0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),F(-n))^{\vee}\otimes
F(-n)\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
165.1304pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
189.1304pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
87.0027pt\raise-18.76389pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\xi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
96.0652pt\raise-32.02777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
189.1304pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{U}|_{\\{S\\}\times
X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
251.98824pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
207.86487pt\raise-32.69446pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
251.98824pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{O}_{S}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
76.98766pt\raise-42.52777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{F(-n)\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
115.14276pt\raise-42.52777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
198.10516pt\raise-42.52777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
198.10516pt\raise-42.52777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E_{[\xi]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
250.59933pt\raise-42.52777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
250.59933pt\raise-42.52777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{O}_{S}(m).}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

The homomorphism $\xi$ in (88) is non-zero, since the extension (82) is non-
trivial. Thus, $\xi$ is an epimorphism. The class $[\xi]$ of the epimorphism
$\xi$ modulo automorphisms of the space $\mathbf{k}$ is a point of
projectivization
$\mathbb{P}(\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),$
$\mathcal{O}_{X}(-n))^{\vee})$. We have the subset $\mathcal{Y}_{k,m,n}$ in
$\mathbb{P}(\mathcal{A}^{\vee})$ defined as
$\mathcal{Y}_{k,m,n}:=\\{[\xi]\in\mathbb{P}(\mathcal{A}^{\vee})\ |\ \\\
\textit{the extension}\ E_{[\xi]}\ \textit{in}\ \eqref{diag 98}\ \textit{is
stable}\\}$, with the modular morphism $\Theta:\ \mathcal{Y}_{k,m,n}\to
M_{X}(v),\ \ \ [\xi]\mapsto[E_{[\xi]}],$ defined verbatim as in (52). At the
same time, as in the case of extension (24), due to (83), the extension (82)
defined by the sheaf $E$ is unique, that is, the class $[\xi]$ can be
reconstructed from the point $[E]$. Thus, $\Theta$ is a bijection of the
variety $\mathcal{Y}_{k,m,n}$ onto its image $M_{k,m,n}$ in $M_{X}(v)$, and by
analogy with (54) we have the equalities

(90) $\begin{split}&\dim
M_{k,m,n}=\dim\mathcal{Y}_{k,m,n}=\dim\mathbb{P}(\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),F(-n))^{\vee})+\dim
W+\\\
&+\dim|\mathcal{O}_{X}(k)|=\mathrm{ext}^{1}(\mathcal{O}_{S}(m),F(-n))+\dim
W+N-1.\end{split}$

Applying the bifunctor $\mathbf{R}\operatorname{Hom}(-,-)$ to the triple (82)
and taking into account the equalities (83), we get a commutative diagram:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
51.81544pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\underset{{}}{\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),F(-n)}/\mathbf{k}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
51.81546pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
75.81544pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
75.81544pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\underset{{}}{\operatorname{Ext}\nolimits^{1}(E,F(-n))}/\mathbf{k}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
163.20117pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
187.20116pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
119.5083pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
119.5083pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
187.20116pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\underset{{}}{\operatorname{Ext}\nolimits^{1}(F(-n),F(-n))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
232.8535pt\raise-17.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.28406pt\hbox{$\scriptstyle{=}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 232.8535pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),E)\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
39.0226pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
90.55275pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-70.63776pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\lower-3.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
90.55275pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Ext}\nolimits^{1}(E,E)\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
196.79955pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
119.5083pt\raise-70.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\lower-3.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
196.79955pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Ext}\nolimits^{1}(F(-n),E)}$}}}}}}}{\hbox{\kern-45.75632pt\raise-82.27551pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{S}(m))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
58.371pt\raise-77.99146pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.28406pt\hbox{$\scriptstyle{=}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
82.43013pt\raise-82.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
82.43013pt\raise-82.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Ext}\nolimits^{1}(E,\mathcal{O}_{S}(m)).}$}}}}}}}{\hbox{\kern
229.8535pt\raise-82.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

From this diagram and the equalities
$\mathrm{ext}^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{S}(m))=N$ and
$\operatorname{ext}^{1}(F(-n),\\\ F(-n))=\dim W$ it follows that
$\mathrm{ext}^{1}(E,E)=\mathrm{ext}^{1}(\mathcal{O}_{S}(m),F(-n))-1+\dim W+N$,
from which, due to (90), we find
$\dim\Theta(\mathcal{Y}_{k,m,n})=\mathrm{ext}^{1}(E,E)=\dim_{[E]}M_{X}(v).$
This means that $M_{k,m,n}$ is a smooth component of the moduli scheme
$M_{X}(v)$. From here, as in (57) and (73), we obtain an isomorphism $\Theta:\
\mathcal{Y}_{k,m,n}\to M_{k,m,n}$, that is, $M_{k,m,n}$ is a smooth variety.
Moreover, from the diagram (43) we obtain that $M_{k,m,n}$ is a fine moduli
space. Exact formulas for the dimensions of the varieties $M_{k,m,n}$ are
obtained using (90) by analogy with the formulas (33). Thus, the following
theorem holds.

######  Theorem 4.9.

Let $X$ be one of varieties $X_{1}$, $X_{4}$, $X_{5}$, let $k\geq 1$,
$n=\lfloor\frac{k}{2}\rfloor+1$, $m<-n$, and $M_{X}(v)$ be the Gieseker-
Maruyama moduli scheme of semistable sheaves $E$ on $X$ with Chern character
$v=\mathrm{ch}(E)$ determined from the triple (82). Then
(i) the family

$\begin{split}&M_{k,m,n}=\\{[E]\in M_{X}(v)\ |\ E\ \textit{is specified by the
extension \eqref{extension F} in which}\ S\in|\mathcal{O}_{X}(k)|,\\\
&\textit{and the sheaf}\ F\ \textit{is determined by the above conditions
(I)-(III)}\\}\end{split}$

is a smooth irreducible component of the scheme $M_{X}(v)$ and is described as
the projectivization $\pi:\ \mathbb{P}(\mathcal{A}^{\vee})\to
W\times\mathbb{P}^{N}$ of the bundle $\mathcal{A}^{\vee}$, where the variety
$W$ is defined by conditions (I)-(III) above, and $\mathcal{A}$ is the locally
free sheaf on $W\times\mathbb{P}^{N}$ defined by the formula (84), where
$\mathbb{P}^{N}=|\mathcal{O}_{X}(k)|$; this component is a rational variety
for $X=X_{1},\ X_{2},\ X_{5}$, and non-rational for $X=X_{4}$; Moreover,
$M_{k,m,n}$ is a fine moduli space, and all sheaves from $M_{k,m,n}$ are
stable;
(ii) the dimension of the variety $M_{k,m,n}$ is given by the formulas:
(ii.1) $\dim
M_{k,m,n}=\binom{k+3}{3}+3\binom{-m-n+k+3}{3}-\binom{-m-n+k+2}{3}-3\binom{-m-n+3}{3}+\binom{-m-n+2}{3}+1$
for $X=X_{1}$;
(ii.2) $\dim
M_{k,m,n}=4[\binom{k-m-n+4}{4}-\binom{k-m-n+2}{4}-\binom{-m-n+4}{4}+\binom{-m-n+2}{4}]$
for $X=X_{4}$;
(ii.3) $\dim
M_{k,m,n}=\frac{5}{3}(k-m-n+1)^{3}+\frac{5}{2}(k-m-n+1)^{2}+\frac{5}{6}(k-m-n+1)-\frac{5}{3}(-m-n+1)^{3}-\frac{5}{2}(-m-n+1)^{2}-\frac{5}{6}(-m-n+1)+\frac{k^{3}}{6}+\frac{k^{2}}{2}+\frac{8}{15}k-1$
for $X=X_{5}$.

###  4.4. Reflexive stable sheaves of general type with $c_{1}=0$ on $X_{4}$
and $X_{5}$.

To conclude this section, we prove a theorem concerning reflexive sheaves in
the components of the Gieseker-Maruyama scheme constructed in Theorems
4.2-4.9.

For $X=X_{4}$ or $X_{5}$ we denote by $B(X)$ the base of the family of lines
on $X$. As is known, $B(X_{4})$ is a smooth abelian surface, and
$B(X_{5})\simeq\mathbb{P}^{2}$. Let us give the following definition, which
will be key for use in section 6.

######  Definition 4.10.

A reflexive sheaf $E$ of rank 2 with first Chern class $c_{1}(E)=0$ on
$X=X_{4}$ or $X=X_{5}$ is called a sheaf of general type if for any line $l\in
B(X)$ not passing through points of $\operatorname{Sing}E$ we have either
$E|_{l}\cong\mathcal{O}_{\mathbb{P}^{1}}^{\oplus 2}$, and such lines
constitute a dense open set in $B(X)$, or
$E|_{l}\cong\mathcal{O}_{\mathbb{P}^{1}}(m)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-m)$,
where $m>0$, and the set $B_{2}(X):=\\{l\in B(X)\ |\
E|_{l}\cong\mathcal{O}_{\mathbb{P}^{1}}(m)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-m),\
m\geq 2\\}$ has dimension $\leq 0$.

The following theorem gives examples of infinite series of irreducible
components of Gieseker-Maruyama moduli schemes for varieties $X_{4}$ and
$X_{5}$ such that general points of these components are reflexive sheaves of
general type.

######  Theorem 4.11.

Consider the following infinite series of components of the Gieseker–Maruyama
moduli schemes of rank two semistable sheaves on varieties $X_{4}$ and
$X_{5}$:
(I) series of components $M_{2,m,1}$ from Theorem 4.2 for the case $k=2$,
$m\leq-2$, $n=1$;
(II) series of components $M_{1,m,1}$ from Theorem 4.9 for the case $k=n=1$,
$m\leq-1$.
Then the general sheaf in each of these components is a reflexive stable sheaf
of general type.

###### Proof.

(I) For $[E]\in M_{2,m,1}$ we have the triple (24) for $k=2,\ m\leq-2,\ n=1$
and $S\in\mathbb{P}^{N}=|\mathcal{O}_{X}(2)|$:

(91) $0\to\mathcal{O}_{X}(-1)^{\oplus 2}\to E\to\mathcal{O}_{S}(m)\to 0,\ \ \
\ \ \ \ \ \ \ \ S\in|\mathcal{O}_{X}(2)|.$

Consider the variety $\Pi:=\\{(\mathbb{P}^{1},S)\in B(X)\times\mathbb{P}^{N}\
|\ \mathbb{P}^{1}\subset S\\}$ with projections
$B(X)\xleftarrow{pr_{1}}\Pi\xrightarrow{pr_{2}}\mathbb{P}^{N}$. An easy
calculation shows that the dimension of a fiber of the projection $pr_{1}$ is
equal to $N-3$. On the other hand, it is known [1] that $\dim B(X)=2$ for
$X=X_{4}$. Hence $\dim\Pi=N-1<N=\dim\mathbb{P}^{N}$. Therefore, let us take in
the triple (91) a surface $S$ such that
$S\in\mathbb{P}^{N}\smallsetminus(pr_{2}(\Pi))$. By the definition of $\Pi$ it
means that there are no projective lines on the surface $S$. Therefore, taking
an arbitrary projective line $\mathbb{P}^{1}\subset X$ disjoint from
$\operatorname{Sing}E$, we have a scheme $Z_{2}:=S\cap\mathbb{P}^{1}$ of
length $\ell(Z_{2})=2$. Restricting the triple (91) to $\mathbb{P}^{1}$ we
obtain an exact triple

(92) $0\to\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus 2}\to
E|_{\mathbb{P}^{1}}\to\mathcal{O}_{Z_{2}}\to 0.$

This restriction can be considered as a homomorphism of spaces of extensions
$r:\
\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-1)^{\oplus
2}))\to\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{Z_{2}},\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus
2}))$, for which $r(\xi)=\xi_{\mathbb{P}^{1}}$, where the elements $\xi$ and
$\xi_{\mathbb{P}^{1}}$ define extensions (91) and (92) respectively. The
homomorphism $r$ decomposes into a composition $r:\
\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n)^{\oplus
2}))\underset{\cong}{\xrightarrow{\varphi}}H^{0}(\mathcal{O}_{S}(1-m)^{\oplus
2})\xrightarrow{\otimes\mathcal{O}_{\mathbb{P}^{1}}}H^{0}(\mathcal{O}_{Z_{2}}^{\oplus
2})\underset{\cong}{\xrightarrow{\varphi_{\mathbb{P}^{1}}^{-1}}}\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{Z_{2}},\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus
2}))$, where $\varphi$ is an isomorphism (49) for $k=2,\ n=1$, and the
isomorphism $\varphi_{\mathbb{P}^{1}}$ is constructed by analogy with
$\varphi$. Let us check that $r$ is an epimorphism. For this it is enough to
verify that the homomorphism $\otimes\mathcal{O}_{\mathbb{P}^{1}}$ is
epimorphic. Consider the embedding $X=X_{q}\hookrightarrow\mathbb{P}^{q+1}$,
$q\in\\{4,5\\}$, and let $L\cong\mathbb{P}^{q-1}$ be a subspace of codimension
2 in $\mathbb{P}^{q+1}$, containing the subscheme $Z_{2}$ and intersecting $S$
along some 0-dimensional scheme $Z$. (A simple calculation of parameters shows
that for a general $L$ passing through $Z_{2}$ the condition $\dim Z=0$ is
satisfied.) Then we have the $\mathcal{O}_{S}$-Koszul resolution of the sheaf
$\mathcal{O}_{Z}^{\oplus 2}$: $0\to\mathcal{O}_{S}(-m-1)^{\oplus
2}\to\mathcal{O}_{S}(-m)^{\oplus 4}\to\mathcal{O}_{S}(-m+1)^{\oplus
2}\xrightarrow{\otimes\mathcal{O}_{Z}}\mathcal{O}_{Z}^{\oplus 2}\to 0$. Using
the triple (30) and the condition $m\leq-2$, we find
$H^{2}(\mathcal{O}_{S}(-m-1))=H^{1}(\mathcal{O}_{S}(-m))=0$, and the
resolution gives an epimorphism $h^{0}(\varepsilon):\
H^{0}(\mathcal{O}_{S}(-m+1)^{\oplus 2})\to H^{0}(\mathcal{O}_{Z}^{\oplus 2})$.
Thus, $\otimes\mathcal{O}_{\mathbb{P}^{1}}:\
H^{0}(\mathcal{O}_{S}(-m+1)^{\oplus
2})\xrightarrow{h^{0}(\otimes\mathcal{O}_{Z})}H^{0}(\mathcal{O}_{Z}^{\oplus
2})\overset{h^{0}(\otimes\mathcal{O}_{Z_{2}})}{\twoheadrightarrow}H^{0}(\mathcal{O}_{Z_{2}}^{\oplus
2})$ is an epimorphism.

Since $\mathbb{P}^{1}\cap\operatorname{Sing}E=\varnothing$, then the sheaf
$E|_{\mathbb{P}^{1}}$ is locally free, so the triple (92) shows that
$E|_{\mathbb{P}^{1}}\cong\mathcal{O}_{\mathbb{P}^{1}}(a)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-a),\
0\leq a\leq 1$. On the other hand, for general
$\xi_{\mathbb{P}^{1}}\in\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{Z_{2}},\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus
2}))$ in (92) we have
$E|_{\mathbb{P}^{1}}\cong\mathcal{O}_{\mathbb{P}^{1}}^{\oplus 2}$. From here
in view of surjectivity of $r$ it follows that for general
$\xi\in\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-1)^{\oplus
2}))$ the sheaf $E$ in (91) is a sheaf of general type.

(II) Consider the case $[E]\in M_{1,m,1}$ from Theorem 4.9. We have a triple
(24) for $k=n=1$, $m\leq-1$, and $S\in\mathbb{P}^{N}=|\mathcal{O}_{X}(1)|$:

(93) $0\to F(-1)\to E\to\mathcal{O}_{S}(m)\to 0,\ \ \ \ \ \ \ \ \ \
\in|\mathcal{O}_{X}(1)|.$

In this case, a general surface $S\in|\mathcal{O}_{X}(1)|$ for both varieties
$X=X_{4}$ and $X=X_{5}$ is a smooth del Pezzo surface containing a finite
number of lines and satisfying the condition
$S\cap\operatorname{Sing}E=\varnothing$. Then the sheaf $E|_{\mathbb{P}^{1}}$
is locally free, and the triple (93) restricted to any line
$\mathbb{P}^{1}\subset S$ gives an epimorphism
$E|_{\mathbb{P}^{1}}\twoheadrightarrow\mathcal{O}_{\mathbb{P}^{1}}(m)$, hence
$E|_{\mathbb{P}^{1}}\cong\mathcal{O}_{\mathbb{P}^{1}}(a)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-a),\
a\leq m$. Let now $\mathbb{P}^{1}$ be any line on $X$ such that
$\mathbb{P}^{1}\not\subset S$ and
$\mathbb{P}^{1}\cap\operatorname{Sing}E=\varnothing$. Then, as before, the
sheaf $E|_{\mathbb{P}^{1}}$ is locally free, and restricting the triple (93)
to $\mathbb{P}^{1}$ we obtain an exact triple

$0\to\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-1)\to
E|_{\mathbb{P}^{1}}\to\mathbf{k}_{x}\to 0,\ \ \ \ \ \ x=\mathbb{P}^{1}\cap S.$

By analogy with the case (I), it suffices for us to check the surjectivity of
the homomorphism $r:\
\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\SS(-1))\to\operatorname{Ext}\nolimits^{1}(\mathbf{k}_{x},\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-1))$,
which is in turn equivalent to the surjectivity of the homomorphism
$h^{0}(\otimes\mathcal{O}_{Z}):\ H^{0}(\SS(-m)|_{S})\to
H^{0}(\mathcal{O}_{Z}^{\oplus 2})$ for a 0-dimensional scheme $Z=L\cap S$
containing the point $x$, where, as above, $L$ is a subspace of codimension 2
in $\mathbb{P}^{q+1}$. The surjectivity of $h^{0}(\otimes\mathcal{O}_{Z})$ for
$m\leq-1$ is checked using the resolution $0\to F(-m-2)|_{S}\to
F(-m-1)^{\oplus 2}|_{S}\to
F(-m)|_{S}\xrightarrow{\otimes\mathcal{O}_{Z}}\mathcal{O}_{Z}^{\oplus 2}\to
0$.

Note that the reflexivity and the stability of general $E$ in (91) and (93)
follow from Theorem 4.1. ∎

##  5\. Moduli spaces of rank two semistable sheaves with maximal third Chern
class on the three-dimensional quadric $X_{2}$

###  5.1. General case of rank two semistable sheaves with $-1\leq c_{1}\leq
0$ and maximal class $c_{3}$ on the quadric $X_{2}$.

Stable sheaves $E$ of rank 2 with maximal third Chern class, described in
Theorem 3.1, are special cases of sheaves included in the exact triples (24),
(25), (60), (76). Namely, for a sheaf $E$ with
$v=\ch(E)=(2,cH,dH^{2},e[\mathrm{pt}])$ we have:
(1) For $c=-1$, non-integer $d\leq-\frac{3}{2}$ and $e=d^{2}-2d+\frac{5}{12}$,
according to the statement (1.4) of Theorem 3.1, the sheaf $E$ is included in
the exact triple (24) with $k=n=1,\ m=d-\frac{1}{2}$, $S=Q_{2}$.
(2) For $c=-1$, integer $d\leq-1$ and $e=d^{2}-2d+\frac{1}{6}$, according to
the statement (1.3) of Theorem 3.1, the sheaf $E$ is included in the exact
triple (25) with $m=d$, $S=Q_{2}$.
(3) For $c=0$, non-integer $d\leq-\frac{3}{2}$ and $e=d^{2}+\frac{1}{4}$,
according to the statement (2.4) of Theorem 3.1, the sheaf $E$ is included in
the exact triple (60) with $k=n=1,\ m=d+\frac{1}{2}$, $S=Q_{2}$.
(4) For $c=0$, integer $d\leq-3$ and $e=d^{2}$, according to the statement
(2.6) of Theorem 3.1, the sheaf $E$ is in the exact triple (76) with $m=d$,
$S=Q_{2}$.

In this section we prove the following theorem, which gives a complete
description of moduli spaces of rank 2 semistable sheaves with maximal third
Chern class on the quadric $X_{2}$ for of all possible values of the second
Chern class $c_{2}$ in cases (1)-(4) above.

######  Theorem 5.1.

Let $X=X_{2}$ be the quadric and $M_{X}(v)$ be the Gieseker-Maruyama moduli
scheme of rank 2 semistable sheaves $E$ on $X$ with Chern classes
$(c_{1},c_{2},c_{3})$, where $c_{1}\in\\{-1,0\\}$, $c_{2}\geq 0$,
$c_{3}=c_{3\mathrm{max}}$ is maximal for every $c_{1}$ and $c_{2}$, and

$v=\ch(E)=(2,c_{1}H,\frac{1}{2}(c_{1}^{2}-c_{2})H^{2},\frac{1}{2}(c_{3\mathrm{max}}+\frac{2}{3}c_{1}^{3}-c_{1}c_{2})[\mathrm{pt}])$

according to (9). Then the scheme $M_{X}(v)$ is an irreducible and smooth
rational projective variety, all sheaves from $M_{X}(v)$ are stable, a general
sheaf from $M_{X}(v)$ is reflexive, and $M_{X}(v)$ is a fine moduli space.
Moreover, we have the following statements.
(i) For $c_{1}=-1$, an even $c_{2}=2p$, $p\geq 2$, and
$c_{3\mathrm{max}}=\frac{1}{2}c_{2}^{2}$ the variety $M_{X}(v)$ is the
grassmannization of 2-dimensional quotient spaces of the vector bundle
$\mathcal{A}^{\vee}$ of rank
$\operatorname{{rk}}\mathcal{A}=\frac{1}{4}(c_{2}+2)^{2}$ on the space
$\mathbb{P}^{4}$, defined by the first formula (38) for $n=1$ and $m=-p$. In
this case $\dim M_{X}(v)=\frac{1}{2}(c_{2}+2)^{2}$.
(ii) For $c_{1}=-1$, an odd $c_{2}=2p+1$, $p\geq 1$, and
$c_{3\mathrm{max}}=\frac{1}{2}(c_{2}^{2}-1)$ the variety $M_{X}(v)$ is the
grassmannization of 2-dimensional quotient spaces of the vector bundle
$\widetilde{\mathcal{A}}^{\vee}$ of rank
$\operatorname{{rk}}\widetilde{\mathcal{A}}^{\vee}=\frac{1}{4}(c_{2}+1)(c_{2}+3)$
on the grassmannian $\mathbb{G}$ defined by the second formula (38) for
$m=-p$. In this case $\dim M_{X}(v)=\frac{1}{2}(c_{2}+1)(c_{2}+3)$.
(iii) For $c_{1}=0$, an odd $c_{2}=2p+1$, $p\geq 1$, and
$c_{3\mathrm{max}}=\frac{1}{2}(c_{2}^{2}+1)$ the variety $M_{X}(v)$ is the
projectivization of the vector bundle $\mathcal{A}_{\SS}^{\vee}$ of rank
$\operatorname{{rk}}\mathcal{A}_{\SS}^{\vee}=\frac{1}{2}(c_{2}+1)(c_{2}+3)$ on
the space $\mathbb{P}^{4}$, defined by the formula (61) for $n=1$ and $m=-p$.
In this case $\dim M_{X}(v)=\frac{1}{2}c_{2}^{2}+2c_{2}+\frac{9}{2}$.
(iv) For $c_{1}=0$, an even $c_{2}=2p$, $p\geq 3$, and
$c_{3\mathrm{max}}=\frac{1}{2}c_{2}^{2}$ the variety $M_{X}(v)$ is the
projectivization of the vector bundle $\widetilde{\mathcal{A}}_{\SS}^{\vee}$
of rank
$\operatorname{{rk}}\widetilde{\mathcal{A}}_{\SS}^{\vee}=\frac{1}{2}c_{2}^{2}+2c_{2}+1$
on the grassmannian $\mathbb{G}$, defined by the formula (77) for $n=1$ and
$m=1-p$. In this case $\dim M_{X}(v)=\frac{1}{2}c_{2}^{2}+2c_{2}+4$ .

###### Proof.

(i)-(ii) Let us show that any sheaf $E$ defined by the extension (24) for the
case $k=n=1,\ m=-p$, $S\in|\mathcal{O}_{X}(1)|$:

(94) $0\to\mathcal{O}_{X}(-1)^{\oplus 2}\to E\to\mathcal{O}_{S}(-p)\to 0,\ \ \
\ \ p\geq 2,$

or extension (25) for $m=-p$:

(95) $0\to\mathcal{O}_{X}(-1)^{\oplus 2}\to
E\to\mathcal{J}_{\mathbb{P}^{1},S}(-p)\to 0,\ \ \ \ \ p\geq 1,$

is stable, that is, according to the definitions (45) and (66), that

(96) $\mathcal{Y}_{1,m,1}={\mathcal{G}}r(\mathcal{A}^{\vee},2),\ \ \ m\leq-2;\
\ \ \ \ \ \ \ \ \ \ \
\widetilde{\mathcal{Y}}_{m}={\mathcal{G}}r(\widetilde{\mathcal{A}}^{\vee},2),\
\ \ m\leq-1.$

Indeed, specifying an extension (94) is equivalent to specifying an element of
the space
$\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(-p),\mathcal{O}_{X}(-1)^{\oplus
2})$. For a semistable sheaf $E$, the triple (94) gives its tilt
Harder–Narasimhan filtration in the sense of [7, Lem. 3.2.4] if $E$ is tilt-
unstable. The Harder–Narasimhan tilt filtration factors are unique, that is,
$E$ determines a surface $S$ and a subsheaf $\mathcal{O}_{X}(-1)^{\oplus 2}$.
In this case, the group $\mathop{\mathrm{GL}}\nolimits(2)$ acts by
automorphisms on the sheaf $\mathcal{O}_{X}(-1)^{\oplus 2}$ without changing
the isomorphism class of the sheaf $E$. This means that there is a subspace in
$\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(-p),\mathcal{O}_{X}(-1))$. If
this subspace is zero-dimensional, then $E$ is a direct sum and, therefore, is
unstable. Let us assume that this subspace is one-dimensional. Then the
morphism $\mathcal{O}_{S}(-p)\to\mathcal{O}_{X}(-1)^{\oplus 2}[1]$ factorizes
through $\mathcal{O}_{X}(-1)[1]$. Then the octahedron axiom implies the
existence of a mapping $E\twoheadrightarrow\mathcal{O}_{X}(-1)$, contrary to
stability of $E$.

Conversely, let a 2-dimensional subspace in
$\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(-p),\mathcal{O}_{X}(-1))$ be
given. A choice of two basis vectors in this subspace gives an extension (94).
Here $E$ is strictly semistable along the induced wall $W$. Its Jordan-Hölder
filtration factors along the wall are two copies of the sheaf
$\mathcal{O}_{X}(-1)$ and one copy of the sheaf $\mathcal{O}_{S}(-p)$. Then
the Jordan–Hölder factors of any destabilizing subobject for $E$ are contained
among the specified sheaves. By construction we have
$\operatorname{Hom}(E,\mathcal{O}_{X}(-1))=0$. Therefore, neither the sheaf
$\mathcal{O}_{S}(-p)$ nor the extension of the sheaf $\mathcal{O}_{S}(-p)$ by
$\mathcal{O}_{X}(-1)$ cannot be a subobject of $E$.

In the case of the extension (95) the reasoning is similar to the case of the
extension (94). Now applying statements (i) and (ii.2) of Theorems 4.2 and 4.6
to the varieties $\mathcal{Y}_{1,m,1}={\mathcal{G}}r(\mathcal{A}^{\vee},2)$
and
$\widetilde{\mathcal{Y}}_{m}={\mathcal{G}}r(\widetilde{\mathcal{A}}^{\vee},2)$
from (96), respectively, we obtain the statements (i) and (ii) of the Theorem.

(iii)-(iv) Let us show that any sheaf $E$ defined by extensions (60) and (76)
for the case $k=n=1,\ m=-p$, $S\in|\mathcal{O}_{X}(1)|$:

(97) $0\to\SS(-1)\to E\to\mathcal{O}_{S}(-p)\to 0,\ \ \ \ \ p\geq 1,$ (98)
$0\to\SS(-1)\to E\to\mathcal{J}_{\mathbb{P}^{1},S}(-p)\to 0,\ \ \ \ \ p\geq
2.$

is stable, that is, according to the definitions of $\mathcal{Y}_{\SS,1,m,1}$
and $\widetilde{\mathcal{Y}}_{\SS,1,m,1}$ (see Remarks 4.4 and 4.7
respectively) that

(99) $\mathcal{Y}_{\SS,1,m,1}=\mathbb{P}(\mathcal{A}_{\SS}^{\vee}),\ \ \ \ \
m\leq-1;\ \ \ \ \ \ \ \ \
\widetilde{\mathcal{Y}}_{\SS,m}=\mathbb{P}(\widetilde{\mathcal{A}}_{\SS}^{\vee}),\
\ \ \ \ m\leq-2.$

Indeed, an extension (97) corresponds to an element from
$\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(-p),\SS(-1))$. For semistable
objects this extension gives tilt Harder–Narasimhan filtration for $E$
provided that $E$ is tilt-unstable. Harder–Narasimhan factors in this case are
unique. This means that $E$ determines a two-dimensional quadric $S$ and the
sheaf $\SS(-1)$ as a subobject of $E$. Multiplying the morphism $\SS(-1)\to E$
by a scalar does not change the isomorphism class of the sheaf $E$. In
addition, if $E$ is a direct sum, it is not stable.

Conversely, let a one-dimensional subspace in
$\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(-p),\SS(-1))$ be given.
Choice of any nonzero vector in this subspace gives a nontrivial extension
(97). The object $E$ in it is strictly semistable along the induced wall $W$.
The Jordan–H?lder factors of the object $E$ are uniquely defined, and the only
subobjects that can destabilize $E$ above the wall are either $\SS(-1)$ or
$\mathcal{O}_{S}(-p)$. However, $\SS(-1)$ does not destabilize $E$ for
numerical reasons. On the other hand, the non-splitability of the previous
exact triple means that $\mathcal{O}_{S}(-p)$ cannot be a subobject of $E$.

In the case of the extension (98) the argument is similar to the case of the
extension (97). Now applying the statements (i) and (ii.2) of Theorems 4.5 and
4.8 to the varieties
$\mathcal{Y}_{\SS,1,m,1}={\mathcal{G}}r(\mathcal{A}_{\SS}^{\vee},2)$ and
$\widetilde{\mathcal{Y}}_{\SS,1,m,1}={\mathcal{G}}r(\widetilde{\mathcal{A}}_{\SS}^{\vee},2)$
from (99) respectively, we obtain the statements (iii) and (iv) of the
Theorem. ∎

###  5.2. Special cases of rank two semistable sheaves with small values of
class $c_{2}$ and maximal class $c_{3}\geq 0$ on $X_{2}$.

Theorem 5.1 covers most of the possible values of Chern classes of rank 2
semistable sheaves $E$ on the quadric $X_{2}$. The remaining special cases for
$c_{3}\geq 0$ and small values of class $c_{2}$ are indicated in statements
(1.1), (1.2), (2.1), (2.3) and (2.5) of Theorem 3.1. Below in Theorem 5.2 we
describe the moduli of sheaves $E$ for the first three of the indicated cases,
and in Theorems 5.3 and 5.4 — for the last two cases respectively.

######  Theorem 5.2.

Under the conditions and notation of Theorem 5.1 the following statements are
true:
(1) For $c_{1}=-1$, $c_{2}=1$ and $c_{3\mathrm{max}}=0$ the variety $M_{X}(v)$
is a point $[\SS(-1)]$.
(2) For $c_{1}=c_{2}=c_{3\mathrm{max}}=0$ the variety $M_{X}(v)$ is a point
$[\mathcal{O}_{X}^{\oplus 2}]$.
(3) For $c_{1}=-1$, $c_{2}=2$ and $c_{3\mathrm{max}}=2$ we have
$M_{X}(v)\simeq\mathrm{Gr}(2,5)$.

###### Proof.

Statements (1) and (2) follow directly from statements (1.1) and (2.1) of
Theorem 3.1. Let us consider the last case.

(3) In this case, according to statement (1.2) of Theorem 3.1 the sheaf $E$ is
determined by an exact triple

(100) $0\to\mathcal{O}_{X}(-2)\xrightarrow{f}\mathcal{O}_{X}(-1)\otimes U\to
E\to 0.$

The morphism $f$ can be considered as a 3-dimensional quotient $U$ of the
space
$H^{0}(\mathcal{O}_{X}(1))^{\vee}$, that is, as a point of the grassmannian
$\mathrm{Gr}(2,5)$. Let us consider the mapping $\varphi:\mathrm{Gr}(2,5)\to
M:(H^{0}(\mathcal{O}_{X}(1))^{\vee}\twoheadrightarrow U)\mapsto E$, where $E$
is defined in (100).

Let us check that the function $\varphi$ is correctly defined. To do this, it
is enough to make sure that $E$ is $\mu$-stable. By Theorem 3.1 there is a
unique wall $W$ for the above objects $E$, given by the exact triple

(101) $0\to\mathcal{O}_{X}(-1)^{\oplus 3}\to E\to\mathcal{O}_{X}(-2)[1]\to 0.$

Therefore, checking that an object $E=\varphi(U)$ is $\mu$-stable is
equivalent to checking $\nu_{\alpha,\beta}$-stability of $E$ in the
neighborhood of this wall $W$ above it. If $E$ is not semistable over $W$,
then there is a destabilizing semistable quotient $E\twoheadrightarrow G$.
Since $E$ is strictly semistable along $W$, the quotient satisfies the
equality $\nu_{\alpha,\beta}(E)=\nu_{\alpha,\beta}(G)$ for $(\alpha,\beta)$ on
$W$. As is known, any Jordan–Hölder filtration for $E$ has three stable
factors $\mathcal{O}_{X}(-1)$ and one stable factor $\mathcal{O}_{X}(-2)[1]$.
Thus, the stable factors of the object $G$ are contained among these sheaves.
The factor $\mathcal{O}_{X}(-2)[1]$ does not destabilize $E$ over $W$ by
numerical reasons. The vector space
$\operatorname{Hom}(E,\mathcal{O}_{X}(-1))$ is the kernel of a homomorphism
$\operatorname{Hom}(\mathcal{O}_{X}(-1)\otimes
U,\mathcal{O}_{X}(-1))\to\operatorname{Hom}(\mathcal{O}_{X}(-2),\mathcal{O}_{X}(-1))$,
which is injective. Therefore, $G\not\cong\mathcal{O}_{X}(-1)^{\oplus a}$ for
$a\in\\{1,2,3\\}$. If $G$ is an extension of $\mathcal{O}_{X}(-1)$ by
$\mathcal{O}_{X}(-2)[1]$, then the corresponding subobject is
$\mathcal{O}_{X}(-1)^{\oplus 2}$ and it does not destabilize $E$ above the
wall. Likewise, if $G$ is an extension of $\mathcal{O}_{X}(-1)^{\oplus 2}$ by
$\mathcal{O}(-2)[1]$, then the corresponding subobject is given by the sheaf
$\mathcal{O}_{X}(-1)$ that does not destabilize $E$ above the wall. Thus $E$
is $\mu$-stable.

Let us check that $\varphi$ is a bijection. According to Theorem 3.1.(1.2) any
semistable object $E$ with $c_{1}=-1$, $c_{2}=2$ and $c_{3\mathrm{max}}=0$
satisfies the exact triple (101). Specifying such an extension is equivalent
to specifying an element
$\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{X}(-2)[1],\\\
\mathcal{O}_{X}(-1)^{\oplus 3})=H^{0}(\mathcal{O}_{X}(1))^{\oplus 3}$ . In
Theorem 3.1 we showed that this extension is the tilt Harder–Narasimhan
filtration for $E$ under the wall. The factors of this filtration are unique.
This means that $E$ determines the subobject $\mathcal{O}_{X}(-1)^{\oplus 3}$.
However, the group $GL(3)$ acts by automorphisms on
$\mathcal{O}_{X}(-1)^{\oplus 3}$ without changing the isomorphism class of
$E$. This means that we get a unique subspace in $H^{0}(\mathcal{O}_{X}(1))$.
However, if this subspace is not three-dimensional, then there is a
destabilizing morphism $E\twoheadrightarrow\mathcal{O}_{X}(-1)$. This
simultaneously proves surjectivity and injectivity of $\varphi$.

Let us show that $\varphi$ is a morphism of schemes. To do this, it is enough
to construct a universal family $\mathcal{E}$ of sheaves $E$ on
$\mathrm{Gr}(2,5)\times\mathbb{P}^{3}$. Universal property of the scheme
$M_{X}(v)$ then shows that $\varphi$ is a morphism. We have two projections
$\mathrm{Gr}(2,5)\xleftarrow{p_{1}}\mathrm{Gr}(2,5)\times
X\xrightarrow{p_{2}}X$ and the rank 3 tautological quotient bundle of
$\mathcal{O}_{\mathrm{Gr}(2,5)}\otimes
H^{0}(\mathcal{O}_{X}(1))^{\vee}\twoheadrightarrow\mathcal{Q}$ on
$\mathrm{Gr}(2,5)$. The required universal family $\mathcal{E}$ is obtained as
the cokernel of the composition of morphisms $p_{2}^{*}\mathcal{O}_{X}(-2)\to
H^{0}(\mathcal{O}_{X}(1))^{\vee}\otimes
p_{2}^{*}\mathcal{O}_{X}(-1)\to\mathcal{Q}\boxtimes\mathcal{O}_{X}(-1)$.

To prove the smoothness of $M_{X}(v)$, it is enough to show that
$\operatorname{Ext}\nolimits^{2}(E,E)=0$. This is obtained by applying the
three functors ${\mathbf{R}}\operatorname{Hom}(-,\mathcal{O}_{X}(-2)[1])$,
${\mathbf{R}}(-,\\\ \mathcal{O}_{X}(-1))$, and
${\mathbf{R}}\operatorname{Hom}(E,-)$ to the exact triple (101). From the
smoothness of $M_{X}(v)$ and bijectivity of $\varphi$ and [31, Ch. 2, §4.4,
Thm. 2.16] it follows that $\varphi$ is an isomorphism. ∎

######  Theorem 5.3.

For $c_{1}=0$, $c_{2}=2$ and $c_{3\mathrm{max}}=2$ the scheme $M_{X}(v)$ is
irreducible and has dimension 9 and is not smooth.

###### Proof.

Let $E$ be a Gieseker-semistable sheaf with $c_{1}=0$, $c_{2}=2$ and
$c_{3\mathrm{max}}=2$, that is, $[E]\in M_{X}(v)$, where
$v=(2,0,-H^{2},\\{\mathrm{pt}\\})$. By Theorem 3.1.(2.3) $E$ is destabilized
by an exact triple

(102) $0\to\SS(-2)^{\oplus 2}\to\mathcal{O}_{X}(-1)^{\oplus 6}\to E\to 0.$

Let us prove that there exist stable sheaves $E$ included in (102). Let $E$ be
properly semistable, then there exists a subsheaf $F\subset E$ of rank 1 such
that that $\chi(F(m))=\frac{1}{2}\chi(E(m))$ for $m\gg 0$. From get thathere
we obtain $F\in M(1,0,-\frac{1}{2}H^{2},\frac{1}{2})$. Since $F$ is stable, by
the proof of Lemma 3.4 we find an exact triple
$0\to\SS(-2)\to\mathcal{O}_{X}(-1)^{\oplus 3}\to E\to 0.$ Since
$\mathrm{Ext}^{1}(\mathcal{O}_{X}(-1)^{\oplus 6},\SS(-2)^{\oplus 2})$, the
embedding $F\to E$ induces a morphism $\mathcal{O}_{X}(-1)^{\oplus
3}\to\mathcal{O}_{X}(-1)^{\oplus 6}$. Let us denote
$W=H^{0}(\SS(1))\cong\mathbf{k}^{16}$ and
$A=H^{0}(\alpha^{\vee}(-1)):\mathbf{k}^{6}\to\mathbf{k}^{2}\otimes W$, then we
get a commutative diagram with exact rows

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
38.51666pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
38.51666pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbf{k}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
96.02777pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
45.55556pt\raise-29.2211pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
96.02777pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbf{k}^{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
103.06667pt\raise-19.58276pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{A}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
103.06667pt\raise-27.41666pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
153.53888pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
153.53888pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbf{k}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
160.57777pt\raise-19.58276pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
200.63333pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
200.63333pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-5.5pt\raise-39.16554pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbf{k}\otimes
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
85.61111pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
85.61111pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbf{k}^{2}\otimes
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
144.52222pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
144.52222pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbf{k}\otimes
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
200.63333pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
200.63333pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

In other words, the matrix of the mapping $A$ in some bases of the spaces
$\mathbf{k}^{6}$ and $\mathbf{k}^{2}$ (consistent with the choice of subspaces
$\mathbf{k}^{3}\subset\mathbf{k}^{6},\mathbf{k}\subset\mathbf{k}^{2}$) has the
form

$\begin{pmatrix}*&*&*&*&*&*\\\ 0&0&0&*&*&*\end{pmatrix}$

Consider the incidence variety

$X=\\{(A_{i})_{i=1}^{16}\in\mathrm{Hom}(\mathbf{k}^{6},\mathbf{k}^{2})^{\oplus
16},V_{1}\in\mathrm{Gr}(3,6),V_{2}\in\mathrm{Gr}(1,2)|A_{i}(V_{1})\subset
V_{2}\\}$

with the projection of $X\to
Y=\mathrm{Hom}(\mathbf{k}^{6},\mathbf{k}^{2})^{\oplus 16}$. Because
$\dim\mathrm{Gr}(3,6)=9,\linebreak\dim\mathrm{Gr}(1,2)=1$ and on each of the
16 matrices $A_{i}$ three independent linear conditions are imposed, $\dim
X=16\cdot 12+9+1-16\cdot 3<16\cdot 12=\dim Y$. It follows that there is a
dense open subset of the variety $Y$ that does not intersect the image of $X$
under the projection $X\to Y$. This proves the existence of Gieseker-stable
sheaves $E=\operatorname{coker}\alpha$.

Since $E$ is tilt-semistable over the semicircular wall, then
$\operatorname{Hom}(E,\mathcal{O}_{X}(-1))=0$. In addition,
$\operatorname{Ext}\nolimits^{1}(\SS(-2),\SS(-2))=0$ (since $\SS(-1)$ is an
exceptional object of the category $D^{b}(X)$) and
$\operatorname{Ext}\nolimits^{2}(\mathcal{O}_{X}(-1),\SS(-2))=0$ (since
$\SS(-1)$ is an ACM sheaf). Now from (102) the equality
$\operatorname{Ext}\nolimits^{2}(E,\SS(-2)^{\oplus 2})=0$ follows, so we have
an exact sequence

(103)
$\begin{split}&0\to\operatorname{Hom}(E,E)\to\operatorname{Ext}\nolimits^{1}(E,\SS(-2)^{\oplus
2})\to\\\ &\operatorname{Ext}\nolimits^{1}(E,\mathcal{O}_{X}(-1)^{\oplus
6})\to\operatorname{Ext}\nolimits^{1}(E,E)\to 0.\end{split}$

If now the sheaf $E$ is stable, then $\hom(E,E)=1$, and a calculation by (103)
and (102) gives $\operatorname{ext}^{1}(E,E)=9$. On the other hand, from the
triple (102) it follows that $M_{X}(v)$ is irreducible and $\dim
M_{X}(v)=12h^{0}(\SS)-\dim(GL(2,\mathbf{k})\times
GL(6,\mathbf{k})/\mathbf{k}^{*})=12\cdot 4-4-36+1=9$, so $M_{X}(v)$ is smooth
at the point $[E]$.

In the properly semistable case we use the following result [23, p. 294,
Lemma]:

######  Lemma 5.1.

Let $E$ be a semistable sheaf of rank 2, $E\cong E_{1}\oplus E_{2}$, where
$\chi(E_{1}(m))=\chi(E_{2}(m))=\frac{1}{2}\chi(E(m))$ and $E_{1}\not\cong
E_{2}$. If $\operatorname{Ext}\nolimits^{2}(E,E)=0$, then the tangent space to
the moduli scheme of semistable sheaves at the point $[E]$ is isomorphic to
$\operatorname{Ext}\nolimits^{1}(E_{1},E_{1})\oplus(\operatorname{Ext}\nolimits^{1}(E_{1},E_{2})\oplus\operatorname{Ext}\nolimits^{1}(E_{2},E_{1}))\oplus\operatorname{Ext}\nolimits^{1}(E_{2},E_{2}).$

In our case, we apply this lemma to the sheaf $E\cong E_{1}\oplus E_{2}$,
where $E_{1}\not\cong E_{2}$, the sheaves $E_{1}$ and $E_{2}$ are stable and
included in exact triples of the form

(104) $0\to\SS(-2)\to\mathcal{O}_{X}(-1)^{\oplus 3}\to E_{i}\to 0.$

Since $\operatorname{Ext}\nolimits^{1}(\SS(-2),\mathcal{O}_{X}(-1)))\cong
H^{1}(\SS)=0$ and $\operatorname{Ext}\nolimits^{2}(\mathcal{O}_{X}(-1),\
OO_{X}(-1))=0$, then, taking into account (104) we have
$\operatorname{Ext}\nolimits^{2}(E,\mathcal{O}_{X}(-1)^{\oplus 6})=0$. On the
other side, since
$\operatorname{Ext}\nolimits^{2}(\SS(-2),\SS(-2))=\operatorname{Ext}\nolimits^{3}(\mathcal{O}_{X}(-1),\SS(-2))=0$
due to the exceptionality of the pair $(\SS(-1),\mathcal{O}_{X})$ in
$D^{b}(X)$, then $\operatorname{Ext}\nolimits^{3}(E,\SS(-2))=0$. From here and
from (102) we get $\operatorname{Ext}\nolimits^{2}(E,E)=0$. Now by Lemma 5.1
the dimension of the tangent space to $M_{X}(v)$ at the point
$[E]=[E_{1}\oplus E_{2}]$ is equal to $3+2+2+3=10\neq 9$, so $M_{X}(v)$ is not
smooth. ∎

######  Theorem 5.4.

For $c_{1}=0$, $c_{2}=4$ and $c_{3\mathrm{max}}=8$, the scheme $M_{X}(v)$ is
the union of two irreducible components $M_{1}$ and $M_{2}$. These components
are described as follows.
(i) $M_{1}$ is a smooth rational variety of dimension 21, which is the
projectivization of a locally free sheaf of rank 17 on $\mathbb{G}$. $M_{1}$
is a fine moduli space, and all sheaves from $M_{1}$ are stable. The scheme
$M_{X}(v)$ is nonsingular along $M_{1}$.
(ii) the scheme $M_{2}$ is irreducible, has dimension 21, and polystable
sheaves from $M_{2}$ constitute a closed subset of dimension 12 in $M_{2}$ in
which the scheme $M_{X}(v)$ is not smooth.
(iii) The scheme $M_{X}(v)$ is disconnected:

$M_{X}(v)=M_{1}\sqcup M_{2}.$

###### Proof.

Let $E$ be a Gieseker-semistable sheaf with $c_{1}=0$, $c_{2}=2$ and
$c_{3\mathrm{max}}=2$, that is, $[E]\in M_{X}(v)$, where
$v=(2,0,-2H^{2},4\\{\mathrm{pt}\\})$. By Theorem 3.1.(2.5) the sheaf $E$ is
destabilized by an exact triple

(105) $0\to\SS(-1)\to E\to\mathcal{I}_{\mathbb{P}^{1},S}(-1)\to 0,$

or an exact triple

(106) $0\to\mathcal{O}_{X}(-2)^{\oplus 2}\to\mathcal{O}_{X}(-1)^{\oplus 4}\to
E\to 0.$

(i) According to Lemma 3.8 any sheaf $E$ in the triple (105) is tilt-stable
for $\beta<0,\alpha\gg 0$, and therefore is stable according to Proposition
2.1. In other words, in the notation of Remark 4.7 we have an equality
$\widetilde{\mathcal{Y}}_{\SS,1,m,1}=\mathbb{P}(\widetilde{\mathcal{A}}_{\SS}^{\vee})$,
similar to the second equality (99) (in our case $m=-1$). Therefore, applying
Theorem 4.8 for $k=n=1,\ m=-1$, we find that the bundles $E$ included in the
triple (105) constitute a smooth irreducible rational component
$M_{1}:=M_{1,-1,1}$ of the scheme $M_{X}(v)$, described as the
projectivization of $\widetilde{\pi}:\ \mathbb{P}(\widetilde{\mathcal{A}}_{\
SS}^{\vee})\to\Pi$, where $\mathcal{A}$ is a locally free sheaf of rank 17 on
$\Pi$. In this case $\dim M_{1}=21$, $M_{1}$ is a fine moduli space, and
according to the above, all sheaves from $M_{1}$ are stable.

(ii) Let us denote $M_{2}=\\{[E]\in M_{X}(v)\ |\ E\ \text{is included in the
triple}\ \eqref{tr 2.2}\\}$. From (106) we obtain that $M_{2}$ is irreducible
as a quotient of an open subset of the space
$\operatorname{Hom}(\mathcal{O}_{X}(-2)^{\oplus 2},\mathcal{O}_{X}(-1)^{\oplus
4})$ by the group $\dim(GL(2,\mathbf{k})\times
GL(4,\mathbf{k})/\mathbf{k}^{*}$, and $\dim
M_{2}=\hom(\mathcal{O}_{X}(-2)^{\oplus 2},\mathcal{O}_{X}(-1)^{\oplus
4})-\dim(GL(2,\mathbf{k})\times GL(4,\mathbf{k})/\mathbf{k}^{*})=2\cdot 4\cdot
5-4-16+1=21$. Next we will carry out for the sheaves $E$ from $M_{2}$ an
argument similar to the argument with the triple (102), carried out in the
proof of Theorem 5.3. In particular, it is verified that $M_{2}$ contains
stable sheaves $E$. For stable sheaves $E$ we have
$\operatorname{Hom}(E,\mathcal{O}_{X}(-1)^{\oplus 4})=0$, and also
$\operatorname{Ext}\nolimits^{2}(E,\mathcal{O}_{X}(-2)^{\oplus 2})=0$ due to
the exceptionality of line bundles on $X$ and the ACM property. Thus, in this
case there is a similar to (103) exact sequence

$0\to\operatorname{Hom}(E,E)\to\operatorname{Ext}\nolimits^{1}(E,\mathcal{O}_{X}(-2)^{\oplus
2})\to\operatorname{Ext}\nolimits^{1}(E,\mathcal{O}_{X}(-1)^{\oplus
4})\to\operatorname{Ext}\nolimits^{1}(E,E)\to 0.$

A standard calculation with this sequence using (106) shows that the dimension
$\operatorname{ext}^{1}(E,E)$ of the tangent space to $M_{X}(v)$ at the point
$[E]$ for stable $E$ is equal to 21, which coincides with $\dim M_{2}$. Thus,
$M_{2}$ is an irreducible component of the scheme $M_{X}(v)$.

Let us now consider a point $[E]=[E_{1}\oplus E_{2}]$, where $E_{1}\not\cong
E_{2}$ and $E_{i}$ are included into exact triples of the form

(107) $0\to\mathcal{O}_{X}(-2)\to\mathcal{O}_{X}(-1)^{\oplus 2}\to E_{i}\to
0,\ \ \ \ \ i=1,2,$

A simple calculation using these triples shows that
$\operatorname{Ext}\nolimits^{2}(E_{i},E_{j})=0,\ 1\leq i,j\leq 2,$
$\operatorname{ext}^{1}(E_{i},E_{i})=6,\ i=1,2$,
$\operatorname{ext}^{1}(E_{i},E_{j})=5,\ i\neq j$. Hence we have
$\operatorname{Ext}\nolimits^{2}(E,E)=0$, and by Lemma 5.1 $\dim
T_{[E]}M_{X}(v)=6+5+5+6=22\neq 21$, therefore $M_{X}(v)$ is not smooth at the
point $[E]$.

(iii) Let us give two proofs of this statement. (First proof is more
conceptual, but the second is much shorter than the first.)
First proof. It follows directly from statements (i) and (ii) that the
varieties $M_{1}$ and $M_{2}$ do not coincide and, moreover,

(108) $M_{2}\not\subset M_{1}.$

Assume that the scheme $M_{X}(v)=M_{1}\cup M_{2}$ is connected and take an
arbitrary point $x\in M_{1}\cap M_{2}$. Let us use the following result — see
[2, Lemma 24].

######  Lemma 5.2.

Let $W$ be a projective scheme, and let $Y$ and $Z$ be its two irreducible
closed subsets of positive dimension, and suppose that there is a point
$x_{0}\in Y\cap Z$ such that $W$ is smooth at the point $x_{0}$ and $\dim
Y=\dim T_{x_{0}}W$. Then $Z\subset Y$.

Applying this lemma to the case $W=M_{X}(v)$, $Y=M_{1}$, $Z=M_{2}$, $x_{0}=x$,
we obtain that $M_{2}\subset M_{1}$, contrary to (108).
Second proof. According to the last statement of Lemma 3.8, $M_{1}\cap
M_{2}=\varnothing$, that is, the union $M_{1}\cup M_{2}$ is disjoint. ∎

##  6\. Boundedness of the third Chern class of rank two stable reflexive
sheaves of general type with $c_{1}=0$ on the varieties $X_{4}$ and $X_{5}$

In this section we consider rank two stable reflexive sheaves of general type
with $c_{1}=0$ (see Definition 4.10) on varieties $X_{4}$ and $X_{5}$. For any
such sheaf $E$ we prove boundedness from above of the third Chern class
$c_{3}$ of the sheaf $E$ by a quadratic polynomial in the second Chern class
$c_{2}$ – see Theorems 6.1 and 6.2 below.

###  6.1. Rank two stable reflexive sheaves of general type with $c_{1}=0$ on
the variety $X_{5}$

In this subsection everywhere $X=X_{5}$.

Let us recall some well-known facts about the variety $X$ (see, for example,
[1, Th. 4.2(iii) and Cor. 6.6(ii)]).
1) The base $B=B(X)$ of the family of lines on $X$ is isomorphic to
${\mathbb{P}^{2}}$.
2) For an arbitrary line $l\in B$, the set $B_{l}=\\{l^{\prime}\in B\ |\
l^{\prime}\cap l\neq\varnothing\\}$ is a line in $\mathbb{P}^{2}$.
3) For an arbitrary line $l\in B$, either
$N_{l/X}\cong\mathcal{O}_{\mathbb{P}^{1}}^{\oplus 2}$ (general case), or
$N_{l/X}\cong\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-1)$.
4) Let $\varphi:\ X\dasharrow{\mathbb{P}^{4}}$ be the linear projection of the
variety $X\subset{\mathbb{P}^{6}}$ into the space ${\mathbb{P}^{4}}$ from a
line $l\subset X$, for which
$N_{l/X}\cong\mathcal{O}_{{\mathbb{P}^{1}}}^{\oplus 2}$. Then
$\varphi(X)=Q\cong X_{2}$ is a smooth quadric in $\mathbb{P}^{4}$, the
morphism $\varphi:X\dasharrow Q$ is birational and decomposes into a
Hironaka’s roof

(109)
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widetilde{X}}$$\scriptstyle{\delta}$$\scriptstyle{\sigma}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{Q.}$

Here $\delta^{-1}:\ X\dasharrow\widetilde{X}$ is the blow-up of $X$ centered
at $l$, $S=\delta^{-1}(l)\simeq{\mathbb{P}^{1}}\times{\mathbb{P}^{1}}$,
$\sigma|_{S}:S\xrightarrow{\simeq}\sigma(S)=Q_{2}$ is an isomorphism of $S$
onto a smooth two-dimensional quadric $Q_{2}\subset Q$, and $\sigma^{-1}:\
Q\dasharrow\widetilde{X}$ is the blow-up of $Q$ centered in a smooth rational
cubic curve $C$ of type (2,1) on $Q_{2}$, so
$\widetilde{X}\simeq{\mathbb{P}^{(}}\mathcal{I}_{C,Q})$. In particular, a
triple $0\to N_{C,Q_{2}}\to N_{C/Q}\to N_{Q_{2}/Q}|_{C}\to 0$ is exact, where
due to isomorphisms $Q_{2}\simeq{\mathbb{P}^{1}}\times{\mathbb{P}^{1}}$ and
$C\simeq{\mathbb{P}^{1}}$ we obtain
$N_{C/Q_{2}}\cong\mathcal{O}_{{\mathbb{P}^{1}}}(3)$,
$N_{Q_{2}/Q}|_{C}\cong\mathcal{O}_{Q}(H)|_{C}\cong\mathcal{O}_{{\mathbb{P}^{1}}}(3)$,
hence $\det N_{C/Q}\cong\mathcal{O}_{{\mathbb{P}^{1}}}(6)$.
5) Let $\Gamma=\\{(x,l)\in X\times B\ |\ x\in l\\}$ be an incidence graph with
projections $X\xleftarrow{p_{1}}\Gamma\xrightarrow{p_{2}}B$. As we know [1,
Prev. 5.2], $p_{1}:\Gamma\to X$ is a morphism, which is finite at a general
point, and we have $\dim\\{x\in X\ |\ \dim p_{1}^{-1}(x)\geq 1\\}\leq 0$.

Consider an extension

(110) $\xi:\ 0\to\mathcal{O}_{Q}(nH)\to
F\xrightarrow{\varepsilon}\mathcal{I}_{C,Q}\to 0.$

The sheaf
$\mathop{\mathcal{H}om}\nolimits(\mathcal{I}_{C,Q},\mathcal{O}_{Q}(nH))\cong\mathcal{O}_{Q}(nH)$
on $Q$ and the sheaf
$\mathcal{M}=\mathop{\mathcal{E}xt}\nolimits^{1}(\mathcal{I}_{C,Q},\\\
\mathcal{O}_{Q}(nH))\cong\mathop{\mathcal{E}xt}\nolimits^{2}(\mathcal{O}_{C},\mathcal{O}_{Q}(nH))\cong\det
N_{C/Q}\otimes\mathcal{O}_{Q}(nH)\cong\mathcal{O}_{{\mathbb{P}^{1}}}(6+3n)$ on
$C$ for $n\gg 0$ are very ample. Therefore, we can assume that:
1)
$H^{i}(\mathop{\mathcal{H}om}\nolimits(\mathcal{I}_{C,Q},\mathcal{O}_{Q}(nH)))=0,\
i=1,2$, which implies that the long exact sequence of
$\operatorname{Ext}\nolimits$-groups for a pair of sheaves
$\mathcal{I}_{C,Q},\mathcal{O}_{Q}(nH)$ gives isomorphisms
$\operatorname{Ext}\nolimits^{1}(\mathop{\mathcal{H}om}\nolimits(\mathcal{I}_{C,Q},\mathcal{O}_{Q}(nH))\cong
H^{0}(\mathop{\mathcal{E}xt}\nolimits^{1}(\mathcal{I}_{C,Q},\mathcal{O}_{Q}(nH))\cong
H^{0}(\mathop{\mathcal{E}xt}\nolimits^{2}(\mathcal{O}_{C},\mathcal{O}_{Q}(nH))=H^{0}(\mathcal{M})$;
2) The triple (110) as an extension
$\xi\in\operatorname{Ext}\nolimits^{1}(\mathop{\mathcal{H}om}\nolimits(\mathcal{I}_{C,Q},\mathcal{O}_{Q}(nH)))=H^{0}(\mathcal{M})$
is a section of the sheaf $\mathcal{M}$, the scheme of zeros $(\xi)_{0}$ of
which is reduced, i.e. it is a simple divisor $D_{\xi}=x_{1}+...+x_{3n+6}$.
This means by Serre’s construction (see [16, Section 4]), that the sheaf $F$
in (110) is a reflexive sheaf of rank two with $\operatorname{Sing}F=D_{\xi}$
and the simplest singularities at the points $x_{i}$, i.e.
$\mathop{\mathcal{E}xt}\nolimits^{1}(F,\mathcal{O}_{Q})\cong\oplus_{i=1}^{3n+6}\mathbf{k}_{x_{i}}$.
Thus, according to [4, Cor. 2.8], the projective spectrum $Y=\mathbb{P}(F)$ of
the sheaf $F$ is a smooth variety. The epimorphism $\varepsilon$ in (110)
corresponds to an embedding $i$ of the divisor
$\widetilde{X}=\mathbb{P}(\mathcal{I}_{C,Q})$ into $Y$, and let $p:Y\to Q$ be
the natural projection, so that $\sigma=p\circ i$. Note also that the sheaf

(111) $L=\mathcal{O}_{Y/Q}(1)\otimes p^{*}\mathcal{O}(mH)$

is ample for $m\gg 0$, where $\mathcal{O}_{Y/Q}(1)$ is the Grothendieck sheaf
on $Y=\mathbb{P}(F)$.

Let

(112) $\begin{split}&E\ \textit{be a rank 2 stable reflexive sheaf of general
type with}\ c_{1}(E)=0\\\ &\textit{on}\ X;\ \textit{in particular,}\
H^{0}(E)=0.\end{split}$

Since $\dim\operatorname{Sing}E\leq 0$, it follows from properties 1) – 5)
that we can choose for $l$ a general line on $X$ for which
$N_{l/X}\cong\mathcal{O}_{{\mathbb{P}^{1}}}^{\oplus 2}$ such that the
following conditions are satisfied.
a) $B_{l}\cap p_{2}(p_{1}^{-1}(\operatorname{Sing}E))=\varnothing$, so the
surface $S_{l}=p_{1}(p_{2}^{-1}(B_{l}))$ does not intersect the set
$\operatorname{Sing}E$. Since by construction $S_{l}=\delta(S)$, then this
means that the sheaf

(113) $\widetilde{E}=\delta^{*}E$

satisfies the condition

(114) $\operatorname{Sing}\widetilde{E}\cap S=\varnothing.$

Note that from the projection formula for the blow-up $\delta$ it follows that
$E=\delta_{*}\widetilde{E}$, $R^{i}\delta_{*}\widetilde{E}=0,\ i>0,$ Therefore

(115) $\chi(E)=\chi(\widetilde{E})$

and, in addition, (112) implies

(116) $H^{0}(\widetilde{E})=0.$

b) Since $E$ is a sheaf of general type, the set $G_{C}=\\{x\in C\ |\
\widetilde{E}|_{\sigma^{-1}(x)}\cong\mathcal{O}_{{\mathbb{P}^{1}}}(1)\oplus\mathcal{O}_{{\mathbb{P}^{1}}}(-1)\\}$
is a proper subset of the curve $C$, and
$\widetilde{E}|_{\sigma^{-1}(x)}\cong\mathcal{O}_{{\mathbb{P}^{1}}}^{\oplus
2}$ for $x\in C\smallsetminus G_{C}$.

In addition, we will assume that the section $\xi$ of the sheaf $\mathcal{M}$
is sufficiently general, so that

(117) $D_{\xi}\cap G_{C}=\varnothing.$

Below we will need another sufficiently general section $\xi^{\prime}\in
H^{0}(\mathcal{L})$ such that

(118) $D_{\xi^{\prime}}\cap G_{C}=\varnothing,\ \ \ \ \ \ D_{\xi}\cap
D_{\xi^{\prime}}=\varnothing.$

Put

(119)
$\begin{split}&Q^{0}:=Q\smallsetminus(p(\operatorname{Sing}\widetilde{E})\cup
D_{\xi}),\ \ \ \ \ \ C^{0}:=C\smallsetminus D_{\xi}\hookrightarrow Q^{0},\\\
&X^{0}:=\sigma^{-1}(Q^{0})=\widetilde{X}\smallsetminus(\operatorname{Sing}\widetilde{E}\cup\sigma^{-1}D_{\xi}),\\\
&Y^{0}:=Y\smallsetminus p^{-1}(Q^{0}),\ \ \ \ \ \
\widetilde{E}^{0}:=\widetilde{E}|_{X^{0}},\ \ \ \ \ \
\overline{E}^{0}:=\sigma_{*}\widetilde{E}^{0},\end{split}$

where $X^{0}$ becomes a divisor in $Y^{0}$ via the embedding $i$ defined
above, so that the following diagram is commutative:

(120) $\textstyle{X^{0}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{i}$$\textstyle{Y^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Q^{0}.}$

Note that, by (119), $\operatorname{Sing}F=D_{\xi}$ and $Q_{0}$ are disjoint,
so that the $\mathcal{O}_{Q^{0}}$-sheaf $F|_{Q^{0}}$ is locally free and,
therefore, $p:Y^{0}=\mathbb{P}(F|_{Q^{0}})\to Q^{0}$ is a locally trivial
${\mathbb{P}^{1}}$-bundle. Therefore, by Serres theorem, for sufficiently
large natural numbers $m,s,r$ there is an epimorphism $e:\
\mathcal{L}_{0}:=(L^{\otimes-s}|_{Y^{0}})^{\oplus
r}\twoheadrightarrow\widetilde{E}^{0}$. Since $\overline{E}^{0}$ is a locally
free $\mathcal{O}_{X^{0}}$-sheaf on a smooth divisor $X^{0}$ in a smooth
variety $Y^{0}$, then, considered as $\mathcal{O}_{Y^{0}}$-sheaf,
$\widetilde{E}^{0}$ has homological dimension 1, so $\mathcal{L}_{1}:=\ker e$
is a locally free $\mathcal{O}_{Y^{0}}$-sheaf. Thus, there is an exact triple

(121)
$0\to\mathcal{L}_{1}\to\mathcal{L}_{0}\xrightarrow{e}\widetilde{E}^{0}\to 0.$

From the definition (119) of the curve $C^{0}$ it follows, that

(122)
$\widetilde{E}^{0}|_{\sigma^{-1}(x)}\cong\left\\{\begin{array}[]{cc}\mathcal{O}_{{\mathbb{P}^{1}}}^{\oplus
2},&\ \ \ \ \ \ \ x\in C^{0}\smallsetminus G_{C},\\\
\mathcal{O}_{{\mathbb{P}^{1}}}(1)\oplus\mathcal{O}_{{\mathbb{P}^{1}}}(-1),&x\in
G_{C},\\\ \mathbf{k}^{2}_{\sigma^{-1}(x)},&\ \ \ \ \ \ \ x\in
Q^{0}\smallsetminus
C^{0},\end{array}\right.\chi(\widetilde{E}^{0}|_{\sigma^{-1}(x)})=2.$

Therefore, applying [13, Prop. 1.13] to the projection $\sigma:X^{0}\to
Q^{0}$, we obtain that

(123) $\overline{E}^{0}|_{Q^{0}\smallsetminus G_{C}}\ \textit{is a locally
free}\ {\mathcal{O}_{Q^{0}\smallsetminus G_{C}}-}\textit{sheaf}.$

and

(124)
$\widetilde{E}^{0}|_{X^{0}\smallsetminus\sigma^{-1}(G_{C})}=\sigma^{*}(\overline{E}^{0}|_{Q^{0}\smallsetminus
G_{C}}),\ \ \ \ \ \ \textrm{Supp}(R^{1}\sigma_{*}\widetilde{E}^{0})\subset
G_{C}.$

Since $x\in Q^{0}$, the fiber $p^{-1}(x)$ is isomorphic to ${\mathbb{P}^{1}}$,
and

(125)
$\mathcal{L}_{0}|_{p^{-1}(x)}\cong\mathcal{O}_{{\mathbb{P}^{1}}}(-s)^{\oplus
r},$

so $p_{*}\mathcal{L}_{0}=0$, and taking into account the relative Serre
duality for the locally trivial ${\mathbb{P}^{1}}$-bundle $p:Y^{0}\to Q^{0}$
we obtain that

(126) $\begin{split}&\mathcal{M}_{0}=R^{1}p_{*}\mathcal{L}_{0}\ \textit{is a
locally free sheaf of rank}\\\
&\operatorname{{rk}}\mathcal{M}_{0}=h^{1}(\mathcal{L}_{0}|_{p^{-1}(x)})=-\chi(\mathcal{L}_{0}|_{p^{-1}(x)})=r(s-1),\
\ \ \ \ \ \ \ \ x\in Q^{0}.\end{split}$

Restricting the exact triple (121) to the fiber $p^{-1}(x)$, we obtain an
exact triple

(127)
$0\to\mathcal{L}_{1}|_{p^{-1}(x)}\to\mathcal{O}_{{\mathbb{P}^{1}}}(-s)^{\oplus
r}\to\widetilde{E}^{0}|_{\sigma^{-1}(x)}\to 0.$

From (122) and (127) it follows that
$\chi(\mathcal{L}_{1}|_{p^{-1}(x)})=\chi(\mathcal{L}_{0}|_{p^{-1}(x)})-\chi(\widetilde{E}^{0}|_{\sigma^{-1}(x)})=-r(s-1)-2$,
and by analogy with (126) we find that

(128) $\begin{split}&\mathcal{M}_{1}=R^{1}p_{*}\mathcal{L}_{1}\ \textit{is a
locally free sheaf of rank}\\\
&\operatorname{{rk}}\mathcal{M}_{1}=h^{1}(\mathcal{L}_{1}|_{p^{-1}(x)})=-\chi(\mathcal{L}_{1}|_{p^{-1}(x)})=r(s-1)+2,\
\ \ \ \ \ \ \ \ x\in Q^{0}.\end{split}$

As a consequence, applying the functor $R^{i}p_{*}$ to (121), we obtain an
exact subsequence

(129) $0\to\overline{E}^{0}\to\mathcal{M}_{1}\to\mathcal{M}_{0}\to\kappa\to
0,\ \ \ \ \ \kappa:=R^{1}\sigma_{*}\widetilde{E}^{0},$

and in view of (124) we have the inclusion

(130) $\textrm{Supp}(\kappa)\subset G_{C}.$

Since $\mathcal{M}_{0}$ and $\mathcal{M}_{1}$ are locally free sheaves on a
smooth three-dimensional variety $Q^{0}$, it follows (see, for example, [16,
Prop. 1.1]) that

(131) $\overline{E}^{0}\ \textit{is a reflexive}\
\mathcal{O}_{Q^{0}}-\textit{sheaf}.$

On the other hand, again due to the local freeness of $\mathcal{M}_{0}$ and
$\mathcal{M}_{1}$, applying to (129) the functor
$\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q^{0}}}^{1}(-,\mathcal{O}_{Q^{0}})$
and using the fact that $\kappa$ is a sheaf of dimension $\leq 0$, we obtain
an isomorphism of sheaves

(132)
$\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q^{0}}}^{1}(\overline{E}^{0},\mathcal{O}_{Q^{0}})\cong\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q^{0}}}^{3}(\kappa,\mathcal{O}_{Q^{0}}).$

Since by virtue of (130) the sheaf $\kappa$ is either zero or an artinian
sheaf, that is $\kappa$ has a finite filtration with factors - residue fields
$\mathbf{k}_{x}$ of points $x\in\textrm{Supp}(\kappa)$. Since
$\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q^{0}}}^{3}(\mathbf{k}_{x},\mathcal{O}_{Q^{0}})\cong\mathbf{k}_{x},\
x\in\textrm{Supp}(\kappa),$ therefore
$\chi(\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q^{0}}}^{3}(\kappa,\mathcal{O}_{Q^{0}}))=\chi(\kappa)$.
From this and from (132) it follows, that

(133)
$\textrm{Supp}(\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q^{0}}}^{1}(\overline{E}^{0},\mathcal{O}_{Q^{0}}))=\textrm{Supp}(\kappa),$
(134)
$d:=\chi(\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q^{0}}}^{1}(\overline{E}^{0},\mathcal{O}_{Q^{0}}))=\chi(\kappa).$

Since $\sigma:\widetilde{X}\smallsetminus S\xrightarrow{\cong}Q\smallsetminus
C$ is an isomorphism and
$\textrm{Sing}\widetilde{E}\subset\widetilde{X}\smallsetminus S$, then,
setting

$\overline{E}:=\sigma_{*}\widetilde{E},$

we have

$\sigma(\textrm{Sing}\widetilde{E})=\textrm{Sing}(\overline{E}|_{Q\smallsetminus
C})\subset\textrm{Sing}\overline{E}.$

We also obtain an isomorphism of artinian sheaves

(135) $\sigma_{*}:\
\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{\widetilde{X}}}^{1}(\widetilde{E},\mathcal{O}_{\widetilde{X}})\xrightarrow{\cong}\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q}}^{1}(\overline{E},\mathcal{O}_{Q})|_{Q\smallsetminus
C},$

where

(136)
$\textrm{Supp}(\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q}}^{1}(\overline{E},\mathcal{O}_{Q})|_{Q\smallsetminus
C})=\textrm{Sing}(\overline{E}|_{Q\smallsetminus C}).$

Moreover, setting $c_{3}:=c_{3}(E)$, we have

(137)
$c_{3}=c_{3}(\widetilde{E})=\chi(\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{\widetilde{X}}}^{1}(\widetilde{E},\mathcal{O}_{\widetilde{X}}))=\chi(\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q}}^{1}(\overline{E},\mathcal{O}_{Q})|_{Q\smallsetminus
C}).$

Here in (137) the first equality follows from (114), the second equality is
proven in [16, Prop. 2.6], and the third equality follows from (135).

Put

(138) $Q_{\xi}:=Q\smallsetminus
D_{\xi}=Q^{0}\cup\operatorname{Sing}\overline{E},\ \ \
X_{\xi}:=\sigma^{-1}(Q_{\xi}),\ \ \ \ \ \
\widetilde{E}_{\xi}:=\widetilde{E}|_{X_{\xi}},\ \ \
\overline{E}_{\xi}:=\sigma_{*}\widetilde{E}_{\xi}.$

By construction, $\overline{E}^{0}=\overline{E}_{\xi}|_{Q^{0}}$, therefore
from (117), (123), (131) and (138) follows, that $\overline{E}_{\xi}$ is a
reflexive sheaf, locally free at the points of the curve
$C_{\xi}=C\smallsetminus D_{\xi}$, in particular, at the points of the divisor
$D_{\xi^{\prime}}$ on $C$:

(139) $\overline{E}_{\xi}\ \textit{is a reflexive sheaf, locally free at the
points of}\ D_{\xi^{\prime}}.$

Taking now $\xi^{\prime}$ instead of $\xi$ and assuming by analogy with (138)

(140) $Q_{\xi^{\prime}}:=Q\smallsetminus D_{\xi^{\prime}},\
X_{\xi^{\prime}}:=\sigma^{-1}(Q_{\xi^{\prime}}),\ \ \ \ \ \
\widetilde{E}_{\xi^{\prime}}:=\widetilde{E}|_{X_{\xi^{\prime}}},\ \ \
\overline{E}_{\xi^{\prime}}:=\sigma_{*}\widetilde{E}_{\xi^{\prime}},$

we get similarly to (139):

(141) $\overline{E}_{\xi^{\prime}}\ \textit{is a reflexive sheaf, locally free
at the points of}\ D_{\xi}.$

Consider the sheaf

$\overline{E}=\sigma_{*}\widetilde{E}.$

From (116) we have:

(142) $H^{0}(\overline{E})=0.$

Note that due to (138) and (140) $E_{\xi}=\overline{E}|_{Q_{\xi}}$,
$E_{\xi^{\prime}}=\overline{E}|_{Q_{\xi^{\prime}}}$. From this and from (139)
and (141) we find that $\overline{E}$ is a reflexive sheaf. Moreover, for any
line $l^{\prime}$ on $X$ that does not intersect a line $l$ we have an
isomorphism
$\beta:=(\sigma\circ\delta^{-1}|_{l}):l^{\prime}\xrightarrow{\sim}m:=\beta(l^{\prime})$
such that $E|_{l^{\prime}}\cong\overline{E}|_{m}$, hence
$c_{1}(\overline{E})=c_{1}(\overline{E}|_{m})=c_{1}(E|_{l^{\prime}})=0$. Thus,
by virtue of (142) $E$ is stable. So,

(143) $\overline{E}\ \textit{is a stable reflexive sheaf with}\
c_{1}(\overline{E})=0.$

Note also that from (130), (138), (140) and the fact that
$\overline{E}^{0}=\overline{E}|_{Q^{0}}$, it follows that
$\kappa=R^{1}\sigma_{*}\widetilde{E}^{0}=R^{1}\sigma_{*}\widetilde{E}$, hence

(144)
$\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q}}^{1}(\overline{E},\mathcal{O}_{Q})|_{Q_{0}}=\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q^{0}}}^{1}(\overline{E}^{0},\mathcal{O}_{Q^{0}})$

and due to (133)
$\textrm{Supp}(\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q^{0}}}^{1}(\overline{E}^{0},\mathcal{O}_{Q^{0}}))\subset
G_{C}$. Since $G_{C}\cap(Q\smallsetminus C)=\varnothing$, then from (144) the
equality follows:

(145)
$\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q}}^{1}(\overline{E},\mathcal{O}_{Q})=\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q^{0}}}^{1}(\overline{E}^{0},\mathcal{O}_{Q^{0}})\oplus\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q}}^{1}(\overline{E},\mathcal{O}_{Q})|_{Q\smallsetminus
C}.$

Note that due to [16, Prop. 2.6] and (143) we have
$\overline{c}_{3}:=c_{3}(\overline{E})=\chi(\mathop{\mathcal{E}xt}\nolimits_{\mathcal{O}_{Q}}^{1}(\overline{E},\mathcal{O}_{Q}))$,
therefore (134), (137), (145) and the equality
$\kappa=R^{1}\sigma_{*}\widetilde{E}$ give:

(146) $\overline{c}_{3}=d+c_{3},\ \ \ \ \ \
d=\chi(R^{1}\sigma_{*}\widetilde{E}).$

Since, obviously, $R^{i}\sigma_{*}\widetilde{E}=0,\ i\geq 2,$ then the Leray
spectral sequence for the morphism $\sigma:\widetilde{X}\to Q$ gives
$\chi(\widetilde{E})=\chi(\overline{E})-\chi(R^{1}\sigma_{*}\widetilde{E})$.
Hence, taking into account (146) and (115) we have:

(147)
$\chi(\overline{E}))=\chi(E)+\chi(R^{1}\sigma_{*}\widetilde{E})=\chi(E)+d.$

Denote

(148) $c_{2}:=c_{2}(E),\ \ \ \ \ \overline{c}_{2}:=c_{2}(\overline{E}).$

Recall the general Riemann–Roch formula for an arbitrary coherent sheaf
$\mathcal{E}$ of rank $r$ with Chern classes $\tilde{c}_{1},\ \tilde{c}_{2},\
\tilde{c}_{3}$ on a smooth projective three-dimensional Fano variety $X$ with
canonical class $\omega_{X}$:

(149)
$\chi(\mathcal{E})=r\chi(\mathcal{O}_{X})+\frac{1}{6}\tilde{c}_{1}^{3}+\frac{1}{2}(\tilde{c}_{3}-\tilde{c}_{1}\tilde{c}_{2})-\frac{1}{4}\omega_{X}(\tilde{c}_{1}^{2}-2\tilde{c}_{2})+\frac{\tilde{c}_{1}}{12}(c_{2}(\Omega_{X})+\omega_{X}^{2}).$

Applying this formula to the sheaf $E$ on $X=X_{5}$ (see (112) and,
respectively, to the sheaf $\overline{E}$ on $X=Q$, we get:

(150) $\chi(E)=2+\frac{1}{2}c_{3}-c_{2},$ (151)
$\chi(\overline{E})=2+\frac{1}{2}\overline{c}_{3}-\frac{3}{2}\overline{c}_{2}.$

In addition, according to Theorem 3.1.(4) we have

(152)
$\overline{c}_{3}\leq\left\\{\begin{array}[]{cc}\frac{1}{2}\overline{c}_{2}^{2}\
\ \ ,&\text{if}\ c_{2}\ \text{is even},\\\
\frac{1}{2}(\overline{c}_{2}^{2}+1),&\ \ \ \text{if}\ c_{2}\ \text{is
odd}.\end{array}\right.$

From (147), (150) and (151) we find:
$\overline{c}_{2}=\frac{2}{3}c_{2}-\frac{1}{3}d$. Substituting this relation
and the equality $c_{3}=\overline{c}_{3}-d$, following from (146), in (153),
we obtain the inequality

(153)
$c_{3}=\overline{c}_{3}-d\leq\left\\{\begin{array}[]{cc}\frac{1}{2}(\frac{2}{3}c_{2}-\frac{1}{3}d)^{2}-d,&\text{if}\
c_{2}\ \text{is even},\\\
\frac{1}{2}((\frac{2}{3}c_{2}-\frac{1}{3}d)^{2}+1)-d,&\ \ \ \text{if}\ c_{2}\
\text{is odd}.\end{array}\right.$

Since $\overline{c}_{2}=\frac{2}{3}c_{2}-\frac{1}{3}d\geq 0$ and $d\geq 0$,
then, assuming in (153) $d=0$, we obtain the main result of this subsection —
the following theorem.

######  Theorem 6.1.

Let $E$ be a rank 2 stable reflexive sheaf of general type with $c_{1}(E)=0$,
$c_{2}=c_{2}(E)>0$, $c_{3}=c_{3}(E)$ on the variety $X=X_{5}$. Then the
following inequalities are valid:

(154) $c_{3}\leq\left\\{\begin{array}[]{cc}\frac{2}{9}c_{2}^{2},&\text{if}\
c_{2}\ \text{is even},\\\ \frac{2}{9}c_{2}^{2}+\frac{1}{2},&\ \ \ \text{if}\
c_{2}\ \text{is odd}.\end{array}\right.$

###  6.2. Rank two stable reflexive sheaves of general type on $X_{4}$

In this subsection everywhere $X=X_{4}$.

Let us list some known facts about the variety $X$ (see, for example, [1,
Theor. 4.2(iii)]).
1) The base $B=B(X)$ of the family of lines on $X$ is isomorphic to the
Jacobian $J(C)$ of a smooth curve $C$ of genus 2.
2) For an arbitrary line $l\in B$ the set $B_{l}=\\{l^{\prime}\in B\ |\
l^{\prime}\cap l\neq\varnothing\\}$ is a curve in $B$ isomorphic to the curve
$C$.
3) For an arbitrary line $l\in B$ or
$N_{l/X}\cong\mathcal{O}_{{\mathbb{P}^{1}}}^{\oplus 2}$ (general case), or
$N_{l/X}\cong\mathcal{O}_{{\mathbb{P}^{1}}}(1)\oplus\mathcal{O}_{{\mathbb{P}^{1}}}(-1)$.
4) Let $\varphi:\ X\dasharrow{\mathbb{P}^{4}}$ be the linear projection of the
variety $X\subset{\mathbb{P}^{5}}$ to the space ${\mathbb{P}^{3}}$ from a line
$l\subset X$ for which $N_{l/X}\cong\mathcal{O}_{{\mathbb{P}^{1}}}^{\oplus
2}$. Then $\varphi(X)={\mathbb{P}^{3}}$, the morphism
$\varphi:X\dasharrow{\mathbb{P}^{3}}$ is birational and decomposes into a
Hironaka’s roof

(155)
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widetilde{X}}$$\scriptstyle{\delta}$$\scriptstyle{\sigma}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{Q.}$

Here $\delta^{-1}:\ X\dasharrow\widetilde{X}$ is the blow-up of $X$ centered
at $l$, $S=\delta^{-1}(l)\simeq{\mathbb{P}^{1}}\times{\mathbb{P}^{1}}$,
$\sigma|_{S}:S\xrightarrow{\simeq}\sigma(S)=Q_{2}$ is an isomorphism of $S$
onto a smooth two-dimensional quadric $Q_{2}\subset{\mathbb{P}^{3}}$, and
$\sigma^{-1}:\ Q\dasharrow\widetilde{X}$ is the blow-up of $Q$ centered in a
smooth curve $C$ of type (2,3) on $Q_{2}$, so that
$\widetilde{X}\simeq{\mathbb{P}^{(}}\mathcal{I}_{C,{\mathbb{P}^{3}}})$, which
is isomorphic to the curve $C$ from the statement 1) above. In particular, the
triple $0\to N_{C,Q_{2}}\to N_{C/{\mathbb{P}^{3}}}\to
N_{Q_{2}/{\mathbb{P}^{3}}}|_{C}\to 0$ is exact, where, due to the isomorphism
$Q_{2}\simeq{\mathbb{P}^{1}}\times{\mathbb{P}^{1}}$, we obtain
$N_{C/Q_{2}}\cong\mathcal{O}_{C}(D_{I})$, $\deg D_{I}=12$,
$N_{Q_{2}/{\mathbb{P}^{3}}}|_{C}\cong\mathcal{O}_{{\mathbb{P}^{3}}}(2H)|_{C}\cong\mathcal{O}_{C}(D_{II})$,
$\deg D_{II}=10$, hence $\det
N_{C/{\mathbb{P}^{3}}}\cong\mathcal{O}_{C}(D_{I}+D_{II})$. Note that the
divisor $D_{I}+D_{II}$ on $C$ is very ample, since $C$ is a curve of genus 2.
5) Let $\Gamma=\\{(x,l)\in X\times B\ |\ x\in l\\}$ be an incidence graph with
projections $X\xleftarrow{p_{1}}\Gamma\xrightarrow{p_{2}}B$. It is known [1,
Prop. 5.2] that $p_{1}:\Gamma\to X$ is a finite at a general point morphism
and $\dim\\{x\in X\ |\ \dim p_{1}^{-1}(x)\geq 1\\}\leq 0$.

Consider an extension

(156) $\xi:\ 0\to\mathcal{O}_{{\mathbb{P}^{3}}}(nH)\to
F\xrightarrow{\varepsilon}\mathcal{I}_{C,{\mathbb{P}^{3}}}\to 0.$

The sheaf
$\mathop{\mathcal{H}om}\nolimits(\mathcal{I}_{C,{\mathbb{P}^{3}}},\mathcal{O}_{{\mathbb{P}^{3}}}(nH))\cong\mathcal{O}_{{\mathbb{P}^{3}}}(nH)$
on ${\mathbb{P}^{3}}$ and the sheaf
$\mathcal{M}=\mathop{\mathcal{E}xt}\nolimits^{1}(\mathcal{I}_{C,{\mathbb{P}^{3}}},\\\
\mathcal{O}_{{\mathbb{P}^{3}}}(nH))\cong\mathop{\mathcal{E}xt}\nolimits^{2}(\mathcal{O}_{C},\mathcal{O}_{{\mathbb{P}^{3}}}(nH))\cong\det
N_{C/{\mathbb{P}^{3}}}\otimes\mathcal{O}_{{\mathbb{P}^{3}}}(nH)\cong\mathcal{O}_{C}(D_{I}+D_{II}+nH)$
on $C$ with $n\gg 0$ are very ample. Therefore, we can assume that:
1)
$H^{i}(\mathop{\mathcal{H}om}\nolimits(\mathcal{I}_{C,{\mathbb{P}^{3}}},\mathcal{O}_{{\mathbb{P}^{3}}}(nH)))=0,\
i=1,2$, which means that the long exact sequence of
$\operatorname{Ext}\nolimits$-groups for the pair of sheaves
$\mathcal{I}_{C,{\mathbb{P}^{3}}},\mathcal{O}_{{\mathbb{P}^{3}}}(nH)$ gives
isomorphisms
$\operatorname{Ext}\nolimits^{1}(\mathop{\mathcal{H}om}\nolimits(\mathcal{I}_{C,{\mathbb{P}^{3}}},\mathcal{O}_{{\mathbb{P}^{3}}}(nH))\cong
H^{0}(\mathop{\mathcal{E}xt}\nolimits^{1}(\mathcal{I}_{C,{\mathbb{P}^{3}}},\mathcal{O}_{{\mathbb{P}^{3}}}(nH))\cong
H^{0}(\mathop{\mathcal{E}xt}\nolimits^{2}(\mathcal{O}_{C},\mathcal{O}_{{\mathbb{P}^{3}}}(nH))=H^{0}(\mathcal{M})$;
2) the triple (156) as an extension
$\xi\in\operatorname{Ext}\nolimits^{1}(\mathop{\mathcal{H}om}\nolimits(\mathcal{I}_{C,{\mathbb{P}^{3}}},\mathcal{O}_{Q}(nH))=H^{0}(\mathcal{M})$
is a section of the sheaf $\mathcal{M}$, the scheme of zeros $(\xi)_{0}$ of
which is reduced, i.e. it is a simple divisor $D_{\xi}=x_{1}+...+x_{5n+22}$.
This means by Serre’s construction (see [16, Section 4]) that in (156) the
sheaf $F$ is a reflexive sheaf of rank two with $\operatorname{Sing}F=D_{\xi}$
and has the simplest singularities in the points $x_{i}$, i.e.
$\mathop{\mathcal{E}xt}\nolimits^{1}(F,\mathcal{O}_{Q})\cong\oplus_{i=1}^{3n+6}\mathbf{k}_{x_{i}}$.
Therefore, according to [4, Cor. 2.8], the projective spectrum
$Y=\mathbb{P}(F)$ of the sheaf $F$ is a smooth variety. The epimorphism
$\varepsilon$ in (156) corresponds to an embedding $i$ of the divisor
$\widetilde{X}=\mathbb{P}(\mathcal{I}_{C,{\mathbb{P}^{3}}})$ into $Y$, and let
$p:Y\to{\mathbb{P}^{3}}$ be the natural projection, so $\sigma=p\circ i$. Note
also that the sheaf

(157) $L=\mathcal{O}_{Y/{\mathbb{P}^{3}}}(1)\otimes p^{*}\mathcal{O}(mH)$

is ample for $m\gg 0$, where $\mathcal{O}_{Y/{\mathbb{P}^{3}}}(1)$ is the
Grothendieck sheaf on $Y=\mathbb{P}(F)$.

Let

$\begin{split}&E\ \textit{be a rank 2 stable reflexive sheaf of general type
with}\ c_{1}(E)=0\\\ &\textit{on}\ X_{4};\ \textit{in particular}\
H^{0}(E)=0.\end{split}$

Repeating for the sheaf $E$ verbatim all the arguments from (113) – (153) with
$Q$ and $Q^{0}$ replaced by ${\mathbb{P}^{3}}$ and $({\mathbb{P}^{3}})^{0}$
respectively, and using Hartshornes result for stable reflexive sheaves on
${\mathbb{P}^{3}}$ (see [16, Thm. 8.2(b)]), we obtain the following analogue
of theorem 6.1 for $X=X_{4}$.

######  Theorem 6.2.

Let $E$ be a rank 2 stable reflexive sheaf of general type with $c_{1}(E)=0$,
$c_{2}=c_{2}(E)>0$, $c_{3}=c_{3}(E)$ on the variety $X=X_{4}$. Then the
following inequality is true:

(158) $c_{3}\leq c_{2}^{2}-c_{2}+2.$

## References

  * [1] Iskovskih, V. A. (1977), Fano threefolds. I, Math. USSR Izv., 11 (3): 485527.
  * [2] Almeida C., Jardim M., Tikhomirov A. S. Irreducible components of the moduli space of rank 2 sheaves of odd determinant on projective space. Advances in Mathematics. 402 (2022). Article 108363. https://doi.org/10.1016/j.aim.2022.108363
  * [3] V. Antonelli, G. Casnati, O. Genc. Even and odd instanton bundles on Fano threefolds. Asian J. Math. 26, No. 1, (2022), 081–118.
  * [4] E. Ballico, J. A. Wisniewski. On Bănică sheaves and Fano manifolds. Compositio mathematica. t. 102, no. 3 (1996), 313–335.
  * [5] C. Banica, M. Putinar, G. Schumacher. Variation der globalen Ext in Deformationen kompakter komplexer Räume. Math. Ann., 250 (1980), 135–155.
  * [6] W. Barth, K. Hulek. Monads and moduli of vector bundles. Manuscr. Math. 25 (1978), 323–347.
  * [7] A. Bayer, E. Macri, Y. Toda. Bridgeland Stability conditions on threefolds I: Bogomolov-Gieseker type inequalities. J. Algebraic Geom. 23 (2014), 117–163.
  * [8] T. Bridgeland. Stability conditions on $K3$ surfaces. Duke Math. J., 141(2) (2008), 241–291.
  * [9] G. Comaschi, M. Jardim, C. Martinez, D. Mu. Instanton sheaves: the next frontier. São Paulo J. Math. Sci. (2023). https://doi.org/10.1007/s40863-023-00387-3
  * [10] L. Costa, R. M. Miró-Roig. Monads and instanton bundles on smooth hyperquadrics. Mathematische Nachrichten, 282(2) (2009), 169–179.
  * [11] L. Ein, I. Sols. Stable vector bundles on quadric hypersurfaces. Nagoya Math. J. 96 (1984), 11–22.
  * [12] L. Ein. Generalized null correlation bundles. Nagoya Math. J. 111, (1988), 13–24.
  * [13] G. Ellingsrud and S.A. Strømme. Stable rank 2 vector bundles on ${\mathbb{P}^{3}}$ with $c_{1}=0$ and $c_{2}=3$. Math. Ann. 255(1) (1981), 123–135.
  * [14] D. Faenzi. Even and odd instanton bundles on Fano threefolds of Picard number one. manuscripta math. 144 (2014), 199–239.
  * [15] R. Hartshorne. Stable vector bundles. Math. Ann. 238 (1978), 229–280.
  * [16] R. Hartshorne. Stable reflexive sheaves. Math. Ann. 254 (1980), 121–176.
  * [17] V. A. Iskovskikh and Yu. G. Prokhorov. Fano varieties. Algebraic geometry. V., volume 47 of Encyclopaedia Math. Sci. Springer, Berlin, 1999.
  * [18] M. Jardim, D. Markushevich, A. S. Tikhomirov. Two infinite series of moduli spaces of rank 2 sheaves on $\mathbb{P}^{3}$. Annali di Matem. Pura ed Appl. 196. No. 4 (2017), 1573–1608.
  * [19] A. Kuznetsov. Instanton bundles on Fano threefolds. Cent. Eur. J. Math., 10(4) (2012), 1198–1231.
  * [20] A. A. Kytmanov, A. S. Tikhomirov, S. A. Tikhomirov. Series of rational moduli components of stable rank two vector bundles on P3. Selecta Mathematica, New Series. Vol. 25 (2019), 1–47.
  * [21] H. Lange. Universal families of extensions. J. Algebra, 83(1) (1983), 101–112.
  * [22] Emanuele Macrì, Benjamin Schmidt. Derived categories and the genus of space curves. Algebraic Geometry 7(2) (2020), 153–191.
  * [23] Miro-Roig R. M., Trautmann G. The moduli scheme M(0,2,4) over $\mathbb{P}^{3}$. Math. Zeitschrift, 216(2) (1994), 283–316.
  * [24] Ch. Okonek, M. Schneider and H. Spindler, Vector bundles on complex projective spaces. Progr. Math., vol. 3, Birkhaäuser, Boston, Mass. 1980, vii+389 pp.
  * [25] G. Ottaviani. Some extensions of Horrocks criterion to vector bundles on grassmannians and quadrics. Annali di Matem. Pura ed Appl. 155 (1989), 317–341.
  * [26] A. P. Rao. A note on cohomology modules of rank two bundles. J. Algebra 86, 2334 (1984).
  * [27] B. Schmidt. A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold. Bull. Lond. Math. Soc. 46, no. 5, (2014), 915–923.
  * [28] B. Schmidt. Rank two sheaves with maximal third Chern character in three-dimensional projective space. Matematica Contemporanea, Vol. 47 (2018), 228–270.
  * [29] B. Schmidt. Bridgeland Stability on Threefolds – Some Wall Crossings. J. Algebraic Geom. 29 (2020), 247–283.
  * [30] B. Schmidt. Sheaves of low rank in three-dimensional projective space. Eur. J. Math. 9:103 (2023), 3-71.
  * [31] I. R. Shafarevich. Book 1, Basic algebraic geometry. 3rd ed., Springer, Heidelberg, 2013.
  * [32] V. K. Vedernikov. Moduli of stable vector bundles of rank 2 on P3 with fixed spectrum. Math. USSR–Izv. 25 (1985), 301–313.
  * [33] V. K. Vedernikov. The moduli of super-null-correlation bundles on ${\bf P}_{3}$. Math. Ann. 276 (1987), 365–383.

Alexander S. Tikhomirov
Faculty of Mathematics
National Research University
Higher School of Economics
Usacheva St., 6
119048 Moscow, Russia
E-mail<EMAIL_ADDRESS>
Danil A. Vassiliev
Faculty of Mathematics
National Research University
Higher School of Economics
Usacheva St., 6
119048 Moscow, Russia
E-mail<EMAIL_ADDRESS>