1,1)$ corresponds
therefore to a deformation of
$[{\mathbb{P}}^{1}(H^{0}(E(1)))]\in{\mathcal{Z}}$ along a direction normal to
${\mathcal{Z}}$ (the smoothness of ${\mathcal{B}}$ and ${\mathcal{Z}}$ implies
that such a deformation is always possible). ∎

##  6\. The moduli space ${\mathcal{M}}_{X}(2;-1,2,0)$

In this section we provide a full description of the moduli space
${\mathcal{M}}$. In the previous section we proved that the closure
$\overline{{\mathcal{L}}(2)}$ of the instanton moduli space is an irreducible
component of ${\mathcal{M}}$; to complete our description of ${\mathcal{M}}$
we pass then to the study the closed subscheme
${\mathcal{C}}:={\mathcal{M}}\setminus{\mathcal{L}}(2)\subset{\mathcal{M}}$
consisting of the non instanton sheaves in ${\mathcal{M}}$. Such sheaves can
be characterized as follows.

######  Proposition 61.

Each sheaf $E$ corresponding to a point $[E]\in{\mathcal{C}}$ is obtained by
elementary transformation of a $\mu$-stable reflexive sheaf $F$ with Chern
classes $(-1,2,2)$ along a point. Conversely, for each sheaf $F$ such that
$[F]\in{\mathcal{M}}_{X}(-1;2,2,2)$ the kernel of a surjection
$F\twoheadrightarrow{\mathcal{O}}_{p}$ locates a point in ${\mathcal{C}}$

###### Proof.

Let us take a non instanton sheaf $E$ and consider $E^{\vee\vee}$. This latter
must be a $\mu$-stable reflexive sheaf having $c_{1}(E^{\vee\vee})=-1$ and by
Lemma 41, either $c_{2}(E^{\vee\vee})=1$ and $E^{\vee\vee}\simeq{\mathcal{S}}$
or $c_{2}(E^{\vee\vee})=2$. In the first case $E^{\vee\vee}/E$ is a one-
dimensional sheaf $T$ with Hilbert polynomial $n+1$; we denote by $T_{0}$ the
maximal zero-dimensional subsheaf of $T$ and by $T_{1}$ the quotient
$T_{1}:=T/T_{0}$. $T_{1}$ is thus a line bundle on a line $l\subset X$ and
since ${\mathcal{S}}$ surjects onto $T_{1}$,
${\mathcal{S}}|_{l}\simeq{\mathcal{O}}_{l}(-1)\oplus{\mathcal{O}}_{l}$ and
$[E]\not\in{\mathcal{L}}(2)$, we conclude that
$T_{1}\simeq{\mathcal{O}}_{l}(-1)$ and that $T_{0}\simeq{\mathcal{O}}_{p},\
p=\operatorname{Supp}(T_{0})$. Denote by $F$ the kernel of the surection
${\mathcal{S}}\twoheadrightarrow{\mathcal{O}}_{l}(-1)$; this is a $\mu$-stable
sheaf with Chern classes $(-1,2,2)$ and from the commutative diagram:

${0}$${E}$${{\mathcal{S}}}$${T}$${0}$${0}$${F}$${{\mathcal{S}}}$${{\mathcal{O}}_{l}(-1)}$${0}$

we see that $E\simeq\ker(F\twoheadrightarrow{\mathcal{O}}_{p})$. Suppose now
that $c_{2}(E^{\vee\vee})=2$. In this case $T:=E^{\vee\vee}/E$ is a zero-
dimensional and has Chern character
$\operatorname{ch}(T)=(0,0,0,\frac{c_{3}(E)}{2})$. Applying [10, Theorem 2.2]
the spectrum of $E^{\vee\vee}$ can only consists of the integer
$k=-2=-1-\frac{c_{3}}{2}$. This implies that $c_{3}(E)=2$ hence that
$T\simeq{\mathcal{O}}_{p},\ p=\operatorname{Supp}(T)$.

For the converse implication we just need to check that the elementary
transformation $E$ of a sheaf $F$, $[F]\in{\mathcal{M}}_{X}(2;-1,2,2)$ along a
point $p$ is indeed semistable. Arguing as above, for
$[F]\in{\mathcal{M}}_{X}(2;-1,2,2)$ we have that $F^{\vee\vee}$ is reflexive
with $c_{1}(F^{\vee\vee})=-1$ and $c_{2}(F^{\vee\vee})=1$ or $2$. In the first
case $F^{\vee\vee}={\mathcal{S}}$; in the second, applying again [10, Theorem
2.2], we get $c_{3}(F^{\vee\vee})=2$ hence $F\simeq F^{\vee\vee}$. In both
cases $F^{\vee\vee}$ is $\mu$-stable therefore, if
$E=\ker(F\twoheadrightarrow{\mathcal{O}}_{p})$, $E$ is $\mu$-stable as well
since $E^{\vee\vee}\simeq F^{\vee\vee}$. ∎

Once again we will use the Serre correspondence to deduce the geometric
properties of ${\mathcal{C}}$ from the geometry of the family of the
corresponding curves; these curves will still belong to the Hilbert scheme
$\operatorname{Hilb}_{2t+2}(X)$ but this time they wont be l.c.m.

The study of ${\mathcal{C}}$ will lead us to prove the main result of this
section:

######  Theorem 62.

The moduli space ${\mathcal{M}}$ is connected and consists of two irreducible
components:

  1. (1)

a 6-dimensional component $\overline{{\mathcal{L}}(2)}$ given by the closure
of the open subset of instanton sheaves;

  2. (2)

a 10-dimensional irreducible component ${\mathcal{C}}$ consisting of non
instanton sheaves.

In addition, ${\mathcal{M}}$ is generically smooth along both components.

###  6.1. The moduli space ${\mathcal{M}}_{X}(2;-1,2,2)$

In order to better understand the geometry of ${\mathcal{C}}$, we first need
to study ${\mathcal{M}}_{X}(2;-1,2,2)$. From the proof of Proposition 61 we
have already learnt that we have two families of sheaves in
${\mathcal{M}}_{X}(2;-1,2,2)$:

######  Lemma 63.

Let $F$ be a rank 2 semistable sheaf $F$ with Chern classes $(-1,2,2)$. Then
$F$ is $\mu$-stable and either $F$ is reflexive and $\operatorname{Sing}(F)$
is zero dimensional of length 2, or $F^{\vee\vee}\simeq{\mathcal{S}}$ and
${\mathcal{S}}/F\simeq{\mathcal{O}}_{l}(-1)$ for a line $l\subset X$.

We are going to prove the following:

######  Theorem 64.

${\mathcal{M}}_{X}(2;-1,2,2)$ is a smooth 6-dimensional irreducible variety
isomorphic to $\mathbb{G}(1,{\mathbb{P}}(V)$.

Also in this occasion, our main tool to study sheaves in
${\mathcal{M}}_{X}(2;-1,2,2)$ is the Serre correspondence. Towards the rest of
the section we denote by ${\mathcal{R}}(2;-1,2,2)$ the open subset of
${\mathcal{M}}_{X}(2;-1,2,2)$ parameterizing reflexive sheaves.

######  Lemma 65.

Let $[F]$ be a point in ${\mathcal{M}}_{X}(2;-1,2,2)$. Then $h^{0}(F(1))=3$
and for each $s\in H^{0}(E(1)),\ s\neq 0$,
$\operatorname{coker}(s)\simeq{\mathcal{I}}_{C}(1)$ for a conic $C\subset X$.

###### Proof.

Let us start by considering the case $[F]\in{\mathcal{R}}(2;-1,2,2)$. We show
that $F(1)$ always admits global sections. By Riemann–Roch $\chi(F(1))=3$ and
by stability $h^{3}(F(1))=0$. As already claimed in the proof of Proposition
61, the spectrum of $F$ consists just of the integer $-2$ (due to [10, Theorem
2.2]) which implies the following:

(44) $h^{1}(F(j))=h^{0}({\mathcal{O}}_{{\mathbb{P}^{1}}}(k+j+1)),\
\forall\>j\leq-2,\
h^{2}(F(j))=h^{1}({\mathcal{O}}_{{\mathbb{P}^{1}}}(k+j+1)),\ \forall\>j\geq
0.$

This means that $h^{2}(F(1))=0$ hence we necessarily have $h^{0}(F(1))>0$.
Now, since $H^{0}(F)=0$, $\forall\ s\in H^{0}(F(1)),\>s\neq 0$
$\operatorname{coker}(s)$ is torsion free and of the form
${\mathcal{I}}_{C}(1)$ for a l.c.m. curve $C\subset X$.

A Chern class computation leads to $P_{C}(n)=2n+1$ hence $C$ is a plane conic
and from the short exact sequence

(45) $0\rightarrow{\mathcal{O}}_{X}\rightarrow
F(1)\rightarrow{\mathcal{I}}_{C}(1)\rightarrow 0$

we compute that $h^{0}(F(1))=3$ hence $h^{1}(F(1))=0$. If
$F^{\vee\vee}\simeq{\mathcal{S}}$ instead, from the short exact sequence

$0\rightarrow
F\rightarrow{\mathcal{S}}\rightarrow{\mathcal{O}}_{l}(-1)\rightarrow 0,$

we get $h^{i}(F(1))=h^{i}({\mathcal{S}}(1))=0,\ i=2,3$; as moreover
$H^{0}({\mathcal{S}}(1))\rightarrow H^{0}({\mathcal{O}}_{l})$ can not be the
zero map (for $l^{\prime}\subset X$ general, there are no surjection
${\mathcal{I}}_{l^{\prime}}(1)\twoheadrightarrow{\mathcal{O}}_{l})$ we
conclude that $h^{0}(F(1))=3$ and $h^{1}(F(1))=0$. Since any global section of
${\mathcal{S}}$ has torsion free cokernel, for any non-zero $s\in
H^{0}(F(1))$, $\operatorname{coker}(s)\simeq{\mathcal{I}}_{Z}(1)$ fits in

(46)
$0\rightarrow{\mathcal{I}}_{Z}(1)\rightarrow{\mathcal{I}}_{l^{\prime}}(1)\rightarrow{\mathcal{O}}_{l}\rightarrow
0.$

with $l^{\prime}=\operatorname{coker}(\iota(s)),\
\iota:=H^{0}(F(1))\hookrightarrow H^{0}({\mathcal{S}}(1)).$ Therefore $Z$ is a
reducible conic supported on $l\cup l^{\prime}$ (note that for $s$ general, as
${\mathcal{I}}_{l^{\prime}}$ surjects onto ${\mathcal{O}}_{l}(-1)$,
$l^{\prime}$ will meet $l$ at a point). ∎

It is straightforward to check that this construction can be “reversed”,
leading to the following claim.

######  Lemma 66.

Serre correspondence establishes a 1-1 correspondence between

  * •

pairs $(F,s)$ with $[F]\in{\mathcal{R}}(2;-1,2,2),\
s\in{\mathbb{P}}(H^{0}(F(1)))$ (resp. pairs $(F,s)$ with
$[F]\in{\mathcal{M}}_{X}(2;-1,2,2)\setminus{\mathcal{R}}(2;-1,2,2),\
s\in{\mathbb{P}}(H^{0}(F(1)))$)

  * •

pairs $(C,\xi)$ with $[C]\in\operatorname{Hilb}_{2t+1}(X),$ $\
\xi\in{\mathbb{P}}(H^{0}(\omega_{C}(2)))$ vanishing along 2 points on $C$
(resp. $(C,\xi),\ [C]\in\operatorname{Hilb}_{2t+1}(X),$ reducible $\
\xi\in{\mathbb{P}}(H^{0}(\omega_{C}(2)))$ vanishing along a component of $C$)

Mimicking what we have done for sheaves in ${\mathcal{L}}(2)$, we are now
going to describe in details the linear system
${\mathbb{P}}(H^{0}(F(1)))\simeq{\mathbb{P}^{2}}$ of conics associated to
$[F]\in{\mathcal{M}}_{X}(-2;-1,2,2)$. This will help us to better understand
the geometry of the scheme ${\mathcal{M}}_{X}(2;-1,2,2)$.

######  Proposition 67.

  1. (1)

If $[F]\in{\mathcal{R}}(2;-1,2,2)$ and $\operatorname{Sing}(F)$ consists of
two distinct points $p_{1},\>p_{2}$, ${\mathbb{P}}(H^{0}(F(1)))$ identifies
with the linear system of conics containing $p_{i},\ i=1,2$ and its image
under the isomorphism $\operatorname{Hilb}_{2t+1}(X)\simeq\mathbb{G}(2,V^{*})$
is the Schubert variety of planes containing $\overline{p_{1}p_{2}}.$

  2. (2)

If $[F]\in{\mathcal{R}}(2;-1,2,2)$ and $\operatorname{Sing}(F)_{\rm red}=p$,
there exists a line $l\subset{\mathbb{P}}^{4}$ tangent at $p$ to each conic in
${\mathbb{P}}(H^{0}(F(1)))$ and the image of ${\mathbb{P}}(H^{0}(F(1)))$ under
the isomorphism $\operatorname{Hilb}_{2t+1}(X)\simeq\mathbb{G}(2,V^{*})$ is
the Schubert variety of planes containing $l$.

  3. (3)

If $[F]\in{\mathcal{M}}_{X}(2;-1,2,2)\setminus{\mathcal{R}}(2;-1,2,2)$
${\mathbb{P}}(H^{0}(F(1)))$ identifies with the Schubert variety of planes
containing the line $\operatorname{Sing}(F)$.

###### Proof.

Suppose at first that $F$ reflexive with
$\operatorname{Sing}(F)=\\{p_{1},\>p_{2}\\},\ p_{1}\neq p_{2}$. It is easy
then to compute that the conics in $X$ passing through these points are
parameterised by a plane: a conic $C\subset X$ passes indeed through the
points $p_{1},\ p_{2}$ if and only if $\overline{p_{1}p_{2}}\subset\langle
C\rangle$. This means that the image of the family of conics passing through
the $p_{i}$s, under the isomorphism
$\operatorname{Hilb}_{2t+1}(X)\simeq\mathbb{G}(2,V^{*}),\ C\mapsto\langle
C\rangle$ is the Schubert variety of planes containing $\overline{p_{1}p_{2}}$
that is a plane in $\mathbb{G}(2,V^{*})$. The proposition follows since every
conic in ${\mathbb{P}}(H^{0}(F(1)))$ contains $p_{i},\ i=1,2$. Whenever
$\operatorname{Sing}(F)$ is supported on a single point $p$ we can compute
again that the linear system of conics containing $\operatorname{Sing}(F)$ is
a plane in $\mathbb{G}(2,V^{*})$. The scheme $\operatorname{Sing}(F)$
corresponds indeed to the data of the point $p$ together with a tangent
direction $v\in T_{p}X$ or equivalently, to a line $l$ tangent to $X$ at $p$.
A conic $C$ contains $\operatorname{Sing}(F)$ if and only if $\langle
C\rangle$ contains $l$, hence ${\mathbb{P}}(H^{0}(F(1)))$ identifies with the
Schubert variety ${\mathbb{P}^{2}}\subset\mathbb{G}(2,V^{*})$ of planes
containing $l$. Finally if $F$ is singular along a line $l$, consider the
inclusion $\iota:H^{0}(F(1))\hookrightarrow H^{0}({\mathcal{S}}(1))$. Each
$\iota(s)$ defines a line $l^{\prime}$ giving rise to a short exact sequence
of the form (46). Since ${\mathcal{I}}_{l^{\prime}}$ surjects onto
${\mathcal{O}}_{l}(-1)$ if and only if either $l=l^{\prime}$ or $l\cap
l^{\prime}$ consists of a point, we deduce therefore that
${\mathbb{P}}(\iota(H^{0}(F(1)))$ identifies with the space of lines meeting
$l$ and that ${\mathbb{P}}(H^{0}(F(1))$ identifies therefore with the family
of planes containing $l$. ∎

From now on the family of conics associated to
$[F]\in{\mathcal{M}}_{X}(2;-1,2,2)$ will simply be denoted by
${\mathbb{P}}(H^{0}(F(1)))$ and the line contained in every plane $\langle
C\rangle$, $C\in{\mathbb{P}}(H^{0}(F(1)))$ will be denoted by $l_{F}$ (note
that for $[F]$ belonging to the closed subscheme
${\mathcal{M}}_{X}(2;-1,2,2)\setminus{\mathcal{R}}(2;-1,2,2)$,
$l_{F}=\operatorname{Sing}(F)$).

######  Lemma 68.

$[F]\in{\mathcal{R}}(2;-1,2,2)$ if and only if $l_{F}\not\subset X$.

###### Proof.

If $l_{F}\subset X$, all the conics $C\in{\mathbb{P}}(H^{0}(F(1)))$ contain
$l_{F}$ and $\operatorname{Sing}(F)\subset l_{F}$. This can not happen if $F$
is reflexive, since if ever a section $\xi\in H^{0}(\omega_{C}(2))$,
$[C]\in{\mathbb{P}}(H^{0}(F(1))$ vanishes along 2 points on $l_{F}\subset C$,
it would vanish along the entire $l_{F}$ contradicting the reflexivity of $F$.
The converse implication is obvious since $l_{F}\not\subset X$ ensures that
$\operatorname{Sing}(F)=l_{F}\cap X$ consists of two points hence, by Lemma
63, $[F]\in{\mathcal{R}}(2;-1,2,2)$. ∎

Let us now analyse the local behaviour of the moduli space
${\mathcal{M}}_{X}(2;-1,2,2)$.

######  Proposition 69.

For each point $[F]\in{\mathcal{M}}_{X}(2;-1,2,2)$ we have
$\operatorname{ext}^{2}(F,F)=0$. For each point
$[F]\in{\mathcal{R}}(2;-1,2,2)$ we have $\operatorname{ext}^{1}(F,F)=6$.

###### Proof.

A general section $s\in H^{0}(F(1))$ defines a short exact sequence of the
form (45).

Applying the functor $\operatorname{Hom}(\>\cdot\>,F)$ we get a sequence:

$\operatorname{Ext}^{2}(\mathcal{I}_{C},F)\rightarrow\operatorname{Ext}^{2}(F,F)\rightarrow\operatorname{Ext}^{2}({\mathcal{O}}_{X}(-1),F);$

The r.s.t is zero since, from (44), $H^{2}(F(1))=0$; from (45) we compute the
vanishing of $H^{i}(F)=0,\ i=2,3$ yielding:

$\operatorname{Ext}^{2}(\mathcal{I}_{C},F)\simeq\operatorname{Ext}^{3}({\mathcal{O}}_{C},F)\simeq\operatorname{Hom}(F,{\mathcal{O}}_{C}(-3))^{*}.$

Applying $-\otimes{\mathcal{O}}_{C}$ to (45), we obtain an exact sequence

$0\rightarrow{\mathcal{T}\\!}{\it
or}_{1}^{{\mathcal{O}}_{X}}(F,\mathcal{O}_{C})\xrightarrow{a}{\mathcal{T}\\!}{\it
or}_{1}^{{\mathcal{O}}_{X}}(\mathcal{I}_{C},{\mathcal{O}}_{C})\xrightarrow{b}{\mathcal{O}}_{C}(-1)\xrightarrow{c}F|_{C}\xrightarrow{d}\mathcal{N}_{C/X}^{\vee}\rightarrow
0$

from which we extract the short exact sequences:

(47) $0\rightarrow\ker(d)\rightarrow
F|_{C}\xrightarrow{d}\mathcal{N}_{C/X}^{\vee}\rightarrow 0$

(48)
$0\rightarrow\ker(c)\rightarrow{\mathcal{O}}_{C}(-1)\rightarrow\ker(d)\rightarrow
0$

Applying $\operatorname{Hom}(\ \cdot\ ,{\mathcal{O}}_{C}(-3))$ to (48), we get
that $\operatorname{Hom}(\ker(c),{\mathcal{O}}_{C}(-3))$ injects into
$\operatorname{Hom}({\mathcal{O}}_{C}(-1),{\mathcal{O}}_{C}(-3))\simeq
H^{0}({\mathcal{O}}_{C}(-2))=0$. Therefore, from (47), we get
$\operatorname{Hom}(F|_{C},{\mathcal{O}}_{C}(-6pt))\simeq\operatorname{Hom}(\mathcal{N}_{C/X}^{\vee},{\mathcal{O}}_{C}(-6pt))\simeq
H^{0}(\mathcal{N}_{C/X}(-3)).$ $C$ is a plane section of $X$, therefore
${\mathcal{N}}_{C/X}\simeq{\mathcal{O}}_{C}(1)^{\oplus 2}$ hence
$H^{0}(\mathcal{N}_{C/X}(-3))=0$ implying
$\operatorname{Hom}(F,{\mathcal{O}}_{C}(-3))=0$ and finally,
$\operatorname{Ext}^{2}(F,F)=0$. If we suppose that moreover $F$ is reflexive,
it has homological dimension 1; this allows to compute (again arguing as in
[14, Proposition 3.4])

$\chi(F,F)=-5.$

The stability of $F$ implies that $\hom(F,F)=1$ and that
$\operatorname{ext}^{3}(F,F)=0$; from our previous arguments
$\operatorname{ext}^{2}(F,F)=0$ hence $\operatorname{ext}^{1}(F,F)=6$. ∎

We consider now
$\operatorname{Hilb}_{t^{2}+tn+2}(\mathbb{G}(1,{\mathbb{P}}(V^{*})))$, the
Hilbert scheme of planes in $\mathbb{G}(1,{\mathbb{P}}(V^{*}))$. Recall that
this scheme has two components: a component $\Omega$ parameterizing families
of planes contained in a same hyperplane and a second component $\Lambda$
parameterizing families of planes $\Lambda_{l}$ containing a fixed line $l$.
This latter is isomorphic to $\mathbb{G}(1,{\mathbb{P}}(V))$ via the morphism:

$\mathbb{G}(1,{\mathbb{P}}(V))\longrightarrow\Lambda,l\mapsto\Lambda_{l}\simeq\mathbb{G}(1,{\mathbb{P}}(H^{0}({\mathcal{I}}_{l}(1))).$

We consider now the map:

${\mathcal{M}}_{X}(2;-1,2,2)\xrightarrow{\alpha}\Lambda\simeq\mathbb{G}(1,{\mathbb{P}}(V)),\
[F]\mapsto{\mathbb{P}}(H^{0}(F(1)))\leftrightarrow l_{F}.$

######  Proposition 70.

$\alpha$ is an isomorphism of scheme; it identifies ${\mathcal{R}}(2;-1,2,2)$
(resp. ${\mathcal{M}}_{X}(2;-1,2,2)\setminus{\mathcal{R}}(2;-1,2,2)$) with
$\mathbb{G}(1,{\mathbb{P}}(V))\setminus F(X)$ (resp. $F(X)$).

###### Proof.

We apply verbatim the arguments used in the proof of Proposition 54. Doing so
we show that the sheaf ${\mathcal{G}}:={\mathcal{E}}{\it
xt}^{1}_{p_{1}}({\mathcal{I}}_{\mathbf{C}},{\mathcal{O}}_{X\times\operatorname{{Gr}}}(-1))$
on $\operatorname{Hilb}_{2t+1}\simeq\mathbb{G}(1,{\mathbb{P}}(V)^{*})$, for
$\mathbf{C}\subset\mathbb{G}(1,{\mathbb{P}}(V^{*}))\times X$ the universal
curve, is locally free of rank 3 and that the projective bundle
${\mathbb{P}}({\mathcal{G}})$ carries a family $\hat{\mathbf{F}}\in
Coh({\mathbb{P}}({\mathcal{G}})\times X)$ such that
$[\hat{\mathbf{F}}_{(C,e)}]$ is the sheaf constructed from
$e\in\operatorname{Ext}^{1}({\mathcal{I}}_{C},{\mathcal{O}}_{X}(-1))$.
${p_{1}}_{*}(\hat{\mathbf{F}}(1))$ defines a family of linear systems of
conics over ${\mathbb{P}}({\mathcal{G}})$ inducing a morphism
${\mathbb{P}}({\mathcal{G}})\xrightarrow{\gamma}\Lambda$ such that
${p_{1}}_{*}(\hat{\mathbf{F}}(1))\simeq\gamma^{*}({\mathcal{T}})$ for
${\mathcal{T}}$ the tautological rank 3 bundle over $\Lambda$ (that is to say,
${\mathcal{T}}$ is the bundle whose fibre over $\Lambda_{l}\in\Lambda$ is the
vector space
$\bigwedge^{2}(H^{0}({\mathcal{I}}_{l}(1)))\simeq{\mathbb{C}}^{3}$ of planes
belonging to $\Lambda_{l}$). $\gamma$ is the morphism mapping $(C,e)$ to
$\Lambda_{l_{F}}\simeq{\mathbb{P}}(H^{0}(F(1)))$ for $F$ the sheaf arising
from $e\in Ext^{1}({\mathcal{I}}_{C},{\mathcal{O}}_{X}(-1)).$

The family $\hat{\mathbf{F}}$ induces a morphism
${\mathbb{P}}({\mathcal{G}})\xrightarrow{\beta}{\mathcal{M}}_{X}(2;-1,2,2)$,
and applying the argument used in the proof of Corollary 56 we show that
$\beta$ is, in the étale topology, a ${\mathbb{P}}^{2}$-bundle. In this way we
also deduce that ${\mathcal{M}}_{X}(2;-1,2,2)$ is irreducible of dimension 6
hence, by Proposition 69, we get $\operatorname{ext}^{1}(F,F)=6,\
\forall\>[F]\in{\mathcal{M}}_{X}(2;-1,2,2).$ Reasoning then as in Proposition
58, we show that, due to the properties of $\beta$, $\alpha$ is well defined
as a morphism of scheme. Since $\alpha$ maps bijectively
${\mathcal{M}}_{X}(2;-1,2,2)$ into
$\Lambda\simeq\mathbb{G}(1,{\mathbb{P}}(V))$ and since both schemes are
smooth, we conclude that $\alpha$ is an isomorphism. The fact that
$\alpha({\mathcal{R}}(2;-1,2,2))=\mathbb{G}(1,{\mathbb{P}}(V))\setminus F(X)$
is due to lemma 68. This ends the proof of the proposition. ∎

This completes the proof of Theorem 64.

###  6.2. Description of ${\mathcal{C}}$

We can finally come back to the description of ${\mathcal{C}}$.

######  Proposition 71.

For $[E]\in{\mathcal{C}}$, $h^{0}(E(1))=2$ and for all $s\in H^{0}(E(1)),\
s\neq 0$, $\operatorname{coker}(s)\simeq{\mathcal{I}}_{\Gamma}(1)$, for
$\Gamma$ a curve union of a conic and a point. More precisely, all the curves
$\Gamma$ in ${\mathbb{P}}(H^{0}(E(1)))$ are of the form:

$0\rightarrow{\mathcal{O}}_{p}\rightarrow{\mathcal{O}}_{\Gamma}\rightarrow{\mathcal{O}}_{C}\rightarrow
0$

with $p$ fixed and with $C$ varying in a pencil of conics contained in a fixed
hyperplane.

###### Proof.

From Proposition 61, $E$ always fits in a short exact sequence:

(49) $0\rightarrow E\rightarrow F\rightarrow{\mathcal{O}}_{p}\rightarrow 0$

with $[F]\in{\mathcal{M}}_{X}(2;-1,2,2)$. Twisting (49) and taking global
section, we deduce that $h^{0}(E(1))\neq 0$; moreover the fact that $\forall
s\in H^{0}(F(1)),\operatorname{coker}(s)$ is torsion free, ensures that the
same holds for all non-zero $s\in H^{0}(E(1))$. As usual we denote by $\iota$
the inclusion $\iota:H^{0}(E(1))\hookrightarrow H^{0}(F(1))$. For any non-zero
$s\in H^{0}(E(1))$ we therefore have

${0}$${{\mathcal{O}}_{X}}$${{\mathcal{O}}_{X}}$${0}$${0}$${0}$${E(1)}$${F(1)}$${{\mathcal{O}}_{p}(1)}$${0}$${0}$${{\mathcal{I}}_{\Gamma}(1)}$${{\mathcal{I}}_{C}(1)}$${{\mathcal{O}}_{p}(1)}$${0}$$\scriptstyle{s}$$\scriptstyle{\iota(s)}$$\scriptstyle{id}$

from which we deduce that
$\operatorname{coker}(s)\simeq{\mathcal{I}}_{\Gamma}(1)$, for $\Gamma$ a curve
with Hilbert polynomial $2t+2$ and supported on $C\cup p$. Since we can have
no plane section of $X$ containing $\Gamma$ we have
$h^{0}({\mathcal{I}}_{\Gamma}(1))=1$ hence $h^{0}(E(1))=2$. Now,
${\mathbb{P}}(\iota(H^{0}(E(1)))$ is a pencil in
${\mathbb{P}}(H^{0}(F(1)))\simeq\Lambda_{l_{F}}$, therefore there exists a
unique hyperplane section containing all the conics in
${\mathbb{P}}(\iota(H^{0}(E(1)))$. ∎

######  Remark 72.

Suppose that $p\not\in\operatorname{Sing}(F)$. From the proof of the previous
proposition we learn that each
$f\in{\mathbb{P}}(\operatorname{Hom}(F,{\mathcal{O}}_{p}))$ locates a unique
hyperplane $H\in{\mathbb{P}}(H^{0}({\mathcal{I}}_{\langle
l_{F},p\rangle}(1)))$ containing the curves ${\mathbb{P}}(H^{0}(E(1)))$,
$E:=\ker(F)$. For $C\in{\mathbb{P}}(\iota(H^{0}(E(1))))$ general, this is the
hyperplane generated by $\langle C\rangle$ and $p$. Notice also that for $f,\
f^{\prime}\in{\mathbb{P}}(\operatorname{Hom}(F,{\mathcal{O}}_{p})),\ f\neq
f^{\prime}$, denoting by $E:=\ker(f),\ E^{\prime}:=\ker(f^{\prime})$, the
pencils ${\mathbb{P}}(\iota(H^{0}(E(1))))$ and
${\mathbb{P}}(\iota(H^{0}(E^{\prime}(1))))$ meet just at $\langle
l_{F},p\rangle\cap X$. If ever
${\mathbb{P}}(\iota(H^{0}(E(1))))={\mathbb{P}}(\iota(H^{0}(E^{\prime}(1))))$,
we would indeed have that for all conic $C\in{\mathbb{P}}(\iota(H^{0}(E(1)))$
such that $p\not\in C$, $C\cup
p\in{\mathbb{P}}(H^{0}(E(1)))\cap{\mathbb{P}}(H^{0}(E^{\prime}(1)))$. Since
$\operatorname{Ext}^{1}({\mathcal{I}}_{C\cup
p},{\mathcal{O}}_{X}(-1))\simeq\operatorname{Ext}^{1}({\mathcal{I}}_{C},{\mathcal{O}}_{X}(-1))$,
$E$ and $E^{\prime}$ would then both arise from the unique extension class
image of the element
$\xi\in\operatorname{Ext}^{1}({\mathcal{I}}_{C},{\mathcal{O}}_{X}(-1))$
defining $F$, which would lead to $E\simeq E^{\prime}$, a contradiction.

We describe in this way a pencil (parameterized by
${\mathbb{P}}(\operatorname{Hom}(F,{\mathcal{O}}_{p}))$) of lines in
${\mathbb{P}}(H^{0}(F(1)))$ that identifies with the family of hyperplanes
containing $\langle l_{F},p\rangle$.

Now, since each point $[E]\in{\mathcal{C}}$ uniquely determine a pair
$(F,p)\in{\mathcal{M}}_{X}(2;-1,2,2)$, we have a well defined map (for the
moment just defined at level of sets)

$\delta:{\mathcal{C}}\longrightarrow{\mathcal{M}}_{X}(2;-1,2,2),\
[E]\mapsto(F,p)$

where $(F,p)$ are such that $E$ is obtained by elementary transformation of
$F$ along $p$. Consider now the open subset
$({\mathcal{M}}_{X}(2;-1,2,2)\times X)_{0}$ parameterizing pairs $(F,p)$ such
that $p\not\in\operatorname{Sing}(F)$ and denote by
${{\mathcal{C}}}_{0}\subset{\mathcal{C}}$ its preimage under $\delta$.

######  Proposition 73.

For each point $[E]\in{\mathcal{C}}_{0},\ \operatorname{ext}^{1}(E,E)=10$.

###### Proof.

We know that $E$ always fits in a short exact sequence:

(50) $0\rightarrow E\rightarrow F\rightarrow{\mathcal{O}}_{p}\rightarrow 0$

with $[F]\in{\mathcal{M}}_{X}(2;-1,2,2)$. We apply
$\operatorname{Hom}(\>\cdot\>,E)$ to it. We can see immediately that the
stability of $E$ and of $F$ imposes $\operatorname{Hom}(F,E)=0$ therefore we
get:

(51)
$\displaystyle\begin{split}0\rightarrow\operatorname{Hom}(E,E)\rightarrow\operatorname{Ext}^{1}({\mathcal{O}}_{p},E)\rightarrow\operatorname{Ext}^{1}(F,E)\rightarrow\\\
\rightarrow\operatorname{Ext}^{1}(E,E)\rightarrow\operatorname{Ext}^{2}({\mathcal{O}}_{p},E)\rightarrow\operatorname{Ext}^{2}(F,E)\end{split}$

The term $\operatorname{Ext}^{2}(F,E)$ fits into:

$\operatorname{Ext}^{1}(F,{\mathcal{O}}_{p})\rightarrow\operatorname{Ext}^{2}(F,E)\rightarrow\operatorname{Ext}^{2}(F,F)$

The r.s.t vanishes due to Proposition 69; since moreover we are assuming
$[E]\in{\mathcal{C}}_{0}$, $F$ is locally free at $p$ so that
$\operatorname{Ext}^{1}(F,{\mathcal{O}}_{p})\simeq H^{1}({\mathcal{H}}{\it
om}(F,{\mathcal{O}}_{p}))=0$.

These computations lead to $\operatorname{Ext}^{2}(F,E)=0$. Let us now compute
the dimensions of the spaces
$\operatorname{Ext}^{3-i}({\mathcal{O}}_{p},E)\simeq\operatorname{Ext}^{i}(E,{\mathcal{O}}_{p}(-3))^{*},\
i=1,2$. Again, since $p\not\in\operatorname{Sing}(F),\
\operatorname{Ext}^{i}(F,{\mathcal{O}}_{p}(-3))=0,\ i=1,2$, therefore

$\operatorname{Ext}^{i}(E,{\mathcal{O}}_{p}(-3))\simeq\operatorname{Ext}^{i+1}({\mathcal{O}}_{p},{\mathcal{O}}_{p}(-3))\simeq\operatorname{Ext}^{2-i}({\mathcal{O}}_{p},{\mathcal{O}}_{p})^{*},\
i=1,2.$

For $i=2$ we thus get
$\operatorname{ext}^{1}({\mathcal{O}}_{p},E)=1=\hom({\mathcal{O}}_{p},{\mathcal{O}}_{p})$
whilst for $i=1$ we obtain
$\operatorname{ext}^{2}({\mathcal{O}}_{p},E)=\operatorname{ext}^{1}({\mathcal{O}}_{p}{\mathcal{O}}_{p})=h^{0}({\mathcal{N}}_{p/X})=3$.
From (51) we therefore compute that $\operatorname{ext}^{1}(E,E)=10$. ∎

######  Proposition 74.

${{\mathcal{C}}}_{0}$ is a smooth 10-dimensional irreducible scheme.

###### Proof.

We will construct a ${\mathbb{P}^{1}}$ bundle ${\mathbb{P}}({\mathcal{A}})$
over $({\mathcal{M}}_{X}(2;-1,2,2)\times X)_{0}$ and show that this is endowed
with a morphism ${\mathbb{P}}({\mathcal{A}})\to{\mathcal{M}}$ mapping
${\mathbb{P}}({\mathcal{A}})$ bijectively into ${\mathcal{C}}_{0}$. We
consider the Grassmanniann of lines $\mathbb{G}(1,{\mathbb{P}}(V))$ in
${\mathbb{P}}(V)\simeq{\mathbb{P}^{4}}$. For the ease of notations towards the
rest of the proof this latter will always be denoted simply by
$\operatorname{{Gr}}$. We define $(\operatorname{{Gr}}\times X)_{0}$ as the
open set:

$({\operatorname{{Gr}}\times X})_{0}:=\\{([l],p)\in\operatorname{{Gr}}\times
X\mid p\not\in l\\}.$

This scheme carries a family of planes
$\tilde{\Pi}\subset{(\operatorname{{Gr}}\times X)}_{0}\times X$ such that
$\tilde{\Pi}_{(l,p)}\simeq\langle l,p\rangle$ and the sheaf
${\mathcal{A}}:={p_{1}}_{*}({\mathcal{I}}_{\tilde{\Pi}}(1))$ is a rank 2
vector bundle on $({\operatorname{{Gr}}\times X})_{0}$.

Now, the isomorphism $\beta:{\mathcal{M}}_{X}(2;-1,2,2)\to\operatorname{{Gr}}$
(see Proposition 64) induces an isomorphism ${(\operatorname{{Gr}}\times
X)}_{0}\xrightarrow{\simeq}{({\mathcal{M}}_{X}(2;-1,2,2)\times X)}_{0}$, hence
${\mathbb{P}}({\mathcal{A}})$ is a ${\mathbb{P}^{1}}$-bundle over
${({\mathcal{M}}_{X}(2;-1,2,2)\times X)}_{0}$ as well (the fibre over a point
$(F,p)$ identifies with the pencil ${\mathbb{P}}(H^{0}({\mathcal{I}}_{\langle
l_{F},p\rangle}(1)))$ of hyperplanes containing $\langle l_{F},p\rangle$). In
order to prove that ${\mathbb{P}}({\mathcal{A}})$ admits a morphism to
${\mathcal{M}}$, we consider an étale cover $\mathbf{W}$ of
${\mathcal{M}}_{X}(2;-1,2,2)$ supporting a universal sheaf
${\mathbf{F}}_{{\mathbf{W}}}$. This induces an étale cover
$\tilde{{\mathbf{W}}}\to{({\mathcal{M}}_{X}(2;-1,2,2)\times X)}_{0}$, we
denote by $\tilde{{\mathcal{A}}}_{{\mathbf{W}}}$ the pullback of
${\mathcal{A}}$ to $\tilde{{\mathbf{W}}}$ and by
$\tilde{{\mathbf{F}}}_{{\mathbf{W}}}$ the pullback of
${\mathbf{F}}_{{\mathbf{W}}}$ to $\tilde{{\mathbf{W}}}\times X$.
${\mathbb{P}}(\tilde{{\mathcal{A}}}_{{\mathbf{W}}})$ identifies with the
${\mathbb{P}^{1}}$ subbundle of
$G_{2}({p_{1}}_{*}(\tilde{{\mathbf{F}}}_{{\mathbf{W}}}(1)))$ whose fibre over
a point $(w,p)\in\tilde{{\mathbf{W}}}$ is the 1-dimensional linear space:

$\pi_{\tilde{{\mathcal{A}}}_{{{\mathbf{W}}}}}^{-1}(w,p)=\\{{\mathbb{P}}^{1}\subset{\mathbb{P}}(H^{0}({\mathbf{F}}_{w}(1)))\simeq\Lambda_{l_{{\mathbf{F}}_{w}}}\mid[\langle
l_{{\mathbf{F}}_{w}},p\rangle]\in{\mathbb{P}}^{1}\\}$

Define now $\tilde{\Delta}_{{\mathbf{W}}}$ as the pullback to
$\tilde{{\mathbf{W}}}\times X$ of the diagonal $\Delta\subset X\times X$ and
consider the sheaf $\tau_{{\mathbf{W}}}:={p_{1}}_{*}({\mathcal{H}}{\it
om}(\tilde{{\mathbf{F}}}_{{\mathbf{W}}},{\mathcal{O}}_{\tilde{\Delta}_{{\mathbf{W}}}}))$.
This is a rank 2 vector bundle over $\tilde{{\mathbf{W}}}\times X$ whose fibre
over $(w,p)$ identifies with
$\operatorname{Hom}({\mathbf{F}}_{w},{\mathcal{O}}_{p}).$ We claim that we
have an isomorphism:
${\mathbb{P}}(\tilde{{\mathcal{A}}}_{{{\mathbf{W}}}})\simeq{\mathbb{P}}(\tau_{{\mathbf{W}}}).$
Denote by $\hat{{\mathbf{F}}}_{{\mathbf{W}}},\hat{\Delta}_{{\mathbf{W}}}$ the
pullback to ${\mathbb{P}}(\tau_{{\mathbf{W}}})$ of
$\tilde{{\mathbf{F}}}_{{\mathbf{W}}},\ \tilde{\Delta}_{{\mathbf{W}}}$,
respectively. The image of the identity
$id_{\tau_{{\mathbf{W}}}}\in\operatorname{End}(\tau_{{\mathbf{W}}})$ through
the isomorphism:

$\displaystyle\operatorname{Hom}(\tau_{{\mathbf{W}}},\tau_{{\mathbf{W}}})\simeq
H^{0}(\tau_{{\mathbf{W}}}\otimes\tau_{{\mathbf{W}}}^{\vee})\simeq
H^{0}(\tau_{{\mathbf{W}}}\otimes{\pi_{\tau_{{\mathbf{W}}}}}_{*}{\mathcal{O}}(1))\simeq$
$\displaystyle\simeq$ $\displaystyle\leavevmode\nobreak\
H^{0}({p_{1}}_{*}{\mathcal{H}}{\it
om}(\hat{{\mathbf{F}}}_{{\mathbf{W}}},{\mathcal{O}}_{\hat{\Delta}_{{\mathbf{W}}}})\otimes{\mathcal{O}}_{{\mathbb{P}}(\tau_{{\mathbf{W}}})}(1))\simeq\operatorname{Hom}(\hat{{\mathbf{F}}}_{{\mathbf{W}}},{\mathcal{O}}_{\hat{\Delta}_{{\mathbf{W}}}}\otimes{p_{1}}^{*}{\mathcal{O}}_{{\mathbb{P}}(\tau_{{\mathbf{W}}})}(1))$

defines a morphism
$\hat{{\mathbf{F}}}_{{\mathbf{W}}}\to{\mathcal{O}}_{\hat{\Delta}_{{\mathbf{W}}}}\otimes{p_{1}}^{*}{\mathcal{O}}(1)$
inducing a short exact sequence on ${\mathbb{P}}(\tau_{{\mathbf{W}}})\times
X$:

$0\rightarrow\hat{{\mathbf{E}}}_{{\mathbf{W}}}\rightarrow\hat{{\mathbf{F}}}_{{\mathbf{W}}}\rightarrow{\mathcal{O}}_{\hat{\Delta}_{{\mathbf{W}}}}\otimes{p_{1}}^{*}{\mathcal{O}}_{{\mathbb{P}}(\tau_{{\mathbf{W}}})}(1)\rightarrow
0.$

Twisting and applying ${p_{1}}_{*}$, we obtain a rank 2 vector bundle
${p_{1}}_{*}(\hat{{\mathbf{E}}}_{{\mathbf{W}}}(1))$; this latter induces an
embedding ${\mathbb{P}}(\tau_{{\mathbf{W}}})\to
G_{2}({p_{1}}_{*}\tilde{{\mathbf{F}}}_{{\mathbf{W}}}(1))$ that maps
${\mathbb{P}}(\tau_{{\mathbf{W}}})$ bijectively into
${\mathbb{P}}(\tilde{{\mathcal{A}}}_{{{\mathbf{W}}}})$. This induces an
isomorphism
${\mathbb{P}}(\tau_{{\mathbf{W}}})\simeq{\mathbb{P}}(\tilde{{\mathcal{A}}}_{{\mathbf{W}}})$
mapping a point $f\in\operatorname{Hom}({\mathbf{F}}_{w},{\mathcal{O}}_{p})$
in $\pi_{\tau_{{\mathbf{W}}}}^{-1}$ to ${\mathbb{P}}(H^{0}(E_{w}(1))),\
E_{w}=\ker f$. The sheaves $\hat{{\mathbf{E}}}_{{{\mathbf{W}}}}$ determine
morphisms
$\psi_{\tilde{{\mathbf{W}}}}:{\mathbb{P}}(\tilde{{\mathcal{A}}}_{{\mathbf{W}}})\xrightarrow{\simeq}{\mathbb{P}}(\tau_{{\mathbf{W}}})\to{\mathcal{M}}$
that descend to a morphism ${\mathbb{P}}({\mathcal{A}})\to{\mathcal{M}}$ which
maps ${\mathbb{P}}({\mathcal{A}})$ bijectively to ${\mathcal{C}}_{0}$. This
means that ${\mathcal{C}}_{0}$ is irreducible and has dimension $10$. Since by
Proposition 73, the dimension of the Zarisky tangent space at each point
$[E]\in{\mathcal{C}}_{0}$ is 10, we conclude that ${\mathcal{C}}_{0}$ inherits
from ${\mathbb{P}}({\mathcal{A}})$ the structure of a smooth 10-dimensional
scheme. ∎

From these last results we deduce that $\overline{{\mathcal{C}}_{0}}$ is an
irreducible component of ${\mathcal{C}}$ and that this latter is smooth along
${\mathcal{C}}_{0}$. We finally want to show that actually, we have an
equality ${\mathcal{C}}=\overline{{\mathcal{C}}_{0}}$, that is to say, that
${\mathcal{C}}$ is irreducible.

######  Proposition 75.

${\mathcal{C}}$ is an irreducible 10-dimensional scheme that coincides with
$\overline{{\mathcal{C}}_{0}}$.

###### Proof.

We show that ${\mathcal{C}}_{0}$ is dense in ${\mathcal{C}}$. From Proposition
71, we learn that pairs $(E,s),\
[E]\in{\mathcal{C}}\setminus{\mathcal{C}}_{0},\ s\in{\mathbb{P}}(H^{0}(E(1))$
corresponds to pairs $(\Gamma,\xi)$ with $\Gamma$ a non l.cm. curve consisting
of a conic $C$ with an embedded point $p\in C$ and
$\xi\in\operatorname{Ext}^{1}({\mathcal{I}}_{\Gamma},{\mathcal{O}}_{X}(-1))\simeq\operatorname{Ext}^{1}({\mathcal{I}}_{C},{\mathcal{O}}_{X}(-1))\simeq
H^{0}({\mathcal{O}}_{C}(2pt))$ vanishing at $p$. For a point
$[E]\in{\mathcal{C}}\setminus{\mathcal{C}}_{0}$ let then $\Gamma$ be a curve
defined by a global section $s\in H^{0}(E(1))$ and $\xi$ the corresponding
element in
$\operatorname{Ext}^{1}({\mathcal{I}}_{\Gamma},{\mathcal{O}}_{X}(-1))$. The
projective space
${\mathbb{P}}(\operatorname{Ext}^{1}({\mathcal{I}}_{\Gamma},{\mathcal{O}}_{X}(-1)))\simeq{\mathbb{P}}^{2}$
determines a family $\mathbf{E}$, flat over ${\mathbb{P}}^{2}$ such that
$\mathbf{E}_{[\xi]}\simeq E$. Now, the points
$x\in{\mathbb{P}}(\operatorname{Ext}^{1}({\mathcal{I}}_{\Gamma},{\mathcal{O}}_{X}(-1)))$
such that $\mathbf{E}_{x}\in{\mathcal{C}}\setminus{\mathcal{C}}_{0}$ are
parameterized by a divisor isomorphic to ${\mathbb{P}}^{1}$. We can therefore
always deform $[\xi]$ in a direction normal to this divisor and produce in
this way a family $\mathbf{E}^{\prime}$ whose central fibre is isomorphic to
$[E]$ and whose general fibre lies in ${\mathcal{C}}_{0}$. ∎

###  6.3. Intersection of ${\mathcal{C}}$ and $\overline{{\mathcal{L}}(2)}$

We end our description of the moduli space ${\mathcal{M}}$ addressing the
issue of its connectedness. Since we have already proved that ${\mathcal{M}}$
is union of two irreducible components, this reduces to proving that their
intersection is non-empty. We present here how to construct a 5-dimensional
irreducible family ${\mathcal{P}}$ contained in
${\mathcal{C}}\cap\overline{{\mathcal{L}}(2)}$. To begin with we consider
$\operatorname{Tan}(X)\subset\mathbb{G}(1,{\mathbb{P}}(V))$, the variety of
tangent lines to $X$. This is a smooth 4-dimensional locally closed subvariety
of $\mathbb{G}(1,{\mathbb{P}}(V))$ and it identifies with the subvariety
$\Sigma_{T}\subset\mathbb{G}(1,{\mathbb{P}}(V))\times X$ defined as

$\Sigma_{T}:=\\{([l],p)\in\operatorname{Tan}(X)\times X\mid
l\subset{\mathbb{T}}_{p}X\\}.$

We denote by ${\mathcal{R}}_{T}\subset{\mathcal{R}}(2;-1,2,2)\times X$ the
image of $\Sigma_{T}$ under the isomorphism
$\mathbb{G}(1,{\mathbb{P}}(V))\times
X\xrightarrow{\simeq}{\mathcal{M}}_{X}(2;-1,2,2)\times X$. By definition, for
each $([F],p)\in{\mathcal{R}}_{T}$, $l_{F}$ is tangent to $X$ at
$p=\operatorname{Sing}(F)_{\rm red}$. Now, reasoning exactly as in the proof
of Proposition 74, starting from an étale cover ${\mathbf{W}}$ of
${\mathcal{M}}_{X}(2;-1,2,2)$ supporting a Poincare sheaf
${\mathbf{F}}_{{\mathbf{W}}}$, we construct an étale cover ${\mathbf{W}}_{T}$
of ${\mathcal{R}}_{T}$ and we consider the sheaf
${p_{1}}_{*}({\mathcal{H}}{\it
om}({\mathbf{F}}_{{\mathbf{W}}_{T}},{\mathcal{O}}_{\Delta_{{\mathbf{W}}_{T}}}))$
on ${\mathbf{W}}_{T}$, where ${\mathbf{F}}_{{\mathbf{W}}_{T}}\in
Coh({\mathbf{W}}_{T}\times X)$ and
${\Delta_{{\mathbf{W}}_{T}}}\subset{\mathbf{W}}_{T}\times X$ are,
respectively, the pullback of the universal sheaf and of the diagonal.

The sheaf ${p_{1}}_{*}({\mathcal{H}}{\it
om}({\mathbf{F}}_{{\mathbf{W}}_{T}},{\mathcal{O}}_{\Delta_{{\mathbf{W}}_{T}}}))$
is a rank 3 vector bundle and replying the reasoning presented in the proof of
Proposition 74, on ${\mathbb{P}}({p_{1}}_{*}({\mathcal{H}}{\it
om}({\mathbf{F}}_{{\mathbf{W}}_{T}},{\mathcal{O}}_{\Delta_{{\mathbf{W}}_{T}}}))\times
X$, there exists a family
$\hat{{\mathbf{E}}}_{{\mathbf{W}}_{T}}\hookrightarrow\hat{{\mathbf{F}}}_{{\mathbf{W}}_{T}}$,
$\hat{{\mathbf{F}}}_{{\mathbf{W}}_{T}}$ being the pullback of
${\mathbf{F}}_{{\mathbf{W}}_{T}}$, whose direct image under the projection
$p_{1}$ fits in:

$0\rightarrow{p_{1}}_{*}(\hat{{\mathbf{E}}}_{{\mathbf{W}}_{T}}(1))\rightarrow{p_{1}}_{*}(\hat{{\mathbf{F}}}_{{\mathbf{W}}_{T}}(1))\rightarrow{p_{1}}_{*}({\mathcal{O}}_{\hat{\Delta}_{{\mathbf{W}}_{T}}})\otimes{\mathcal{O}}(1)\rightarrow
0.$

Denote now by
${\mathbf{l}}_{{\mathbf{W}}_{T}}\subset\Sigma_{T}\times_{{\mathcal{R}}_{T}}{\mathbf{W}}_{T}$
the pullback of the universal line ${\mathbf{l}}\subset\Sigma_{T}\times X$. We
observe that the projective bundle
${\mathbb{P}}({p_{1}}_{*}{\mathcal{I}}_{{\mathbf{W}}_{T}}(1))$ identifies with
the Grassmann bundle $G_{2}({p_{1}}_{*}({{\mathbf{F}}}_{{\mathbf{W}}_{T}}(1))$
(since for $F\in{\mathcal{R}}(2;-1,2,2)$ such that $p\in l_{F}$
$\mathbb{G}(1,{\mathbb{P}}(H^{0}(F(1))))\simeq{\mathbb{P}}(H^{0}({\mathcal{I}}_{l_{F}}(1)))\simeq{\mathbb{P}}^{2}$);
as ${p_{1}}_{*}(\hat{{\mathbf{E}}}_{{\mathbf{W}}_{T}}(1))$ is a rank-2
subbundle of ${p_{1}}_{*}(\hat{{\mathbf{F}}}_{{\mathbf{W}}_{T}}(1))$ it
induces therefore a morphism of $\Sigma_{T}$ schemes:

$\epsilon:{\mathbb{P}}({p_{1}}_{*}({\mathcal{H}}{\it
om}({\mathbf{F}}_{{\mathbf{W}}_{T}},{\mathcal{O}}_{\Delta_{{\mathbf{W}}_{T}}}))\rightarrow{\mathbb{P}}({p_{1}}_{*}{\mathcal{I}}_{{\mathbf{l}}_{{\mathbf{W}}_{T}}}(1))$

($\epsilon$ is the morphism mapping $f\in Hom(F,{\mathcal{O}}_{p})$ to the
unique hyperplane containing all curves in ${\mathbb{P}}(H^{0}(E(1)))$ for
$E:=\ker f$). This morphism is bijective hence, by the smoothness of both
${\mathbb{P}}({p_{1}}_{*}({\mathcal{H}}{\it
om}({\mathbf{F}}_{{\mathbf{W}}_{T}},{\mathcal{O}}_{\Delta_{{\mathbf{W}}_{T}}}))$
and
${\mathbb{P}}({p_{1}}_{*}{\mathcal{I}}_{{\mathbf{l}}_{{\mathbf{W}}_{T}}}(1))$
it is an isomorphism.

Also this time, the families of sheaves
$\hat{{\mathbf{E}}}_{{\mathbf{W}}_{T}}$ induces morphisms
${\mathbb{P}}({p_{1}}_{*}{\mathcal{I}}_{{\mathbf{l}}_{{\mathbf{W}}_{T}}}(1))\to{\mathcal{C}}$
that descend to a well defined morphism
$\psi:{\mathbb{P}}({p_{1}}_{*}{\mathcal{I}}_{{\mathbf{l}}}(1))\to{\mathcal{C}}$.
Consider now the variety

$\tilde{\Sigma}_{T}:=\\{([l],p,H)\in\Sigma_{T}\times X^{*}\mid
H={\mathbb{T}}_{p}X\\}$

$\tilde{\Sigma}_{T}$ is isomorphic to $\Sigma_{T}$ and it identifies with a
subset of ${\mathbb{P}}({p_{1}}_{*}({\mathcal{I}}_{{\mathbf{l}}}(1))$; we
finally define ${\mathcal{P}}$ as the scheme theoretic image of
$\tilde{\Sigma}_{T}$ under the morphism $\psi$.

######  Proposition 76.

${\mathcal{P}}$ is a 5 dimensional irreducible scheme contained in
$\overline{{\mathcal{L}}(2)}\cap{\mathcal{C}}$.

###### Proof.

The dimension and the irreducibility of ${\mathcal{P}}$ are immediate
consequences of the fact that $\tilde{\Sigma}_{T}$ is irreducible and of
dimension 5. Let us now prove that ${\mathcal{P}}$ is indeed contained in both
components of ${\mathcal{M}}$. For a general point
$[E]\in\psi(\tilde{\Sigma}_{T})$, $E^{\vee\vee}:=F\in{\mathcal{R}}(2;-1,2,2)$
and $F/E\simeq{\mathcal{O}}_{p},\ \operatorname{Sing}(F)_{\rm red}=p$. Let us
give a geometric interpretation of the family of curves defined by
$H^{0}(E(1))$. Consider the short exact sequence:

$0\rightarrow H^{0}(E(1))\xrightarrow{\iota}H^{0}(F(1))\rightarrow
H^{0}({\mathcal{O}}_{p})\rightarrow 0.$

By definition the pencil ${\mathbb{P}}(\iota(H^{0}(E(1))))$ identifies with
the pencil of planes containing $l_{F}$ and contained in ${\mathbb{T}}_{p}X$;
accordingly it identifies with the locus of singular conics in
${\mathbb{P}}(H^{0}(F(1))$ (to see this just notice that the singular plane
sections in $\Lambda_{l_{F}}\simeq{\mathbb{P}}(H^{0}(F(1)))$ identifies with
the locus of tangents to ${\mathbb{P}}(H^{0}({\mathcal{I}}_{l_{F}}(1)))\cap
X^{*}$ i.e. with the pencils $\langle h,{\mathbb{T}}_{p}X\rangle$ with $h$ an
hyperplane containing $l_{F}$.)

Consider now the singular quadric surface $Q_{p}:={\mathbb{T}}_{p}X\cap X$.
$Q_{p}$ is the cone with vertex $p$ over a smooth conic $C$ and the pencil
${\mathbb{P}}(H^{0}(E(1)))$ uniquely determines a 1-dimensional linear system
${\mathbb{P}}^{1}_{E}\subset{\mathbb{P}}(H^{0}({\mathcal{O}}_{C}(1))$ hence, a
point in the projective bundle

${\mathbb{P}}(\operatorname{Sym}^{2}({\mathcal{T}}^{\vee}))\longrightarrow\mathbb{G}(1,F(X))$

where, as usual, ${\mathcal{T}}$ is the tautological bundle. This projective
bundle is a smooth 6 dimensional variety containing the variety
${\mathcal{B}}\simeq{\mathcal{L}}(2)$ (see section 5.2) as an open subset. We
can therefore always construct a regular affine curve with a marked point
$(\operatorname{Spec}(A),0)$ and a 1-parameter family of pencils
${\mathbb{P}}^{1}_{t}$, flat over $\operatorname{Spec}(A)$, such that
${\mathbb{P}}^{1}_{t}\in{\mathcal{B}}$ for $t$ general and whose central fibre
${\mathbb{P}}^{1}_{0}$ is isomorphic to ${\mathbb{P}}^{1}_{E}$. Define
$E_{t}:=\zeta^{-1}({\mathbb{P}}^{1}_{t}),\ t\neq 0$. This is a family of
instantons, flat over $\operatorname{Spec}(A)\setminus\\{0\\}$ such that the
pencil of curves ${\mathbb{P}}(H^{0}(E_{t}(1)))$ coincides with
${\mathbb{P}}^{1}_{t}.$ $E_{t}$ admits a flat limit
$E_{0}\in\overline{{\mathcal{L}}(2)}$ and since for such a sheaf, the pencil
of curves ${\mathbb{P}}(H^{0}(E_{0}(1)))$ must be the flat limit of
${\mathbb{P}}^{1}_{t}$, we conclude that $E_{0}\simeq E$. This means that
$\psi(\tilde{\Sigma}_{T})\subset\overline{{\mathcal{L}}(2)}\cap{\mathcal{C}}$
therefore the same holds for ${\mathcal{P}}$. ∎

This shows the connectedness of ${\mathcal{M}}$ ending the proof of Theorem
62.

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