# Moduli of Rank Two Semistable Sheaves on Rational Fano Threefolds of the
Main Series

###### Abstract.

In this paper we investigate the moduli spaces of semistable coherent sheaves
of rank two on the projective space ${\mathbb{P}^{3}}$ and the following
rational Fano manifolds of the main series - the three-dimensional quadric
$X_{2}$, the intersection of two 4-dimensional quadrics $X_{4}$ and the Fano
manifold $X_{5}$ of degree 5. For the quadric $X_{2}$, the boundedness of the
third Chern class $c_{3}$ of rank two semistable objects in
$\mathrm{D}^{b}(X_{2})$, including sheaves, is proved. An explicit description
is given of all the moduli spaces of semistable sheaves of rank two on
$X_{2}$, including reflexive ones, with a maximal third class $c_{3}\geq 0$.
These spaces turn out to be irreducible smooth rational manifolds in all
cases, except for the following two: $(c_{1},c_{2},c_{3})=(0,2,2)$ or (0,4,8).
The first example of a disconnected module space of semistable rank two
sheaves with fixed Chern classes on a smooth projective variety is found –
this is the second of these exceptional cases $(c_{1},c_{2},c_{3})=(0,4,8)$ on
the quadric $X_{2}$. Several new infinite series of rational components of the
moduli spaces of semistable sheaves of rank two on ${\mathbb{P}^{3}}$,
$X_{2}$, $X_{4}$ and $X_{5}$ are constructed, as well as a new infinite series
of irrational components on $X_{4}$. The boundedness of the class $c_{3}$ is
proved for $c_{1}=0$ and any $c_{2}>0$ for stable reflexive sheaves of general
type on manifolds $X_{4}$ and $X_{5}$.

Alexander S. Tikhomirov, Danil A. Vassiliev

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Semistable objects of rank two in $D^{b}(X_{2})$ with maximal third Chern class $c_{3}$
  4. 4 Infinite series of components of moduli spaces of semistable sheaves of rank two on varieties $X_{1}$, $X_{2}$, $X_{4}$ and $X_{5}$
  5. 5 Moduli spaces of rank two semistable sheaves with maximal third Chern class on the three-dimensional quadric $X_{2}$
  6. 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 $X_{5}$

##  1\. Introduction

In 1980, R. Hartshorne, while investigating in [16] the spectra of stable
reflexive coherent sheaves of rank two on the projective space
$\mathbb{P}^{3}$, proved the boundedness of the third Chern class $c_{3}$ of
these sheaves for fixed first and second Chern classes $c_{1}$ and $c_{2}$.
The exact estimates he obtained for the class $c_{3}$ have the form (see [16,
Thm. 8.2])

(1) $c_{3}\leq c^{2}_{2}-c_{2}+2,\ \ \ \text{if}\ \ \ c_{1}=0;\ \ \ \ \ \ \ \
\ \ c_{3}\leq c^{2}_{2}\ \ \ \text{if}\ \ \ c_{1}=-1.$

In the same work, the irreducibility, smoothness and rationality of the moduli
spaces of such sheaves with $c_{1}=-1$, arbitrary $c_{2}>0$ and maximum
$c_{3}=c^{2}_{2}$ are proved. In 2018, B. Schmidt in [28], investigating the
properties of tilt stability in the derived category
$\mathrm{D}^{b}(\mathbb{P}^{3})$, proved that the estimates (1) are true for
all semistable sheaves of rank two on $\mathbb{P}^{3}$, and gave an explicit
description of their moduli space for $-1\leq c_{1}\leq 0$, $c_{2}>0$ and
maximal $c_{3}$. As a consequence, he obtained that these spaces are
irreducible smooth rational projective varieties. It is not difficult to see
that the moduli spaces of reflexive sheaves described by Hartshorne are open
subsets of these manifolds (see Theorem 4 below). We also note that quite
recently in the 2023 work [30], Schmidt generalized the above results to the
case of sheaves on $\mathbb{P}^{3}$ of all ranks from 0 to 4.

In this paper, we study the moduli spaces of semistable rank two sheaves on
rational three-dimensional Fano varieties of the main series. There are four
such varieties — these are the projective space $X_{1}=\mathbb{P}^{3}$, the
three-dimensional quadric $X_{2}$, the complete intersection $X_{4}$ of two
quadrics in the space $\mathbb{P}^{5}$, and the section $X_{5}$ of the
grassmannian $\mathrm{Gr}(2,5)$ embedded by Plücker in the space
$\mathbb{P}^{9}$ by a linear subspace $\mathbb{P}^{6}$ (see, e. g., [1] or
[17]). Here the subscript $i$ of the variety $X_{i}$ is its projective degree.
Chern classes of bundles $E$ on $X=X_{i}$ are defined by the integers
$c_{1},c_{2},c_{3}$ (see formulas below), and the corresponding moduli spaces
(Gieseker-Maruyama moduli schemes) of semistable bundles of rank two on $X$
will be denoted by $M_{X}(2;c_{1},c_{2},c_{3})$.

The first direction of research in this paper concerns the question of
boundedness of the third Chern class $c_{3}$ of semistable rank 2 sheaves on
$X$ with fixed $c_{1}\in\\{-1,0\\}$ and $c_{2}\geq 0$ and of getting estimates
for the third Chern class $c_{3}$. Using the tilt stability technique in the
derived category $\mathrm{D}^{b}(X)$, we give a complete answer to this
question for the three-dimensional quadric $X_{2}$ in the following theorem
(see the statements (3.1)-(4.2) in Theorem 3.1 of the paper).

Theorem 1. (i) Let $E$ be a semistable sheaf of rank 2 with $c_{1}=-1$ on the
quadric $X_{2}$. Then $c_{2}\geq 0$ and $c_{3}\leq\frac{1}{2}c_{2}^{2}$ if
$c_{2}$ is even, and, respectively, $c_{3}\leq\frac{1}{2}(c_{2}^{2}-1)$ if
$c_{2}$ is odd.
(ii) Let $E$ be a semistable sheaf of rank 2 on $X_{2}$ with $c_{1}(E)=0$.
Then $c_{2}\geq 0$ and $c_{3}\leq\frac{1}{2}c_{2}^{2}$, if $c_{2}$ is even,
and, respectively, $c_{3}\leq\frac{1}{2}(c_{2}^{2}+1)$, if $c_{2}$ is odd.
These estimates are exact for all $c_{3}\geq 0$.

The proof of this theorem is based on the study of the relationship between
tilt semistability and Bridgland semistability in $\mathrm{D}^{b}(X_{2})$. The
key here is Schmidt’s important technical result (2014) on the description of
a subcategory in $\mathrm{D}^{b}(X_{2})$ generated by a torsion pair in the
sense of Bridgeland - see Proposition 2.4 of this paper.

Unfortunately, no analogues of this result are known to date for varieties
$X_{4}$ and $X_{5}$. Therefore, for these varieties it is not possible to use
the same method to obtain exact upper bounds for the class $c_{3}$ for all
semistable sheaves of rank 2 on $X_{4}$ and $X_{5}$. However, using more
traditional technique considering the behavior of stable sheaves at standard
birational transformations $X_{4}\dashrightarrow X_{1}$ and
$X_{5}\dashrightarrow X_{2}$, we give a partial answer to the question about
boundedness of $c_{3}$ for a sufficiently wide class of sheaves on $X_{4}$ and
$X_{5}$.

Namely, we consider stable reflexive sheaves of rank 2 with $c_{1}=0$, called
in this paper sheaves of general type. These sheaves $E$ on $X_{i},\ i=4.5$,
are such that $E|_{l}\cong\mathcal{O}_{\mathbb{P}^{1}}^{\oplus 2}$ for a
general line $l\subset X_{i}$, and lines for which either
$E|_{l}\cong\mathcal{O}_{\mathbb{P}^{1}}(a)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-a)$
with $a\geq 2$, or $E|_{l}$ is not locally free, constitute a subset of
dimension $\leq 0$ in the base of the family of lines on $X_{i}$ (see
definition 4.10 of this paper). In Theorem 4.11 we give examples of infinite
series of components of moduli spaces of semistable sheaves in which the
general sheaf is a reflexive sheaf of general type. (Presumably the property
of being a sheaf of general type is true for all stable reflexive sheaves of
rank 2 with $c_{1}=0$, that is, perhaps an analogue of the Grauert-Mülich
Theorem holds for them, which is known to be valid for stable reflexive
sheaves of rank 2 on $X_{1}$.) For sheaves of general type we prove the
following theorem (see Theorems 6.2 and 6.1 of this paper).

Theorem 2. Let $E$ be a stable reflexive sheaf of rank 2 of general type with
Chern classes $c_{1}=0$, $c_{2}>0$, $c_{3}$ on the variety $X_{4}$ or $X_{5}$.
Then the following inequalities are true for the class $c_{3}$ of the sheaf
$E$.
(i) On $X_{4}$: $c_{3}\leq c_{2}^{2}-c_{2}+2$.
(ii) On $X_{5}$: $c_{3}\leq\frac{2}{9}c_{2}^{2}$ if $c_{2}$ is even, and,
respectively, $c_{3}\leq\frac{2}{9}c_{2}^{2}+\frac{1}{2}$, if $c_{2}$ is odd.
Whether these estimates are sharp is an open question.

The second direction of research in this paper is the construction of new
infinite series (with growing class $c_{2}$) of moduli components of
semistable sheaves of rank two on the varieties $X_{1}$, $X_{2}$, $X_{4}$ and
$X_{5}$, including an explicit description of general111Here and everywhere
below under a general sheaf in a flat irreducible family of sheaves on $X$
with base $B$ we understand a sheaf corresponding to a point in $B$ belonging
to a dense open subset in $B$. sheaves in these components. First two infinite
series of such components on $X_{1}=\mathbb{P}^{3}$ were constructed R.
Hartshorne in [15] in 1978 for vector bundles (this is a series of instanton
components with $c_{1}=0$ and a parallel series components with $c_{1}=-1$)
and in the work [16] in 1980 for reflexive bundles with $c_{1}=-1$ and maximal
$c_{3}$, and with $c_{1}=0$ and maximal spectrum. In 1978, W. Barth and K.
Hulek [6] built another infinite series of families of stable vector bundles
of rank 2 on $\mathbb{P}^{3}$, and in 1981 G. Ellingsrud and S. A. Strømme
proved [13], that these families are open subsets of irreducible components of
moduli spaces. In [32] (1985) and [33] (1987) V. K. Vedernikov constructed new
infinite series of families of stable vector bundles of rank 2 on
$\mathbb{P}^{3}$, and in 1984 A. P. Rao [26] built a more general series of
families, including the Vedernikov series. L. Ein independently described
these families in 1988 and proved that they are open subsets of irreducible
components of moduli spaces. In 2019 A. A. Kytmanov, A. S. Tikhomirov and S.
A. Tikhomirov in the paper [20] proved the rationality of significant part of
the components of these series.

A large number of new infinite series of components of moduli spaces of
semistable sheaves of rank 2 on $\mathbb{P}^{3}$ were found in the work of M.
Jardim, D. Markushevich and A. S. Tikhomirov [18] in 2017 and in the work of
C. Almeida, M. Jardim and A. S. Tikhomirov [2] in 2022, and general sheaves in
these components were described. B. Schmidt in the above work [28] gave a
complete description of all schemes of stable sheaves of rank 2 with maximal
Chern class $c_{3}$, proving their projectivity, irreducibility for any
admissible values of the Chern classes $c_{1}$ and $c_{2}$, rationality and in
almost all cases smoothness. Note that the last property of smoothness is a
rather unexpected phenomenon for projective Gieseker-Maruyama moduli schemes
of semistable sheaves of rank 2 on three-dimensional varieties.

As for the varieties $X_{2}$, $X_{4}$ and $X_{5}$, by now on each of them only
one infinite series of moduli components of semistable sheaves of rank 2 was
found. These are the series of components containing as open sets the families
of instanton bundles. Instantonic bundles on $X_{2}$ were defined by L. Costa
and R. M. Miro-Roig in [10] in 2009, and on $X_{4}$ and $X_{5}$ and other Fano
varieties by A. Kuznetsov [19] in 2012 and D. Faenzi [14] in 2013. In work
[14] D. Faenzi proved that families of instanton bundles on $X_{2}$, $X_{4}$
and $X_{5}$ are indeed open subsets of irreducible components of moduli
spaces, which are reduced at a general point and have the expected dimension.
In recent years, an extensive number of works were devoted to the study of
instanton series of bundles, a review of which can be found, for example, in
[3] and [9].

In this paper we construct several new infinite series of irreducible rational
components of moduli spaces of semistable sheaves of rank 2 on the varieties
$X_{1}$, $X_{2}$, $X_{4}$ and $X_{5}$. We describe general sheaves in these
components and prove their reflexivity, and also find dimensions of the
constructed components. These results are proven in Theorems 4.2, 4.5, 4.6,
4.8 and 4.9. They are collected in the following theorem.

Theorem 3. Let $X$ be one of the varieties $X_{1}$, $X_{2}$, $X_{4}$, $X_{5}$,
and let $\mathcal{O}_{X}(1)$ be the ample sheaf on $X$ such that
$\mathrm{Pic}(X)=\mathbb{Z}[\mathcal{O}_{X}(1)]$. Consider a sheaf $E$ of rank
2 on $X$ defined by one of nontrivial extensions of the form

(2) $0\to F_{i}\to E\to G_{j}\to 0,\ \ \ \ \ 1\leq i\leq 3,\ \ \ 1\leq j\leq
2,$

where $F_{1}=\mathcal{O}_{X}(-n)^{\oplus 2}$, $F_{2}=F(-n)$, where $F$ is a
rank 2 sheaf of one of the types (I)-(III) described in subsection 4.3,
$G_{1}=\mathcal{O}_{S}(m)$, where $S\in|\mathcal{O}_{X}(k)|$, and the sheaves
$F_{3}$ and $G_{2}$ are defined in the case of the quadric $X=X_{2}$, namely,
$F_{3}=\SS(-n)$, where $\SS$ is the spinor bundle on $X_{2}$ with
$\det\SS=\mathcal{O}_{X}(1)$, and $G_{2}=\mathcal{I}_{\mathbb{P}^{1},S}(m)$,
where $S\in|\mathcal{O}_{X}(1)|$, $\mathbb{P}^{1}$ is a line on the surface
$S$. Let $M_{X}(v)$ be the Gieseker-Maruyama moduli scheme of semistable
sheaves on $X$ with Chern character $v=\mathrm{ch}(E)$, determined from the
triple (2), and let

(3) $M:=\\{[E]\in M_{X}(v)\ |\ E\ \textit{is a Gieseker-stable extension}\
\eqref{eqn 1}\\}.$

Then the following statements are true.
1) For $X_{1},\ X_{2},\ X_{4},\ X_{5}$ in the case $i=j=1$, $k\geq 1$,
$n=\lceil\frac{k}{2}\rceil$, $m<-n$,
2) for $X_{1},\ X_{4},\ X_{5}$ in the case of $i=2$, $j=1$, $k\geq 1$,
$n=\lfloor\frac{k}{2}\rfloor$, $m<-n$,
3) for $X_{2}$ in each of the cases
3.1) $i=1$, $j=2$, $n=1$, $m\leq-1$,
3.2) $i=3$, $j=1$, $k\geq 1$, $n=\lfloor\frac{k}{2}\rfloor+1$, $m\leq-n$,
3.3) $i=3$, $j=2$, $1\leq k\leq 2$, $n=1$, $m\leq-1$,
the set $M$ is a smooth dense open subset of an irreducible component
$\overline{M}$ of the moduli scheme $M_{X}(v)$. Moreover, $M$ is a fine moduli
space, and reflexive sheaves form a dense open set in $M$. Moreover, all
components $\overline{M}$ from the infinite series 1), 2) and 3.1)-3.3) are
rational varieties for each of the varieties $X_{l},\ l=1,2,4,5$, except for
the series 2) for $X=X_{4}$, in which each component is irrational. Moreover,
in all cases the dimensions of the components $\overline{M}$ are found as
polynomials from $\mathbb{Q}[k,m,n]$ or $\mathbb{Q}[m]$, respectively.

A significant part of this paper is devoted to the study of semistable sheaves
of rank 2 with maximal class $c_{3}$ on the quadric $X_{2}$. We show that for
$c_{1}\in\\{-1,0\\}$ and all values of the class $c_{2}$, except for a few
small values, every such sheaf is given by an extension of the form (2), that
is, in the notation (3) we have equality $M=\overline{M}$. In this case the
construction from the proof of Theorem 3 allows for a significant refinement,
giving complete description of all moduli spaces of semistable sheaves with
maximal class $c_{3}$ on $X_{2}$. In the regeneraling cases of small values of
$c_{2}$ and maximal $c_{3}\geq 0$ it is also possible to obtain an explicit
description of moduli spaces. (The only case of negative maximal $c_{3}$ — the
case $(c_{1},c_{2},c_{3\mathrm{max}})=(0,1,-1)$ – is specified in Remark 3.2
of the paper.) These results, proven in Theorems 5.1 – 5.4 are collected in
the following two theorems.

Theorem 4. Let $X=X_{2}$ be a quadric, and $M_{X}(v)$ be the Gieseker-Maruyama
moduli scheme of semistable sheaves $E$ of rank 2 on $X$ with Chern classes
$(c_{1},c_{2},c_{3})$, where $c_{1}\in\\{-1,0\\}$, $c_{2}\geq 0$,
$c_{3}=c_{3\mathrm{max}}\geq 0$ is maximal for each $c_{2}$, and

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

where $H=c_{1}(\mathcal{O}_{X}(1))$. Then the following statements hold.
(1.i) For $c_{1}=-1$, even $c_{2}=2p$, $p\geq 2$, and
$c_{3\mathrm{max}}=\frac{1}{2}c_{2}^{2}$ the variety $M_{X}(v)$ is a
grassmannization of 2-dimensional quotient spaces of the vector bundle of rank
$\frac{1}{4}(c_{2}+2)^{2}$ on the space $\mathbb{P}^{4}$ defined by the first
formula (38) for $n=1$ and $m=-p$. In this case $\dim
M_{X}(v)=\frac{1}{2}(c_{2}+2)^{2}$.
(1.ii) For $c_{1}=-1$, 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 of rank
$\frac{1}{4}(c_{2}+1)(c_{2}+3)$ on the grassmannian
$\mathbb{G}=\mathrm{Gr}(2,4)$ defined by the second formula (38) for $m=-p$.
In this case $\dim M_{X}(v)=\frac{1}{2}(c_{2}+1)(c_{2}+3)$.
(1.iii) For $c_{1}=0$, 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 a
projectivization of the vector bundle of rank $\frac{1}{2}(c_{2}+1)(c_{2}+3)$
on the space $\mathbb{P}^{4}$, defined by the formula (61) with $n=1$ and
$m=-p$. In this case $\dim M_{X}(v)=\frac{1}{2}c_{2}^{2}+2c_{2}+\frac{9}{2}$.
(1.iv) For $c_{1}=0$, even $c_{2}=2p$, $p\geq 3$, and
$c_{3\mathrm{max}}=\frac{1}{2}c_{2}^{2}$ the variety $M_{X}(v)$ is a
projectivization of the vector bundles of rank $\frac{1}{2}c_{2}^{2}+2c_{2}+1$
on the grassmannian $\mathbb{G}$, defined by the formula (77) for $m=1-p$. In
this case $\dim M_{X}(v)=\frac{1}{2}c_{2}^{2}+2c_{2}+4$.
(2) In all the above cases, the scheme $M_{X}(v)$ is irreducible and is a
smooth rational projective variety, all sheaves from $M_{X}(v)$ are stable,
the general sheaf in $M_{X}(v)$ is reflexive, and $M_{X}(v)$ is a fine moduli
space.

Theorem 5. Under the conditions and notation of Theorem 4, 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
$M_{X}(v)\simeq G(2,5)$.
(4) For $c_{1}=0$, $c_{2}=2$ and $c_{3\mathrm{max}}=2$ the scheme $M_{X}(v)$
is irreducible, has dimension 9 and is not smooth.
(5) For $c_{1}=0$, $c_{2}=4$ and $c_{3\mathrm{max}}=8$ the scheme
$M_{X}(v)=M_{X}(2;0,4,8)$ is the union of two irreducible components $M_{1}$
and $M_{2}$. These components are described as follows.
(5.i) $M_{1}$ is a smooth rational variety of dimension 21, which is the
projectivization of a locally free sheaf of rank 17 on grassmannian
$\mathbb{G}$. $M_{1}$ is a fine moduli space and all sheaves in $M_{X}(v)_{1}$
are stable. Moreover, the scheme $M_{X}(v)$ is nonsingular along $M_{1}$.
(5.ii) the $M_{2}$ scheme is irreducible, has dimension 21, and polystable
sheaves in $M_{2}$ form a closed subset of dimension 12 in $M_{2}$, in which
the scheme $M_{X}(v)$ is not smooth.

We highlight the last statement (iii) of Theorem 5.4 as a separate theorem due
to its importance.

Theorem 6. For the quadric $X=X_{2}$, the scheme $M_{X}(2,0,4,8)$ is
disconnected:

$M_{X}(2;0,4,8)=M_{1}\sqcup M_{2},$

and its irreducible components $M_{1}$ and $M_{2}$ are described above in the
statements (5.i)-(5.ii) of Theorem 5.

This result gives the first example of a disconnected moduli scheme of
semistable sheaves of rank two on a smooth projective 3-dimensional variety.
In all the few known so far cases in which the problem of connectivity of the
moduli scheme $M_{X}(2;c_{1},c_{2},c_{3})$ with fixed $c_{1},c_{2},c_{3}$ was
discussed, the union of all known components of the moduli scheme turned out
to be connected. In particular, in the paper [18, Thm. 25, Thm. 27] the
connectedness of the scheme $M_{\mathbb{P}^{3}}(2;0,2,0)$ was proved, as well
as the connectedness of the union of seven known by 2017 irreducible
components of the scheme $M_{\mathbb{P}^{3}}(2;0,3,0)$, and in the same paper
[18, Prop. 24] for an arbitrary positive value $n$ work the connectedness of
the union of some growing with $n$ number of known components of
$M_{\mathbb{P}^{3}}(2;0,n,0)$ was proved. In work [2, Main Thm. 3] the
connectedness of the scheme $M_{\mathbb{P}^{3}}(2;-1,2,m)$ for all admissible
positive values of $m$, namely, for work $m=0,\ 2$, 4, was proved. To our
opinion, one of the possible reasons for the disconnectedness of the scheme
$M_{X_{2}}(2,0,4,8)$ in Theorem 6 may be the fact that the quadric $X_{2}$,
unlike $\mathbb{P}^{3}$, is not a toric variety.

Let us proceed on to a brief summary of the contents of the paper. In section
2 we recall necessary technical means to work in derived categories of
coherent sheaves on Fano varieties, which are used in further. In section 3
the key result of the paper is proved — the theorem 3.1 on the boundedness of
the third Chern class $c_{3}$ of semistable objects, including sheaves, in
$\mathrm{D}^{b}(X_{2})$ and their complete classification is given for maximal
values of $c_{3}\geq 0$. In section 4 we construct new infinite series of
components of moduli spaces of rank two semistable sheaves on the varieties
$X_{1},\ X_{2},\ X_{4}$ and $X_{5}$. As a special case of these series, in
section 5 we describe moduli spaces of rank two semistable sheaves with the
maximal third Chern class on the quadric $X_{2}$. At the end of the same
section we prove the disconnectedness of moduli space $M_{X_{2}}(2,0,4,8)$.
Section 6 is dedicated to a proof of boundedness of the class $c_{3}$ of
semistable rank two reflexive sheaves of general type with $c_{1}=0$ on the
varieties $X_{4}$ and $X_{5}$.

### Notation.

$\mathbf{k}=\mathbb{C}$ | base field
---|---
$X_{1}$ | three-dimensional projective space $\mathbb{P}^{3}$
$X_{2}$ | smooth three-dimensional quadric in ${\mathbb{P}^{4}}$
$X_{4}$ | smooth intersection of two four-dimensional quadrics in $\mathbb{P}^{5}$
$X_{5}$ | smooth section of the grassmannian $G(2,5)$ embedded by
| Plücker into the space ${\mathbb{P}^{9}}$, by a linear subspace
$\mathbb{P}^{6}$
$\SS$ | spinor bundle on the quadric $X_{2}$ with $\det\SS\cong\mathcal{O}_{Q}(1)$

$X$ | one of the varieties $X_{1}$, $X_{2}$, $X_{4}$, $X_{5}$
---|---
$H$ | positive generator of the Picard group $\mathrm{Pic}X\simeq\mathbb{Z}$ –
| class of hyperplane section with embedding
$X=X_{i}\hookrightarrow\mathbb{P}^{2+i}$,
|  $i=1,2$, $X_{i}\hookrightarrow\mathbb{P}^{1+i}$, respectively, $i=4,5$
$\mathrm{Coh}(X)$ | category of coherent sheaves on $X$
$[E]$ |  $S$-equivalence class of an $H$-Gieseker-semistable sheaf
| $E\in\mathrm{Coh}(X)$
$\mathrm{D}^{b}(X)$ | bounded derived category of coherent sheaves on $X$
$\mathcal{H}^{i}(E)$ |  $i$th cohomology group of the complex $E\in\mathrm{D}^{b}(X)$
$\ch(E)$ | Chern character of the object $E\in\mathrm{D}^{b}(X)$
$\ch_{\leq m}(E)$ | $(\ch_{0}(E),\ldots,\ch_{m}(E))$
$M_{X}(v)$ | space (scheme) of $H$-Gieseker-semistable
| sheaves $E$ on $X$ with $\ch(E)=v$
$M_{X}(r;c_{1},c_{2},c_{3})$ | alternative, more traditional, notation of the moduli scheme
|  $M_{X}(v)$, where $r,c_{1},c_{2},c_{3}$ are the rank and Chern classes of
| a sheaf $[E]\in M_{X}(v)$
$\operatorname{ext}^{i}(\mathcal{F},\mathcal{G})$ |  $:=\dim\operatorname{Ext}\nolimits^{i}(\mathcal{F},\mathcal{G})$ for coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X$

##  2\. Preliminaries

This section contains the necessary definitions and results that we will use
below.

Let $X$ be one of the varieties $X_{i},\ i=1,\ 2,\ 4,\ 5$. Cohomology ring
$H^{*}(X,\mathbb{Z})$ is generated by the classes of a hyperplane section
$H\in H^{2}(X,\mathbb{Z})$, a line $L\in H^{4}(X,\mathbb{Z})$ (understood as a
projective line in the space ${\mathbb{P}^{2+i}}\supset X_{i}=X$ for $i=1,2$,
respectively, $X_{i}\hookrightarrow\mathbb{P}^{1+i}$ for $i=4,5$) and a point
$\\{\mathrm{pt}\\}\in H^{6}(X,\mathbb{Z})$ (for simplicity we will also denote
the class of a point by 1). We have

(4) $H^{2}=iL,\ \ H\cdot L=1,\ \ H^{3}=i,\ \ \ \textrm{}\ \ \ X=X_{i},\ \ \
i=1,2,4,5.$

The slope $\mu(E)$ of a coherent sheaf $E\in\operatorname{Coh}(X)$ is defined
as $\mu(E)=(H^{2}\cdot\ch_{1}(E))/(H^{3}\cdot\ch_{0}(E))$ (in the case of
division by 0 we set $\mu(E)=+\infty$). A coherent sheaf $E$ is called
$\mu$-(semi)stable if for any proper subsheaf $0\neq F\hookrightarrow E$ the
inequality $\mu(F)<(\leq)\,\mu(E/F)$ holds.

Let $f,g\in\mathbb{R}[m]$ be polynomials. If $\deg(f)<\deg(g)$, then we set
$f>g$. If $d=\deg(f)=\deg(g)$ and $a$, $b$ are the leading coefficients in
$f$, $g$ respectively, then we put $f<(\leq)g$ if
$\frac{f(m)}{a}<(\leq)\frac{g(m)}{b}$ for all $m\gg 0$. For an arbitrary sheaf
$E\in\operatorname{Coh}(\mathbb{P}^{3})$ we define the numbers $a_{i}(E)$ for
$i\in\\{0,1,2,3\\}$ through the Hilbert polynomial
$P(E,m):=\chi(E(m))=a_{3}(E)m^{3}+a_{2}(E)m^{2}+a_{1}(E)m+a_{0}(E)$. In
addition, we set $P_{2}(E,m):=a_{3}(E)m^{2}+a_{2}(E)m+a_{1}(E)$. The sheaf
$E\in\operatorname{Coh}(\mathbb{P}^{3})$ is called _(Gieseker-)(semi)stable_ ,
respectively, _(Gieseker-)2-(semi)stable_ , if for any its proper subsheaf
$0\neq F\hookrightarrow E$ the inequality $P(F,m)<(\leq)P(E/F,m)$ holds
(respectively, the inequality $P_{2}(F,m)<(\leq)P_{2}(E/F,m)$ holds).
Stability, 2-stability and $\mu$-stability of a sheaf satisfy the relations:

$\mu$-stability$2$-stabilitystability$\mu$-semistability$2$-semistabilitysemistability

Let us recall the concept of tilt stability. Let $\beta\in\mathbb{R}$. Let us
define _twisted Chern character_ as $\ch^{\beta}=e^{-\beta H}\cdot\ch$. Note
that for $\beta\in\mathbb{Z}$ and for any $E\in\mathrm{D}^{b}(X)$ the equality
$\ch^{\beta}(E)=\ch(E(-\beta))$ holds. Let us present explicit formulas for
the components $\ch_{i}^{\beta}=\ch_{i}^{\beta}(E)$:

(5) $\begin{split}&\ch_{0}^{\beta}=r,\ \ch_{1}^{\beta}=\ch_{1}-\beta
H\ch_{0},\ \ch_{2}^{\beta}=\ch_{2}-\beta
H\ch_{1}+\frac{\beta^{2}}{2}H^{2}\ch_{0},\\\ &\ch_{3}^{\beta}=\ch_{3}-\beta
H\ch_{2}+\frac{\beta^{2}}{2}H^{2}\ch_{1}-\frac{\beta^{3}}{6}H^{3}\ch_{0}.\end{split}$

In particular, for

(6) $\begin{split}&X=X_{2}\ \ \ \ \textit{and}\ \ \ \
E\in\mathrm{D}^{b}(X),\\\ &v=\ch(E)=(r,cH,dH^{2},e[\mathrm{pt}]),\ \ \ \
r,c\in\mathbb{Z},\ \ d\in\frac{1}{2}\mathbb{Z},\ \
e\in\frac{1}{6}\mathbb{Z},\\\
&c(E)=(1,c_{1}(E),c_{2}(E),c_{3}(E))=(1,c_{1}H,c_{2}[l],c_{3}[\mathrm{pt}]),\
\ \ c_{1},c_{2},c_{3}\in\mathbb{Z},\end{split}$

we have:

(7) $\begin{split}&\ch_{0}^{\beta}(E)=r,\ \ \ch_{1}^{\beta}(E)=(c-\beta r)H,\
\ \ch_{2}^{\beta}(E)=(d-\beta c+\frac{\beta^{2}}{2}r)H^{2},\\\
&\ch_{3}^{\beta}(E)=e-2\beta d+\beta^{2}c-\frac{\beta^{3}}{3}r,\end{split}$
(8) $c_{1}=c,\ \ \ c_{2}=c^{2}-2d,\ \ \ c_{3}=2e+\frac{1}{3}c^{3}-2cd,\\\ $
(9)
$\ch(E)=(r,c_{1}H,\frac{1}{2}(c_{1}^{2}-c_{2})H^{2},\frac{1}{2}(c_{3}+\frac{2}{3}c_{1}^{3}-c_{1}c_{2})[\mathrm{pt}]).$

Define a _torsion pair_

$\mathcal{T}_{\beta}=\\{E\in\operatorname{Coh}(X)\colon\text{any}\
\text{quotient}\ E\to G\ \text{satisfies}\ \mu(G)>\beta\\},$
$\mathcal{F}_{\beta}=\\{E\in\operatorname{Coh}(X)\colon\text{any}\
\text{subsheaf}\ 0\neq F\to E\ \text{satisfies}\ \mu(F)\leq\beta\\}$

and a category $\operatorname{Coh}^{\beta}(X)$ as the subcategory
$\langle\mathcal{F}_{\beta}[1],\mathcal{T}_{\beta}\rangle$ in $D^{b}(X)$. For
$\alpha\in\mathbb{R}_{+}$ the _tilt-slope_ of an object
$E\in\operatorname{Coh}^{\beta}(X)$ is defined as

$\nu_{\alpha,\beta}(E)=\nu_{\alpha,\beta}(\ch_{0}(E),\ch_{1}(E),\ch_{2}(E))=\frac{H\cdot\ch_{2}^{\beta}(E)-\frac{\alpha^{2}}{2}H^{3}\cdot\ch_{0}^{\beta}(E)}{H^{2}\cdot\ch_{1}^{\beta}(E)}.$

An object $E\in\mathrm{\operatorname{Coh}}^{\beta}(X)$ is called to be
tilt-(semi)stable (or
$\nu_{\alpha,\beta}$-(semi)stable), if for any subobject $0\neq
F\hookrightarrow E$ we have
$\nu_{\alpha,\beta}(F)<(\leq)\,\nu_{\alpha,\beta}(E/F)$. In case where the
last inequality becomes an equality, we will also say that $E$ is destabilized
by the exact triple $0\to F\to E\to E/F\to 0$.

######  Proposition 2.1.

(i) An object $E\in\operatorname{Coh}^{\beta}(X)$ is
$\nu_{\alpha,\beta}$-(semi)stable for $\beta<\mu(E)$ and $\alpha\gg 0$ iff $E$
is a 2-(semi)stable sheaf.
(ii) Let $X=X_{2}$. For any $\nu_{\alpha,\beta}$-semistable object
$E\in\operatorname{Coh}^{\beta}(X_{2})$ with $\ch(E)=(r,cH,dH^{2},e)$ we have
the following inequalities:

$\Delta(E)=\Delta(\ch
E):=\frac{(H^{2}\cdot\ch_{1}^{\beta}(E))^{2}-2(H^{3}\cdot\ch_{0}^{\beta}(E))(H\cdot\ch_{2}^{\beta}(E))}{(H^{3})^{2}}=c^{2}-2rd\geq
0,$
$W_{\alpha,\beta}(E):=\alpha^{2}\Delta(E)+\frac{4(H\cdot\ch_{2}^{\beta}(E))^{2}}{(H^{3})^{2}}-\frac{6(H^{2}\cdot\ch_{1}^{\beta}(E))\ch_{3}^{\beta}(E)}{(H^{3})^{2}}=$
$=(\alpha^{2}+\beta^{2})(c^{2}-2rd)+(3re-2cd)\beta+4d^{2}-3ce\geq 0.$

###### Proof.

The statement (i) is proven in [8, Prop. 14.2] for K3 surfaces, but an
analogous proof holds in our case. For a proof of the first inequality in (ii)
see [22, Thm. 2.1], and for the second one — in [22, Assumption C]. ∎

The set of semistable objects changes with variation of $(\alpha,\beta)$
pairs. Consider $H\cdot\ch_{\leq 2}$ as a mapping to
$\Lambda=\mathbb{Z}^{2}\oplus\frac{1}{2}\mathbb{Z}$. A numerical wall with
respect to a vector $v\in\Lambda$ is a non-empty proper subset $W$ of the
upper half-plane $\\{\alpha,\beta)\ |\ \alpha>0\\}$, given by an equation of
the form $\nu_{\alpha,\beta}(v)=\nu_{\alpha,\beta}(w)$ for some other vector
$w\in\Lambda$. A subset $\mathcal{U}$ of the numerical wall $W$ is called an
actual wall (or simply wall) if the set of semistable objects
$E\in\operatorname{Coh}^{\beta}(X)$ with $\ch(E)=v$ changes when passing
through $\mathcal{U}$.

######  Proposition 2.2 ([28, Theorem 2.9]).

Let $v\in\Lambda$ be a fixed class. Everywhere below, numerical walls are
considered with respect to class $v$.
(i) Numerical walls are either semicircles with centers on the $\beta$-axis,
or rays parallel to the $\alpha$-axis. If $v_{0}\neq 0$, then there is exactly
one vertical numerical wall with the equation $\beta=v_{1}/v_{0}$. If
$v_{0}=0$, then there are no actual vertical walls.
(ii) The curve $\nu_{\alpha,\beta}(v)=0$ is a hyperbola that can degenerate if
$v_{0}=0$ or $\Delta(v)=0$. Moreover, this hyperbola crosses all semicircular
walls at their highest points.
(iii) If two numerical walls defined by the classes $w,u\in\Lambda$ intersect,
then $v$, $w$ and $u$ are linearly dependent. In particular, these two walls
coincide.
(iv) If a numerical wall has at least one point at which it is an actual wall,
then it is an actual wall at every point.
(v) If $v_{0}\neq 0$, then there is a largest semicircular wall on both sides
from a single vertical wall.
(vi) If there is an actual wall, numerically defined by an exact triple of
tilt-semistable objects $0\to F\to E\to G\to 0$ such that $\ch_{\leq 2}(E)=v$,
then $\Delta(F)+\Delta(G)\leq\Delta(E).$ Moreover, the equality is satisfied
if and only if $\ch_{\leq 2}(G)=0$.
(vii) If $\Delta(E)=0$, then $E$ can destabilize only in a single numerical
vertical wall. In particular, line bundles and their shifts by one are tilt-
semistable everywhere.

Throughout this section we will consider the case when

$X=X_{2}$

is a three-dimensional quadric. The following lemma can be verified by direct
calculation.

######  Lemma 2.1.

Let $E\in\mathrm{D}^{b}(X)$ and $\ch(E)=(r,cH,dH^{2},e)$. The equation
$W_{\alpha,\beta}(E)=0$ is equivalent to the relation
$\nu_{\alpha,\beta}(r,cH,dH^{2})=\nu_{\alpha,\beta}(2c,4dH,3eH^{2}).$ In
particular, the equation $W_{\alpha,\beta}(E)=0$ describes a numerical wall in
tilt stability.

Let us denote the radius of the semicircular wall $W_{\alpha,\beta}(E)=0$ by
$\rho_{W}(E)$, and its center by $s_{W}(E)$. Center of the semicircular wall
$\nu_{\alpha,\beta}(E)=\nu_{\alpha,\beta}(F)$ will also be denoted by
$s(E,F)$.

Below we will use the following results about tilt-semistable objects.

######  Lemma 2.2 ([7, Lemma 7.2.1]).

If the object $E\in\operatorname{Coh}^{\beta}(X)$ is
$\nu_{\alpha,\beta}$-semistable for $\alpha\gg 0$, then one of the following
statements is true:
(1) $\mathcal{H}^{-1}(E)=0$ and $\mathcal{H}^{0}(E)$ is a pure
$\mu$-semistable sheaf of dimension $\geq 0$,
(2) $\mathcal{H}^{-1}(E)=0$ and $\mathcal{H}^{0}(E)$ is a sheaf of dimension
$\leq 1$,
(3) $\mathcal{H}^{-1}(E)$ is a $\mu$-semistable torsion-free sheaf and
$\mathcal{H}^{0}(E)$ is a sheaf of dimension $\leq 1$.

######  Proposition 2.3 ([22, Lem. 2.4]).

Suppose that an object $E$ is tilt-semistable and is destabilized by either a
subobject $F\hookrightarrow E$, or a quotient $E\twoheadrightarrow F$ in
$\operatorname{Coh}^{\beta}(X)$, inducing a non-empty semicircular wall $W$.
Let us further assume that $\ch_{0}(F)>\ch_{0}(E)\geq 0$. Then the radius
$\rho_{W}$ of the wall $W$ satisfies the inequality

$\rho_{W}\leq\frac{\Delta(E)}{4\ch_{0}(F)(\ch_{0}(F)-\ch_{0}(E))}.$

Let us recall the construction of Bridgeland stability conditions on $X$. Let

$\begin{split}&\mathcal{T}^{\prime}_{\alpha,\beta}=\\{E\in\operatorname{Coh}^{\beta}(X)\
|\ \text{any quotient}\,E\twoheadrightarrow
G\;\text{satisfies}\;\nu_{\alpha,\beta}(G)>0\\},\\\
&\mathcal{F}^{\prime}_{\alpha,\beta}=\\{E\in\operatorname{Coh}^{\beta}(X)\ |\
\text{any subobject}\ 0\neq F\hookrightarrow E\ \text{satisfies}\\\
&\nu_{\alpha,\beta}(F)\leq 0\\},\end{split}$

and set
$\mathcal{A}^{\alpha,\beta}(X)=\langle\mathcal{F}^{\prime}_{\alpha,\beta}[1],\mathcal{T}^{\prime}_{\alpha,\beta}\rangle$.
For any $s>0$ we define

$\lambda_{\alpha,\beta,s}=\frac{\ch^{\beta}_{3}-s\alpha^{2}H^{2}\cdot\ch^{\beta}_{1}}{H\cdot\ch^{\beta}_{2}-\frac{\alpha^{2}}{2}H^{3}\cdot\ch^{\beta}_{0}}.$

An object $E\in\mathcal{A}^{\alpha,\beta}(X)$ is called
$\lambda_{\alpha,\beta,s}$-(semi)stable if for any nontrivial subobject
$F\hookrightarrow E$ we have
$\lambda_{\alpha,\beta,s}(F)<(\leq)\lambda_{\alpha,\beta,s}(E)$.

$D^{b}(X)$ has a full strong exceptional collection
$(\mathcal{O}_{X}(-1),\SS(-1),$
$\mathcal{O}_{X},\mathcal{O}_{X}(1))$, where $\SS$ is a spinor bundle on $X$.

######  Proposition 2.4 ([27],[29, Thm. 6.1(2)]).

(i) Let $\alpha<\frac{1}{3},\beta\in[-\frac{1}{2}.0],s=\frac{1}{6}.$ For any
$\gamma\in\mathbb{R}$ we define a torsion pair

$\begin{split}&\mathcal{T}^{\prime\prime}_{\gamma}=\\{E\in\mathcal{A}^{\alpha,\beta}(X)\
|\ \text{any quotient}\,E\twoheadrightarrow
G\,\text{satisfies}\,\lambda_{\alpha,\beta,s}(G)>\gamma\\},\\\
&\mathcal{F}^{\prime\prime}_{\gamma}=\\{E\in\mathcal{A}^{\alpha,\beta}(X)\ |\
\text{any subobject}\,0\neq F\hookrightarrow E\,\text{satisfies}\,\\\
&\lambda_{\alpha,\beta,s}(F)\leq\gamma\\}.\end{split}$

There is a $\gamma\in\mathbb{R}$ such that

$\langle\mathcal{T}^{\prime\prime}_{\gamma},\mathcal{F}^{\prime\prime}_{\gamma}[1]\rangle=\mathfrak{C}:=\langle\mathcal{O}_{X}(-1)[3],\SS(-1)[2],\mathcal{O}_{X}[1],\mathcal{O}_{X}(1)\rangle.$

(ii) Let $v$ be the Chern character of an object from $\mathrm{D}^{b}(X),$ and
$\alpha_{0}>0,\beta_{0}\in\mathbb{R},$ and $s>0$ such that
$\nu_{\alpha_{0},\beta_{0}}(v)=0,\ H^{2}\cdot v_{1}^{\beta_{0}}>0,$ and
$\Delta(v)\geq 0$. Let us assume that all
$\nu_{\alpha_{0},\beta_{0}}$-semistable objects of class $v$ are
$\nu_{\alpha_{0},\beta_{0}}$-stable. Then there is a neighborhood $U$ of the
point $(\alpha_{0},\beta_{0})$ such that for all $(\alpha,\beta)\in U$ with
$\nu_{\alpha,\beta}(v)>0$, an object $E\in\operatorname{Coh}^{\beta}(X)$ with
$\ch(E)=v$ is $\nu_{\alpha,\beta}$-semistable if and only if it is
$\lambda_{\alpha,\beta,s}$-semistable.

######  Remark 2.1.

From the proof of [22, Prop. 3.2] it follows that for a tilt-semistable object
$E\in\operatorname{Coh}^{\beta}(X)$ with $\ch(E)=(1,0,-dH^{2},e)$ the
inequality $e\leq d(d+1)$ holds.

In what follows we will use the equalities

(10)
$\ch(\mathcal{O}_{X}(n))=(1,nH,\frac{n^{2}}{2}H^{2},\frac{n^{3}}{3}[\mathrm{pt}]),\
\ \ \ \ch(\SS(-1))=(2,-H,0,\frac{1}{6}[\mathrm{pt}]),$

the exact triple of sheaves on $X$ ([27])

(11) $0\to\SS(-1)\to V\otimes\mathcal{O}_{X}\xrightarrow{\varepsilon}\SS\to
0,\ \ \ \ \ \ \ \ \ \ V\cong\mathbf{k}^{4},$

and the following properties of the spinor bundle $\SS$ on $X$. In the triple
(11) the epimorphism $\varepsilon$ gives an isomorphism in sections
$h^{0}(\varepsilon):\ V\xrightarrow{\cong}H^{0}(\SS)$. Let $B$ be the base of
the family of lines on $X$. Everywhere below we will identify $B$ with
$\mathbb{P}^{3}$ through the isomorphism $f:\
\mathbb{P}^{3}=\mathbb{P}(V)\xrightarrow{\cong}B,\ \mathbf{k}v\mapsto l$,
where line $l$ is the scheme of zeros of the section $h^{0}(\varepsilon)(v)$.
Let $\mathbb{P}^{4}=\langle X\rangle$ be the projective hull of the quadric
$X$, $\check{\mathbb{P}}^{4}=|\mathcal{O}_{X}(1)|$ and let
$\mathbb{S}\subset\check{\mathbb{P}}^{4}\times X$ be the universal family of
two-dimensional quadrics (hyperplane sections) on $X$. Let
$\mathbb{G}:=Gr(2,V)$ be the grassmannian of lines in $\mathbb{P}^{3}$, and
let $l_{x}$ be the line in $\mathbb{P}^{3}$ corresponding to a point
$x\in\mathbb{G}$. Then $f(l_{x})$ is a series of lines on the two-dimensional
quadric $S_{x}\in\check{\mathbb{P}}^{4}$. It is not difficult to see that
$\mu:\ \mathbb{G}\to\check{\mathbb{P}}^{4},\ x\mapsto S_{x}$ is a double
covering ramified in the dual quadric
$\check{X}\subset\check{\mathbb{P}}^{4}$. The covering $\mu:\
\mathbb{G}\to\check{\mathbb{P}}^{4}$ defines on $X$ a family of two-
dimensional quadrics
$\widetilde{\mathbb{S}}:=\mathbb{G}\times_{\check{\mathbb{P}}^{4}}\mathbb{S}\subset\mathbb{G}\times
X$ with the base $\mathbb{G}$. The tautological subbundle $g:\mathcal{K}\to
V\otimes\mathcal{O}_{\mathbb{G}}$ of rank two on $\mathbb{G}$ and the
epimorphism $\varepsilon$ in (11) determine the composition $e:\
\mathcal{K}\boxtimes\mathcal{O}_{X}\xrightarrow{g\boxtimes\mathrm{id}_{X}}V\otimes\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X}\xrightarrow{\mathrm{id}_{\mathbb{G}}\boxtimes\varepsilon}\mathcal{O}_{\mathbb{G}}\boxtimes\SS$,
and let
$\mathbb{I}:=\operatorname{im}(e)\otimes\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X}(-1)$.
It is not hard to see that
$\ker(e)\cong\mathcal{O}_{\mathbb{G}}(-1)\boxtimes\SS(-1)$. Thus, we have the
following exact triple on $X\times\mathbb{G}$:

(12)
$0\to\mathcal{O}_{\mathbb{G}}(-1)\boxtimes\SS(-1)\to\mathcal{K}\to\mathbb{I}\otimes\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X}(1)\to
0.$

For an arbitrary point $x\in\mathbb{G}$ the restriction of this triple to
$\\{x\\}\times X$ gives an exact triple:

(13) $0\to\SS(-1)\to\mathcal{O}^{\oplus 2}_{X}\to\mathcal{I}(1)\to 0,\ \ \ \ \
\ \ \ \ \ \mathcal{I}:=\mathcal{I}_{\mathbb{P}^{1},S}.$

Here $\mathcal{I}_{\mathbb{P}^{1},S}$ is the ideal sheaf of an arbitrary line
$\mathbb{P}^{1}\subset S$ from the series of lines defined by the point $x$ on
a two-dimensional quadric $S=S_{x}$.

##  3\. Semistable objects of rank two in $D^{b}(X_{2})$ with maximal third
Chern class $c_{3}$

In this section we prove one of the main results of the paper — the Theorem
3.1 on the description of rank 2 semistable objects from $D^{b}(X_{2})$,
including sheaves, on the three-dimensional quadric

$X=X_{2},$

having the maximal third Chern class $c_{3}$. Everywhere below, $Q_{2}$
denotes a two-dimensional quadric a hyperplane section of $X$, and
$\mathbb{P}^{1}$ denotes a projective line on the surface $Q_{2}$.

######  Theorem 3.1.

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a tilt-semistable object with
$\ch(E)=(2,cH,dH^{2},e)$, $d\in\frac{1}{2}\mathbb{Z}$,
$e\in\frac{1}{6}\mathbb{Z}$.
(1) If $c=-1$, then $d\leq 0$.
(1.1) If $d=0$, then $e\leq\frac{1}{6}$. In the case $e=\frac{1}{6}$ we have
$E\cong\SS(-1)$.
(1.2) If $d=-\frac{1}{2}$, then $e\leq\frac{5}{3}$. In case of equality $E$ is
destabilized by an exact triple $0\to\mathcal{O}_{X}(-1)^{\oplus 3}\to
E\to\mathcal{O}_{X}(-2)[1]\to 0.$
(1.3) If $d\leq-1$ is an integer, then $e\leq d^{2}-2d+\frac{1}{6}$. In case
of equality $E$ is destabilized by an exact triple
$0\to\mathcal{O}_{X}(-1)^{\oplus 2}\to
E\to\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(d)\to 0.$
(1.4) If $d\leq-\frac{3}{2}$ is non-integer, then $e\leq
d^{2}-2d+\frac{5}{12}$. In in case of equality $E$ is destabilized by an exact
triple $0\to\mathcal{O}_{X}(-1)^{\oplus 2}\to
E\to\mathcal{O}_{Q_{2}}(d-\frac{1}{2})\to 0.$
(2) If $c=0$, then $d\leq 0$.
(2.1) If $d=0$, then $e\leq 0$. In case of equality
$E\cong\mathcal{O}_{X}^{\oplus 2}$.
(2.2) If $d=-\frac{1}{2}$, then $e\leq-\frac{1}{2}$.
(2.3) If $d=-1$, then $e\leq 1$. In the case $e=1$, the sheaf $E$ is
destabilized by an exact triple $0\to\mathcal{O}_{X}(-1)^{\oplus 6}\to
E\to\SS(-2)^{\oplus 2}[1]\to 0.$
(2.4) If $d\leq-\frac{3}{2}$ is non-integer, then $e\leq d^{2}+\frac{1}{4}$.
In case of equality, $E$ is destabilized by an exact triple $0\to\SS(-1)\to
E\to\mathcal{O}_{Q_{2}}(d+\frac{1}{2})\to 0.$
(2.5) If $d=-2$, then $e\leq 4$. In the case $e=4$, the sheaf $E$ is
destabilized by one of exact triples $0\to\SS(-1)\to
E\to\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(-1)\to 0,\ \ \
0\to\mathcal{O}_{X}(-1)^{\oplus 4}\to E\to\mathcal{O}(-2)^{\oplus 2}[1]\to 0$.
(2.6) If $d\leq-3$ is an integer, then $e\leq d^{2}$. In case of equality $E$
destabilized by an exact triple $0\to\SS(-1)\to
E\to\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(d+1)\to 0.$
(3) Let $E\in\operatorname{Coh}(X)$ be a semistable sheaf of rank 2 with
$c_{1}(E)=-H$, $c_{2}(E)=c_{2}[l]$, $c_{3}(E)=c_{3}[\mathrm{pt}]$. Then
$c_{2}\geq 0$ and
(3.1) if $c_{2}$ is even, then

(14) $c_{3}\leq\frac{1}{2}c_{2}^{2};$

(3.2) if $c_{2}$ is odd, then

(15) $c_{3}\leq\frac{1}{2}(c_{2}^{2}-1).$

(4) Let $E\in\operatorname{Coh}(X)$ be a semistable sheaf of rank 2 with
$c_{1}(E)=0$, $c_{2}(E)=c_{2}[l]$, $c_{3}(E)=c_{3}[\mathrm{pt}]$. Then
$c_{2}\geq 0$ and
(4.1) if $c_{2}$ is even, then

(16) $c_{3}\leq\frac{1}{2}c_{2}^{2};$

(4.2) if $c_{2}$ is odd, then

(17) $c_{3}\leq\frac{1}{2}(c_{2}^{2}+1).$

######  Remark 3.2.

In statements (1.1)-(1.4), (2.1), (2.3)-(2.6) of the Theorem, the maximal
value $c_{3\mathrm{max}}(E)$ is non-negative. Statement (2.2) gives the only
case when $c_{3\mathrm{max}}(E)=-1$ is negative. In this case, it is not known
whether an object $E$ with Chern classes $c_{1}(E)=0,\ c_{2}(E)=1,\
c_{3}(E)=c_{3\mathrm{max}}(E)=-1$ is a Gieseker-(semi)stable sheaf.

We will preface the proof of the Theorem with a series of auxiliary statements
— Lemmas 3.1-3.12.

######  Lemma 3.1.

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a tilt-semistable object with
$\ch(E)=(2,0,0,e)$. Then $e\leq 0$, and if $e=0$, then
$E\cong\mathcal{O}_{X}^{\oplus 2}$.

###### Proof.

Since $E\in\operatorname{Coh}^{\beta}(X)$, we get $\beta<0$. We have
$W_{\alpha,\beta}(E)=6e\beta\geq 0$, so $e\leq 0$. If $e=0$, then, following
the arguments from the proof of the statement [29, Prop. 4.1] and using
Proposition 2.4, we find $E\cong\mathcal{O}_{X}^{\oplus 2}$.∎

######  Lemma 3.2.

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a tilt-semistable object with
$\ch(E)=(2,-H,0,e)$. Then $e\leq\frac{1}{6}$, and if $e=\frac{1}{6}$, then
$E\cong\SS(-1)$.

###### Proof.

Since $E\in\operatorname{Coh}^{\beta}(X)$, then $\beta<-\frac{1}{2}$.
Hyperbola $\nu_{\alpha,\beta}(E)=0$ intersects the $\beta$-axis at the point
$\beta=-1$, so if there is an actual semicircular wall, it must intersect the
line $\beta=-1$. In our case, $H^{2}\cdot\ch_{1}^{-1}(E)$ has the smallest
positive value, so if $E$ is $\nu_{\alpha,-1}$-semistable, it must be
$\nu_{\alpha,-1}$-semistable for any $\alpha$ [29, Lem. 3.5]. Therefore, for
$E$ there are no other walls except the unique one vertical wall.

Note that $\nu_{\frac{1}{4},\frac{2-\sqrt{5}}{4}}(E(1))=0$, therefore,
applying Proposition 2.4 to $E(1)$, we obtain that $E(1)$ or $E(1)[1]$ belong
to the category $\mathfrak{C}$. Solving the equation
$\ch(E(1))=a\ch(\mathcal{O}_{X}(-1)[3])+b\ch(\SS(-1)[2])+c\ch(\mathcal{O}_{X}[1])+d\ch(\mathcal{O}_{X}(1))$,
where $a,b,c,d\in\mathbb{Z}$ are simultaneously non-negative or non-positive,
we obtain the required bound for $\ch_{3}(E)$, and for its maximum value we
find $a=d=0,b=-1,c=-4$. We get an exact triple
$0\to\SS(-2)\to\mathcal{O}_{X}(-1)^{\oplus 4}\to E\to 0.$

A calculation shows that the equation
$\nu_{\alpha,\beta}(E)=\nu_{\alpha,\beta}(\mathcal{O}_{X}(-1))$ does not hold
for any $(\alpha,\beta)$, and
$\nu_{\alpha,\beta}(E)>\nu_{\alpha,\beta}(\mathcal{O}_{X}(-1))$. Therefore
there is no morphism $E\twoheadrightarrow\mathcal{O}_{X}(-1)$. Since
$\operatorname{Hom}(\SS(-2),\mathcal{O}_{X}(-1))\cong\mathbb{C}^{4}$, it
follows that any two injective morphisms
$\SS(-2)\to\mathcal{O}_{X}(-1)^{\oplus 4}$ with tilt-semistable cokernel $E$
lie in the same orbit of the $GL(4)$-action. So there is a unique object $E$
up to isomorphism, and the exact triple (11) shows that $E\cong\SS(-1)$. ∎

######  Lemma 3.3.

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a tilt-semistable object with
$\ch(E)=(2,0,-\frac{1}{2}H^{2},e)$. Then $e\leq-\frac{1}{2}$.

###### Proof.

Suppose that there is a semicircular wall intersecting the line $\beta=-1$. We
have $\ch_{\leq 2}^{-1}(E)=(2,2H,\frac{H^{2}}{2})$, so in this case there is a
destabilizing sheaf $F\hookrightarrow E$ with $\ch_{\leq 2}(F)=(R,H,xH^{2})$.
From the condition $\nu_{\alpha,-1}(F)=\nu_{\alpha,-1}(E)$ we obtain
$x-\frac{1}{4}=\alpha^{2}(R-1)$. The case $R=1$ is impossible, since
$x\in\frac{1}{2}\mathbb{Z}$; for $R\geq 2$ we have $x>\frac{1}{4}$, and this
contradicts the condition $\Delta(F)\geq 0$; if $R\leq 0$, then instead of $F$
we can work with the sheaf $E/F$.

Note that from the tilt semistability of $E$ it follows that $H^{0}(E)=0$.
Also $H^{2}(E)\cong\operatorname{Hom}(E,\omega_{X}[1])=0$, since if a nonzero
morphism $E\to\omega_{X}[1]$ exists we would have a semicircular wall
intersecting the line $\beta=-1$. Todd’s class for $X$ has the form
$\mathrm{td}(T_{X})=(1,\frac{3H}{2},\frac{13H^{2}}{12},1).$ By the
Riemann–Roch Theorem we find that
$e+\frac{1}{2}=\chi(E)=-h^{1}(E)-h^{3}(E)\leq 0,$ that is $e\leq-\frac{1}{2}$.
∎

######  Lemma 3.4.

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a tilt-semistable object with
$\ch(E)=(2,0,-H^{2},e)$. Then $e\leq 1$, and if $e=1$ then $E$ is destabilized
by an exact triple $0\to\mathcal{O}_{X}(-1)^{\oplus 6}\to E\to\SS(-2)^{\oplus
2}[1]\to 0.$

###### Proof.

We have $\beta<0$, and any semicircular wall intersects the line $\beta=-1$.
Since $\ch_{\leq 2}^{-1}(E)=(2,2H,0)$, then any destabilizing object
$F\hookrightarrow E$ has $\ch^{-1}(F)=(R,H,xH^{2},e^{\prime})$. Then the
equality $\nu_{\alpha,-1}(F)=\nu_{\alpha,-1}(E)$ takes the form
$x=(R-1)\frac{\alpha^{2}}{2}$. If $R=1$, then $x=0$. Since $\ch_{1}^{-1}(F)$
is minimal, the object $F$ is tilt-semistable for all $\alpha>0,\beta<0$.

Since $F$ is stable, $H^{0}(F)=0$. If $H^{2}(F)\neq 0$, then, by Serre
duality, there is a nontrivial morphism $F\to\mathcal{O}_{X}(-3)[1]$. However,
such a morphism would define a semicircular wall, so $H^{2}(F)=0$.

By the Riemann–Roch Theorem we find that
$e^{\prime}-\frac{1}{2}=\chi(F)=-h^{1}(F)-h^{3}(F)\leq 0.$ Thus,
$e^{\prime}\leq\frac{1}{2}$. Since $\ch_{\leq 2}(E/F)=\ch_{\leq 2}(F)$, also
$\ch_{3}(E/F)\leq\frac{1}{2}$. Therefore, in this case $e\leq 1$, and the
equality is achieved at $e^{\prime}=\frac{1}{2}$. Applying Proposition 2.4, we
get for this case an exact triple $0\to\mathcal{O}_{X}(-1)^{\oplus 3}\to
F\to\SS(-2)[1]\to 0,$ and a similar exact triple for $E/F$, from which the
statement of the lemma follows for $R=1$.

If $R\neq 1$, then, without loss of generality, we can assume that $R\geq 2$,
then $x>0$. However, the equation $\Delta(F)\geq 0$ implies that
$x\leq\frac{1}{4}$, which is impossible, since $x\in\frac{1}{2}\mathbb{Z}$.
Finally, if $E$ has no destabilizing subobjects defining a semicircular wall,
then, carrying out the argument for $E$, similar to the argument for $F$ in
the case $R=1$, we obtain the required statement. ∎

######  Lemma 3.5.

Let $E\in\operatorname{Coh}^{\beta}(X)$ and
$\ch(E)=(2,0,-\frac{3}{2}H^{2},e)$. Then $e\leq\frac{5}{2}$, and if
$e=\frac{5}{2}$, then $E$ is destabilized by an exact triple $0\to\SS(-1)\to
E\to\mathcal{O}_{Q_{2}}(-1)\to 0.$

###### Proof.

We have $W_{0,-1}(E)=15-6e$, and if $e\geq\frac{5}{2}$, then $E$ destabilizes
at $\beta=-1$, or the point $\beta=-1,\alpha=0$ is a limit point of a
semicircular wall. By calculation we get
$\ch^{-1}(E)=(2,2H,-\frac{1}{2}H,e-\frac{7}{3})$, which means that there is a
destabilizing subobject or a quotient $F$ with $\ch_{1}^{-1}(F)\in\\{0,H\\}$.
If $\ch^{-1}(F)=(R,H,xH^{2},e^{\prime})$, then the relation
$\nu_{\alpha,1}(E)=\nu_{\alpha,1}(F)$ is equivalent to the equality
$x+\frac{1}{4}=\alpha^{2}(R-1)$. We can assume that $R>1$; then $\Delta(F)\geq
0$ implies $x=0$. We have
$\ch^{-1}(E/F)=(2-R,H,-\frac{1}{2}H^{2},e^{\prime\prime})$, and from the
inequality $\Delta(E/F)\geq 0$ we obtain $R\in\\{2,3\\}$.

Let $R=2$. From Lemma 3.2 it follows that $e^{\prime}\leq-\frac{1}{6}$, and in
the case of equality $F\cong\SS(-1)$. We have
$\ch(E/F)=(0,H,-\frac{3}{2}H^{2},e^{\prime\prime}+2)$, and any wall for $E/F$
intersects the line $\beta=-\frac{3}{2}$. Since
$\ch_{1}^{-\frac{3}{2}}(E/F)=H$, then any destabilizing subobject
$F^{\prime}\hookrightarrow E/F$ has $\ch_{\leq
2}^{-\frac{3}{2}}(F^{\prime})=(r,\frac{1}{2}H,xH^{2})$. From here we find
$0=2x-\alpha^{2}r$, therefore assuming that $r>0$, we get the only possibility
$r=1,x=\frac{1}{8}$. From Lemma 3.1 it follows that
$\ch_{3}(F^{\prime})\leq-\frac{1}{3}$ (in the case of equality
$F^{\prime}\cong\mathcal{O}_{Q}(-1)$). In addition, $\ch_{\leq
2}((E/F)/F^{\prime})=(-1.2H,-2H^{2})$, so $\ch_{3}$ of this object is less
than or equal to $\frac{8}{3}$. From these inequalities we find
$e\leq\frac{5}{2}$. When equality holds, we can assume that both objects $F$
and $E/F$ are $\nu_{\alpha,\beta}$-semistable, which implies that
$E/F\cong\mathcal{O}_{Q_{2}}(-1)$. In this case we have
$\operatorname{Ext}\nolimits^{1}(F,E/F)=0$, therefore $F$ is indeed a
subobject, and not a quotient of the object $E$. This vanishing is proved
using the definition of an exceptional collection and the exact triple (11).

If $R=3$, we denote the destabilizing subobject by $F^{\prime}\hookrightarrow
E$, and $\ch(F^{\prime})=(3,-2H,\frac{1}{2}H^{2},e^{\prime})$. Any
semicircular wall must intersect the line $\beta=-1$, but
$\ch^{-1}_{1}(F^{\prime})=H$. Hence $F^{\prime}(1)[1]\in\mathfrak{C}$, and a
calculation gives $e^{\prime}\leq-\frac{1}{6}$. In addition,
$\ch(E/F^{\prime})=(-1,2H,-2H^{2},e^{\prime\prime})$, where
$e^{\prime\prime}\leq\frac{8}{3}$. From these bounds we obtain
$e\leq\frac{5}{2}$. In case of equality we obtain exact triples $0\to
F^{\prime}\to E\to\mathcal{O}_{X}(-2)[1]\to 0,$
$0\to\mathcal{O}_{X}(-1)^{\oplus 5}\to F^{\prime}\to\SS(-2)[1]\to 0.$ From the
last exact triple it follows that $F^{\prime}\cong\mathcal{O}_{X}(-1)\oplus
F^{\prime\prime}$, and, as in the proof of the lemma 3.2,
$\nu_{\alpha,\beta}(\mathcal{O}_{X}(-1))\neq\nu_{\alpha,\beta}(F^{\prime\prime})$,
so the object $F^{\prime}$ is not tilt-semistable. We find that
$e<\frac{5}{2}$ in the case of $R=3$.

Finally, if $\ch^{-1}_{\leq 2}(F)=(r,0,xH^{2})$ and
$\nu_{\alpha,\beta}(F)=\nu_{\ alpha,\beta}(E)$ on a semicircular wall having a
limit point $\beta=-1,\alpha=0$, then the slope $\nu_{\alpha,\beta}(F)$ must
tend to a finite limit when we tend to this point along the wall, so $x=0$.
From the inequality $0\leq\Delta(E/F)\leq\Delta(E)$ it follows that $0\leq
r\leq 6$. Cases with $r\leq 5$ reduce to the case $R=2$ discussed above, while
for $r=6$ we get $(E/F)(1)[1]\in\mathfrak{C}$, and a calculation gives
$\ch_{3}((E/F)(1))\leq-\frac{5}{6}$, whence $\ch_{3}(E)\leq\frac{3}{2}$. ∎

######  Lemma 3.6.

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a tilt-semistable object with
$\ch(E)=(2,-H,-\frac{1}{2}H^{2},e)$. Then $e\leq\frac{5}{3}$, and if
$e=\frac{5}{3}$, then $E$ is destabilized by an exact triple
$0\to\mathcal{O}_{X}(-1)^{\oplus 3}\to E\to\mathcal{O}_{X}(-2)[1]\to 0.$

###### Proof.

Assume that $e\geq\frac{5}{3}$. Then $W_{0,-\frac{3}{2}}(E)<0$, and this means
that there is a wall intersecting the line $\beta=-\frac{3}{2}$. In addition,
$\ch_{\leq 2}^{-\frac{3}{2}}(E)=(2,2H,\frac{1}{4}H^{2})$, so there is a
destabilizing subobject or quotient $F$ of the object $E$ with
$\ch_{1}^{-\frac{3}{2}}(F)\in\\{\frac{1}{2}H,H\\}.$ If $\ch_{\leq
2}^{-\frac{3}{2}}(F)=(R,\frac{1}{2}H,xH^{2})$, then from the equality
$\nu_{\alpha,-\frac{3}{2}}(E)=\nu_{\alpha,-\frac{3}{2}}(F)$ we get
$2x-\frac{1}{8}=\alpha^{2}(2R-1)$. Note that $R$ must be odd since
$H^{2}\cdot\ch^{-\frac{3}{2}}_{1}(F)$ is odd. Then the inequalities
$\Delta(F)\geq 0,\Delta(E/F)\geq 0$ imply $R\in\\{-1,1\\}$. If $R=-1$, then
$x=-\frac{1}{8}$ and $\ch_{3}$ is maximized by the object
$F\cong\mathcal{O}_{X}(-2)[1]$. If $R=1$, then $x=\frac{1}{8}$ and
$F\cong\mathcal{O}_{X}(1)$. If $\ch_{\leq 2}^{-\frac{3}{2}}(F)=(R,H,xH^{2})$,
then $x-\frac{1}{8}=\alpha^{2}(R-1)$, $R$ is even, and we can assume that
$R>1$. But then $R=2,\ x=\frac{1}{4},\ F\cong\mathcal{O}_{X}(-1)^{\oplus 2}$.
In any case we obtain the required inequality and the exact triple. ∎

######  Lemma 3.7.

For $d\in\mathbb{Z}$ let $G_{d}\in\operatorname{Coh}^{\beta}(X)$ be a tilt-
semistable object with $\ch(G_{d})=(0,H,dH^{2},e)$. Then the inequality $e\leq
d^{2}-\frac{1}{6}$ holds, and in case of equality we have
$G_{d}\cong\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(d+1)$ for some two-dimensional
quadric $Q_{2}\subset X$ and a line $\mathbb{P}^{1}\subset Q_{2}$.

###### Proof.

The curve $\nu_{\alpha,\beta}(G_{d})=0$ is a straight line $\beta=d$, and
there are no semicircular walls, so $(G_{d})(-d)[1]\in\mathfrak{C}$. A
calculation gives the desired inequality and an exact triple

(18) $0\to\SS(d-1)\to\mathcal{O}_{X}(d)^{\oplus 2}\to G_{d}\to 0.$

As is known, the scheme of zeros $(s)_{0}$ of any section $0\neq s\in
H^{0}(\SS)$ is a reduced projective line $\mathbb{P}^{1}\subset X$, from which
we obtain an exact triple
$0\to\mathcal{O}_{X}\xrightarrow{s}\SS\to\mathcal{I}_{\mathbb{P}^{1}}(1)\to
0$. Hence $\operatorname{coker}(\mathcal{O}_{X}^{\oplus
2}\xrightarrow{s,s^{\prime}}\SS)=\operatorname{coker}(\mathcal{O}_{X}\xrightarrow{\phi}\mathcal{I}_{\mathbb{P}^{1}}(1)$.
Consider the two-dimensional quadric
$Q_{2}=\mathrm{supp}\operatorname{coker}(\phi)$. The morphism
$\mathcal{I}_{Q_{2}}(1)\cong\mathcal{O}_{X}\xrightarrow{\phi}\mathcal{I}_{\mathbb{P}^{1}}(1)$
induces an exact triple
$0\to\mathcal{I}_{Q_{2}}(1)\xrightarrow{\phi}\mathcal{I}_{\mathbb{P}^{1}}(1)\to\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(1)\to
0$. This implies the exactness of the triple $0\to\mathcal{O}_{X}^{\oplus
2}\to\SS\to\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(1)\to 0$. Applying the functor
$\mathbf{R}\mathcal{H}om(\cdot,\mathcal{O}_{X}(d))$ to it and taking into
account the easily verifiable isomophism
$\mathop{\mathcal{E}xt}\nolimits^{1}(\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(1),\mathcal{O}_{X}(d))\cong\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(d+1)$,
we obtain an exact triple $0\to\SS(d-1)\to\mathcal{O}(d)^{\oplus
2}\to\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(d+1)\to 0$. This triple by
construction is isomorphic to the triple (18) with a suitable choice of
sections $s,s^{\prime}$, hence
$G_{d}\cong\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(d+1)$. ∎

######  Lemma 3.8.

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a tilt-semistable object with
$\ch(E)=(2,0,-2H^{2},e)$. Then $e\leq 4$, and if $e=4$, then $E$ is
destabilized by one of the exact triples

(19) $0\to\mathcal{O}_{X}(-1)^{\oplus 4}\to E\to\mathcal{O}_{X}(-2)^{\oplus
2}[1]\to 0,$ (20) $0\to\SS(-1)\to
E\to\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(-1)\to 0.$

In the case when the object $E$ is included in the exact triple (20), it is
tilt-stable for $\beta<0,\alpha\gg 0$ and is not included in the exact triple
(19).

###### Proof.

We have $W_{0,-1}(E)=24-6e$, so the inequality $W_{0,-1}\geq 0$ is equivalent
to $e\leq 4$. Therefore, it is sufficient to check that $e\leq 4$ for objects
that have a semicircular wall at $\beta=-1$. We have $\ch_{\leq
2}^{-1}(E)=(2,2H,-H^{2})$, so in this case there is a destabilizing subobject
or a quotient $F$ of the object $E$ with $\ch_{1}^{-1}(F)=H$. Let $\ch_{\leq
2}^{-1}(F)=(R,H,xH^{2})$, then $\nu_{\alpha,-1}(F)=\nu_{\alpha,-1}(E)$ implies
$x+\frac{1}{2}=\frac{\alpha^{2}}{2}(R-1)$. We can assume that $R\geq 2$, and
then the inequality $\Delta(F)\geq 0$ implies $x\in\\{-\frac{1}{2},0\\}$.

Assume that $x=-\frac{1}{2}$. From the inequality $\Delta(E/F)\geq 0$ we
obtain $R\in\\{2,3\\}$. If $R=2$, then $\ch_{\leq
2}(F)=(2,-H,-\frac{1}{2}H^{2})$ and $\ch_{\leq
2}(E/F)=(0,H,-\frac{3}{2}H^{2})$ (we assume, by abuse of notation, that $F$ is
a subobject of $E$). Therefore, using Lemma 3.6 and the proof of Lemma 3.5, we
obtain the desired inequality and one of the exact triples

(21) $0\to F\to E\to\mathcal{O}_{Q_{2}}(-1)\to 0,$ (22)
$0\to\mathcal{O}_{Q_{2}}(-1)\to E\to F\to 0,$

where $F\in M(2,-H,-\frac{1}{2}H^{2},\frac{5}{3}).$ By Lemma 3.6 we have an
exact triple $0\to\mathcal{O}_{X}(-1)^{\oplus 3}\to
F\to\mathcal{O}_{X}(-2)[1]\to 0$, there is also an exact triple
$0\to\mathcal{O}_{X}(-1)\to\mathcal{O}_{Q_{2}}(-1)\to\mathcal{O}_{X}(-2)[1]\to
0$. A consideration of long exact sequences of groups
$\operatorname{Ext}\nolimits$ shows that
$\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{X}(-1),F)=\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{X}(-1),\mathcal{O}_{Q_{2}}(-1))=0$,
we obtain a morphism $F^{\prime}=\mathcal{O}_{X}(-1)^{\oplus
4}\cong\mathcal{O}_{X}(-1)^{\oplus 3}\oplus\mathcal{O}_{X}(-1)\to E$. By the
Snake Lemma we find that this morphism is injective and
$E/F^{\prime}\cong\mathcal{O}_{X}(-2)^{\oplus 2}[1].$

If $R=3$, then $\ch_{\leq 2}(F)=(3,-2H,0)$ and $\ch_{3}(E/F)$ is maximized by
the object $\mathcal{O}_{X}(-2)[1]$. The curve $\nu_{\alpha,\beta}(F)=0$
intersects the $\beta$-axis at the point $\beta=-\frac{4}{3}$, therefore any
semicircular wall for $F$ intersects the line $\beta=-\frac{4}{3}$. We have
$\ch_{\leq 2}^{-\frac{4}{3}}(F)=(3,2H,0)$, therefore there is a destabilizing
subobject or a quotient $G$ of the object $F$ with
$\ch_{1}^{-\frac{4}{3}}(G)\in\\{\frac{1}{3}H,\frac{2}{3}H,H\\}$. In the first
case we have $G\cong\mathcal{O}_{X}(-1)$, in the second case
$G\cong\mathcal{O}_{X}(-1)^{\oplus 2}$ or $G\cong\mathcal{O}_{X}(-2)[1]$, and
in the third $G\cong\mathcal{O}_{X}(-1)^{\oplus 3}$. In any case, we obtain
the required inequality and the exact triple (19).

Let us now assume that $x=0$. Then $R=2$, and we find $\ch_{\leq
2}(F)=(2,-H,0)$ and $\ch_{\leq 2}(E/F)=(0,H,-2H^{2})$. Lemmas 3.2 and 3.7 give
the required inequality and the exact triple (20).

We have proven that $e\leq 4$. For $e=4$, note that the exact triple (19)
defines the wall $W_{\alpha,\beta}(E)=0$, so this wall is the smallest, and
repeating the arguments from the proof of [29, Theorem 5.1] shows that any
object destabilized by this wall is given by the exact triple (19). Any other
wall intersects the line $\beta=-1$ at a point with $\alpha>0$, therefore,
when $R\geq 2$, the above argument applies for them. If $R=1$, then, due to
the fact that $W_{0,-1}(F)=6-3\ch_{3}(F)$ and $\ch_{1}^{-1}(F)$ is minimal, we
find that $\ch_{3}(F)\leq 2$ and the wall $W_{\alpha,\beta}(F)=0$ is unique.
This wall is given by the exact triple $0\to\mathcal{O}_{X}(-1)^{\oplus 2}\to
F\to\mathcal{O}_{X}(-2)[1]\to 0$, from which, together with a similar exact
triple for $E/F$, by the Snake Lemma we obtain a monomorphism
$\mathcal{O}_{X}(-1)^{\oplus 4}\hookrightarrow E$ and the exact triple (19).

To prove the statement about the tilt stability of an object $E$ from (20) it
is enough to check its tilt stability for $\beta=-1,\alpha>1$. Because
$\nu_{1,-1}(\SS(-1))=\nu_{1,-1}(\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(-1))$, the
object $E$ is $\nu_{1,-1}$-semistable. If $F$ is a subobject that destabilizes
$E$ at $\beta=-1$, then $\ch_{1}^{-1}(F)=H$ (the case $\ch_{1}^{-1}(F)=0$ is
impossible, since in this case $\nu_{1,-1}(F)=+\infty$, which contradicts
$\nu_{1,-1}$-semistability of $E$, and in the case $\ch_{1}^{-1}(F)=2H$ we
have $\nu_{\alpha,-1}(F)<\nu_{\alpha,-1}(E/F)=+\infty$, so $F$ does not
destabilize $E$). Repeating the argument carried out above in the proof of
this Lemma, we obtain that either $F\cong\SS(-1)$, or $E$ is included in an
exact triple of the form (19). Direct calculation shows that $\SS(-1)$ does
not destabilize $E$ when $\beta=-1,\alpha>1$, and for any object $E$ from (19)
we have
$\operatorname{Hom}(\SS(-1),E)=\operatorname{Hom}(E,\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(-1))=0$,
that is, $E$ cannot be included simultaneously in (19) and in (20). ∎

######  Lemma 3.9.

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a tilt-semistable object with
$\ch(E)=(2,-H,-H^{2},e)$. Then $e\leq\frac{19}{6}$, and if $e=\frac{19}{6}$,
then $E$ is destabilized by an exact triple

(23) $0\to\mathcal{O}_{X}(-1)^{\oplus 2}\to
E\to\mathcal{I}_{\mathbb{P}^{1},Q_{2}}(-1)\to 0.$

###### Proof.

Suppose that $e\geq\frac{19}{6}$. Then $W_{0,-\frac{3}{2}}(E)<0$, and we have
$\ch_{\leq 2}^{-\frac{3}{2}}(E)=(2,2H,-\frac{1}{4}H^{2})$, therefore there is
a destabilizing subobject or quotient $F$ of $E$ with
$\ch_{1}^{-\frac{3}{2}}(F)\in\\{\frac{1}{2}H,H\\}$. Suppose that $\ch_{\leq
2}^{-\frac{3}{2}}(F)=(R,H,xH^{2})$; Then the equality
$\nu_{\alpha,-\frac{3}{2}}(F)=\nu_{\alpha,-\frac{3}{2}}(E)$ implies
$x+\frac{1}{8}=\alpha^{2}(R-1)$. Suppose that $R>1$, then
$-\frac{1}{8}<x\leq\frac{1}{4}$. If $x=0$, then $\Delta(E/F)\geq 0$, since
Chern classes are integers it follows that $R=4$, whence we obtain in the
usual way that $e\leq\frac{7}{6}$. The case $x=\frac{1}{8}$ is impossible by
numerical considerations, and if $x=\frac{1}{4}$, then $R=2$, and we get the
required inequality and the exact triple (23).

Suppose that $\ch_{\leq 2}^{-\frac{3}{2}}(F)=(R,\frac{1}{2}H,xH^{2})$. Then
the equality $\nu_{\alpha,-\frac{3}{2}}(F)=\nu_{\alpha,-\frac{3}{2}}(E)$
implies $2x+\frac{1}{8}=\alpha^{2}(2R-1)$. For $R\geq 1$ there is only one
possibility $R=1,\ x=\frac{1}{8}$, and then $\ch_{3}(F)$ is maximized by the
sheaf $F\cong\mathcal{O}_{X}(-1)$, and we also obtain the exact triple (23).
For $R<1$ there is only one possibility $R=-1,x=-\frac{1}{8}$, from where
$F\cong\mathcal{O}_{X}(-2)[1]$. In this case we also get $e\leq\frac{7}{6}$. ∎

Now we can proceed to the general case.

######  Lemma 3.10.

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a tilt-semistable object with
$\ch(E)=(2,cH,dH^{2},e)$. Assume that either

  1. (1)

$c=-1,d\leq-\frac{3}{2}$, and $e\geq d^{2}-2d+\frac{1}{6}$, or

  2. (2)

$c=0,d\leq-\frac{5}{2}$, and $e\geq d^{2}.$

Then $E$ is destabilized along a semicircular wall by a subobject or quotient
of rank not more than two.

###### Proof.

(1) Suppose that $c=-1,\ d\leq-\frac{3}{2}$, and $e\geq d^{2}-2d+\frac{1}{6}$.
Then a calculation gives

$\frac{\rho_{W}(E)}{\Delta(E)}\geq-\frac{9}{64}d-\frac{19}{256}-\frac{9}{128(1-4d)}-\frac{27}{64(1-4d)^{2}}+\frac{81}{256(1-4d)^{3}}.$

Bounding each term, we find $\frac{\rho_{W}(E)}{\Delta(E)}>\frac{1}{12}$,
which means we can apply Proposition 2.3.

(2) Suppose that $c=0,\ d\leq-\frac{5}{2}$, and $e\geq d^{2}.$ A calculation
gives $\frac{\rho_{W}(E)}{\Delta(E)}\geq-\frac{9}{64}d-\frac{1}{4},$ and
applying Proposition 2.3 completes the proof. ∎

######  Lemma 3.11.

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a tilt-semistable object with
$\ch(E)=(2,cH,dH^{2},e)$.
(1) Let $c=-1$ and $d\leq-\tfrac{3}{2}$. If $E$ is destabilized by a subobject
$F$ of rank one, then $e\leq d^{2}-2d+\tfrac{5}{12}$. If
$e=d^{2}-2d+\tfrac{5}{12}$ with non-integer $d$, or $e=d^{2}-2d+\tfrac{1}{6}$
with integer $d$, then $F\cong\mathcal{O}_{X}(-1)$ and $E$ is destabilized by
a subobject of rank two.
(2) Let $c=0$ and $d\leq-\tfrac{5}{2}$. If $E$ is destabilized by a subobject
or quotient $F$ of rank one, then $e<d^{2}$.

###### Proof.

(1) Suppose that $c=-1,\ d\leq-\tfrac{3}{2}$ and $e\geq
d^{2}-2d+\tfrac{1}{6}$. A calculation gives
$W_{0,-\frac{3}{2}}(E)=4d^{2}-6d+\frac{9}{4}-6e\leq-2d^{2}+6d+\frac{5}{4}<0$,
and therefore

$0<H^{2}\cdot\ch_{1}^{-\frac{3}{2}}(F)<H^{2}\cdot\ch_{1}^{-\frac{3}{2}}(E)=2.$

Hence $\ch_{1}(F)\in\\{-H,0\\}$. Assume that $\ch_{1}(F)=-H$. Let us show that
$F$ is a subobject, not a quotient, of the object $E$. We have
$\ch(F)=(1,0,-yH^{2},z)\cdot\ch(\mathcal{O}_{X}(-1))$, and Remark 2.1 shows
that $z\leq y(y+1)$. A calculation gives

$s_{W}(E)=\frac{d+3e}{4d-1}\leq\frac{6d^{2}-10d+1}{8d-2},\ \ \ \
s(E,F)=d+2y-1.$

Since $s(E,F)\leq s_{W}(E)$, we have

$y\leq\frac{d^{2}+1}{2-8d}<-\frac{d}{2}-\frac{1}{4}.$

Using Remark 2.1 for the quotient $E/F$, we find

$e\leq(\frac{1}{2}-d-y)(\frac{3}{2}-d-y)+y(y+1)+2y-\frac{1}{3}.$

The right-hand side of this inequality is a quadratic function of $y$ with a
minimum at the point $y=-\frac{d}{2}-\frac{1}{4}$. Therefore, the maximum is
achieved at the point $y=0$ with $F\cong\mathcal{O}_{Q}(-1)$, and we get
$e\leq d^{2}-2d+\tfrac{5}{12}$. Note also that for $y>0$ we have $e\leq
d^{2}-d+\frac{17}{12}<d^{2}-2d+\frac{1}{6}$, so for the integer $d$ we also
find that $y=0$. Now from the proof of [22, Prop. 3.2] it follows that there
is a destabilizing morphism $\mathcal{O}_{X}(-1)\to G=E/F$, and the quotient
has the Chern character $\ch_{\leq 2}(G^{\prime})=(0,H,(d-1)H^{2})$. The
argument from the proofs of Lemmas 3.5 and 3.8 shows that for the non-integer
$d$ the character $\ch_{3}(G^{\prime})$ is maximized by the sheaf
$G^{\prime}\cong\mathcal{O}_{Q_{2}}(d-\frac{1}{2})$, while for the integer $d$
there is an exact triple

$0\to\SS(d-2)\to\mathcal{O}_{X}(d-1)^{\oplus 2}\to G^{\prime}\to 0.$

A calculation shows that in any case $\operatorname{Ext}\nolimits^{1}(F,G)=0$,
so $F$ is indeed a subobject of $E$ and there is a destabilizing morphism
$\mathcal{O}_{X}(-1)^{\oplus 2}\to E$.

(2) Suppose that $c=0,\ d\leq-\frac{5}{2}$ and $e\geq d^{2}$. By calculation
we get $W_{0,-1}(E)=4d^{2}-4d-6e\leq-2d^{2}-4d<0,$ and therefore

$0<H^{2}\cdot\ch_{1}^{-1}(F)<H^{2}\cdot\ch_{1}^{-1}(E)=2.$

This means that $\ch_{1}(F)=0$, but such $F$ does not define a semicircular
wall. ∎

Proof of Theorem 3.1. The inequalities on $d$ follow from the first inequality
in Proposition 2.1.(ii). Other statements of the Theorem are proved by
induction on $\Delta(E)$, and the start of the induction is given by Lemmas
3.1 – 3.9. Lemma 3.10 shows that $E$ is destabilized by a subobject
$F\hookrightarrow E$, where $F$ has rank $R\in\\{0,1,2\\}$. Lemma 3.11 deals
with the case $R=1$. Let us assume that $R=2$. In what follows we will show
that the case $R=0$ is realized only if $E$ is a direct sum.

(1) Suppose that $c=-1,\ d\leq-\frac{3}{2},$ and $e\geq d^{2}-2d+\frac{1}{6}$.
As in the proof of Lemma 3.11, we have $W_{0,-\frac{3}{2}}(E)<0$, and also
$W_{0,-2}(E)<0$. Then the inequalities $H^{2}\cdot\ch_{1}^{-3/2}>0$ and
$H^{2}\cdot\ch_{1}^{-2}(F)<H^{2}\cdot\ch_{1}^{-2}(E)$ together imply that
$\ch_{1}(F)=-2H$. Therefore, we can assume that
$\ch(F)=(2,0,yH^{2},z)\cdot\ch(\mathcal{O}_{X}(-1))$. If $y=0$, then $z\leq
0$, and the arguments from the proof of Lemma 3.11 give the required
statement. Let us assume that $y\leq-\frac{1}{2}$. By the induction hypothesis
we have $z\leq y^{2}$. A calculation gives:

$s_{W}(E)=\frac{d+3e}{4d-1}\leq\frac{6d^{2}-10d+1}{8d-2},\ \ \ \
s(E,F)=d-y-1.$

Since $s(E,F)\leq s_{W}(E),$ then $y\geq\frac{2d^{2}+1}{8d-2}>\frac{d}{2}.$
Maximizing $\ch_{3}(E/F)$, we get

$e\leq(d-y)^{2}-2(d-y)+y^{2}-2y+C,$

where $C$ is a constant depending on whether $d$ and $y$ are integers or non-
integers. For $y\in\mathbb{Z}$ or $y\in\frac{1}{2}+\mathbb{Z}$ the right hand
side of the inequality is a quadratic function of $y$ with the minimum at the
point $y=\frac{d}{2}$. Therefore the maximum is achieved at $y=-\frac{1}{2}$
or $y=-1$. For $y=-\frac{1}{2}$ by calculation we get $e\leq
d^{2}-d+\frac{11}{12}<d^{2}-2d+\frac{1}{6}$, and for $y=-1$ we find $e\leq
d^{2}+\frac{29}{12}<d^{2}-2d+\frac{1}{6}$.

(2) Suppose that $c=0,\ d\leq-\frac{5}{2}$ and $e\geq d^{2}$. As in the proof
of Lemma 3.11, we have $W_{0,-1}(E)<0,$ which implies that $\ch_{1}(F)=-H$.
Suppose $\ch(F)=(2,-H,yH^{2},z)$. By the induction hypothesis we have $z\leq
y^{2}-2y+\frac{5}{12}.$ A calculation gives:

$s_{W}(E)=\frac{3e}{4d}\leq\frac{3}{4}d,\ \ \ \ s(E,F)=d-y,$

hence $y\geq\frac{d}{4}>\frac{d}{2}+\frac{1}{2}.$ Maximizing $\ch_{3}(E/F)$,
we find

$e\leq 2y^{2}-(2d+2)y+C,$

where $C$ is a constant depending on whether $d$ and $y$ are integers or non-
integers. For $y\in\mathbb{Z}$ or $y\in\frac{1}{2}+\mathbb{Z}$ the right hand
side of the inequality is a quadratic function of $y$ with the minimum at the
point $y=\frac{d}{2}+\frac{1}{2}$. Therefore the maximum is achieved at $y=0$
or $y=-\frac{1}{2}$. For $y=-\frac{1}{2}$ we get $e\leq d^{2}+d+2<d^{2}$,
which implies that $y=0$. It remains to note that the equality
$\operatorname{Ext}\nolimits^{1}(F,E/F)=0$ holds, verified by the exact
triples defining $F$ and $E/F$.

(3) and (4). A direct consequence of the statements (1) and (2), respectively.
∎

##  4\. Infinite series of components of moduli spaces of semistable sheaves
of rank two on varieties $X_{1}$, $X_{2}$, $X_{4}$ and $X_{5}$

###  4.1. First series.

In this susection we consider extensions of one of the following two forms.
(1) Extensions on $X=X_{1},\ X_{2},\ X_{4}$ or $X_{5}$ of the form

(24) $\begin{split}&0\to\mathcal{O}_{X}(-n)^{\oplus 2}\to
E\to\mathcal{O}_{S}(m)\to 0,\ \ \ \ \ \ \ \ \ \ \ S\in|\mathcal{O}_{X}(k)|,\\\
&\text{where}\ \ \ \ \ \ k\geq 1,\ \ \ \ \ n=\lceil\frac{k}{2}\rceil,\ \ \ \ \
\ m+n<0,\end{split}$

(2) or an extension on the quadric $X=X_{2}$ of the form

(25) $\begin{split}&0\to\mathcal{O}_{X}(-1)^{\oplus 2}\to
E\to\mathcal{I}(m)\to 0,\ \ \ \ \ \ \ \ \ \ \ S\in|\mathcal{O}_{X}(1)|,\\\
&\text{where}\ \ \ \ \ \mathcal{I}=\mathcal{I}_{\mathbb{P}^{1},S},\ \ \ \ \ \
m<0,\end{split}$

$\mathbb{P}^{1}$ is a projective line on a two-dimensional quadric $S$, and
the sheaf $\mathcal{I}$ is described in (13). This section mainly deals with
extensions (24). In particular, for $E$ from (24) we have:

(26) $\begin{split}&\operatorname{{rk}}(E)=2,\ \ \ c_{1}(E)=0,\ \ \
c_{2}(E)=(\frac{k^{2}}{4}-km)H^{2},\\\
&c_{3}(E)=(\frac{k^{3}}{4}-k^{2}m+km^{2})H^{3},\ \ \ \ \ \ \ \text{if}\ \ \ k\
\ \ \text{is even},\end{split}$ (27) $\begin{split}&c_{1}(E)=-H,\ \ \
c_{2}(E)=(\frac{(k-1)^{2}}{4}-km)H^{2},\ \ \ c_{3}(E)=\\\
&(\frac{k^{3}}{4}-k^{2}m+km^{2})H^{3},\ \ \ \ \ \ \text{if}\ \ \ k\ \ \
\text{is odd},\end{split}$

where $H$ is the cohomology class of a hyperplane section $X$.

It is well known and verified by standard calculation that

(28) $h^{i}(\mathcal{O}_{X}(\ell))=0\;\ \ \ \ \ \ \ \ \ \ell\in\mathbb{Z},\ \
i=1,2.$

Let us now assume that

(29) $\textit{the sheaf}\ E\ \textit{in}\ \eqref{extension}\ \textit{is
stable}.$

In this case, from (13), (24), (25), (28), the exact triples

(30) $0\to\mathcal{O}_{X}(-k)\to\mathcal{O}_{X}\to\mathcal{O}_{S}\to 0,\ \ \ \
\ \ \ \ $ (31) $\ \ \ \ \ \ \
0\to\mathcal{I}\to\mathcal{O}_{S}\to\mathcal{O}_{\mathbb{P}^{1}}\to 0,\ \ \ \
\ \ \ \ \ \ \ X=X_{2},\ \ \ \ \ \ \ \ ,$

the Serre–Grothendieck duality and the stability of $E$ we get the equalities
(a) for the extension (24):

(32)
$\begin{split}&\hom(E,E)=\hom(\mathcal{O}_{S}(m),\mathcal{O}_{S}(m))=\hom(E,\mathcal{O}_{S}(m))=1,\\\
&\operatorname{ext}^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{S}(m))=N,\end{split}$
(33) $\begin{split}&\hom(\mathcal{O}_{X}(-n)^{\oplus
2},\mathcal{O}_{S}(m))=0,\\\
&\operatorname{ext}^{1}(\mathcal{O}_{X}(-n)^{\oplus
2},E)=\operatorname{ext}^{1}(\mathcal{O}_{X}(-n)^{\oplus
2},E)=\operatorname{ext}^{1}(\mathcal{O}_{X}(-n)^{\oplus
2},\mathcal{O}_{S}(m))=\\\ &\operatorname{ext}^{1}(\mathcal{O}_{X}(-n)^{\oplus
2},\mathcal{O}_{X}(-n)^{\oplus
2})=\operatorname{ext}^{i}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n)^{\oplus
2})=0,\ \ i=0.2.\end{split}$

(b) for the extension (25):

(34)
$\begin{split}&\hom(E,E)=\hom(\mathcal{O}_{S}(m),\mathcal{O}_{S}(m))=\hom(E,\mathcal{O}_{S}(m))=1,\\\
&\operatorname{ext}^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{S}(m))=4,\end{split}$
(35)
$\begin{split}&\hom(\mathcal{I}(m),\mathcal{I}(m))=\hom(E,\mathcal{I}(m))=1,\
\ \hom(\mathcal{O}_{X}(-n)^{\oplus 2},\mathcal{I}(m))=0,\\\
&\operatorname{ext}^{i}(\mathcal{O}_{X}(-1)^{\oplus 2},\mathcal{I}(m))=0,\
i=0,1,\end{split}$ (36) $\hom(\mathcal{O}_{X}(-1)^{\oplus
2},\mathcal{I}(1))=4,\ \ \ \hom(\SS(-1),\mathcal{I}(1))=7,\ \ \ \
\operatorname{ext}^{1}(\mathcal{O}_{X}^{\oplus 2},\mathcal{I}(1))=0,$ (37)
$\begin{split}&\hom(\mathcal{O}_{S},\mathcal{O}_{\mathbb{P}^{1}})=1,\ \ \ \ \
\operatorname{ext}^{1}(\mathcal{O}_{S},\mathcal{O}_{\mathbb{P}^{1}})=k+1,\ \ \
\ \ \operatorname{ext}^{1}(\mathcal{O}_{S},\mathcal{O}_{S})=N,\\\
&\operatorname{ext}^{i}(\mathcal{I}(m),\mathcal{O}_{X}(-n)^{\oplus 2})=0,\ \ \
\ \ i=0,2.\end{split}$

Let $\mathbb{P}^{N}=|\mathcal{O}_{X}(k)|$, let
$\mathbb{S}\subset\mathbb{P}^{N}\times X$ be the universal family of surfaces
of degree $k$ in $X$ and let $\mathbb{P}^{N}\times
X\xrightarrow{p}\mathbb{P}^{N}$ be the projection. Respectively, in the
notation of the triple (12), let $\mathbb{G}\times
X_{2}\xrightarrow{\widetilde{p}}\mathbb{G}$ be the projection. Due to the
smoothness of $\mathbb{P}^{N}$ and $\mathbb{G}$, from (33)-(37), [21, Thm.
1.4], [5, Satz 3.(ii)] there are defined locally free sheaves $\mathcal{A}$
and $\widetilde{\mathcal{A}}$ on $\mathbb{P}^{N}$ and $\mathbb{G}$,
respectively:

(38)
$\begin{split}&\mathcal{A}=\mathcal{E}xt_{p}^{1}(\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(m)|_{\mathbb{S}},\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(-n)),\\\
&\widetilde{\mathcal{A}}=\mathcal{E}xt_{\widetilde{p}}^{1}(\mathbb{I}\otimes\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X_{2}}(m),\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X_{2}}(-1)).\end{split}$

The construction of these sheaves commutes with the base change:

(39) $\begin{split}&\tau_{S}:\
\mathcal{A}\otimes\mathbf{k}_{S}\xrightarrow{\cong}\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n)),\
\ \ \ \ \ \ \ \ \ \ S\in|\mathcal{O}_{X}(k)|,\\\ &\widetilde{\tau}_{x}:\
\widetilde{\mathcal{A}}\otimes\mathbf{k}_{x}\xrightarrow{\cong}\operatorname{Ext}\nolimits^{1}(\mathcal{I}(m),\mathcal{O}_{X}(-1)),\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in\mathbb{G},\ \ \ \ \ \ S=S_{x},\end{split}$

and we have

(40)
$\begin{split}&\mathcal{E}xt_{p}^{0}(\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(m)|_{\mathbb{S}},\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(-n))=0,\\\
&\mathcal{E}xt_{\widetilde{p}}^{0}(\mathbb{I}\otimes\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X_{2}}(m),\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X_{2}}(-1))=0.\end{split}$

Consider the varieties $P=\mathbb{P}(\mathcal{A}^{\vee})$, $X_{P}=P\times X$,
$\widetilde{P}=\mathbb{P}(\widetilde{\mathcal{A}}^{\vee})$,
$X_{\widetilde{P}}=\widetilde{P}\times X_{2}$, with the projections
$P\xrightarrow{\rho}\mathbb{P}^{N}$,
$\widetilde{P}\xrightarrow{\widetilde{\rho}}\mathbb{G}$,
$P\xleftarrow{p_{P}}X_{P}\xrightarrow{\rho_{P}}\mathbb{P}^{N}\times X$,
$\widetilde{P}\xleftarrow{\widetilde{p}_{P}}X_{\widetilde{P}}\xrightarrow{\widetilde{\rho}_{P}}\mathbb{G}\times
X_{2}$. From (39), (40) and [21, Cor. 4.5] it follows that on $X_{P}$ and
$X_{\widetilde{P}}$ there are universal families of extensions, respectively:

$\begin{split}&0\to
p_{P}^{*}\mathcal{O}_{P}(1)\otimes(\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(-n))\to\mathcal{V}\to\rho_{P}^{*}(\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(m)|_{\mathbb{S}})\to
0,\\\
&0\to\widetilde{p}_{P}^{*}\mathcal{O}_{\widetilde{P}}(1)\otimes(\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X_{2}}(-1))\to\widetilde{\mathcal{V}}\to\widetilde{\rho}_{P}^{*}(\mathbb{I}\otimes\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X_{2}}(m))\to
0.\end{split}$

Applying the functors $\rho_{P*}$ and $\widetilde{\rho}_{P*}$ to these
triples, respectively, we obtain the universal extensions on
$\mathbb{P}^{N}\times X$ and on $\mathbb{G}\times X$, where
$\mathcal{W}=\rho_{P*}\mathcal{V}$ and
$\widetilde{\mathcal{W}}=\widetilde{\rho}_{P*}\widetilde{\mathcal{V}}$:

(41) $\begin{split}&0\to
p^{*}\mathcal{A}^{\vee}\otimes(\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(-n))\to\mathcal{W}\to\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(m)|_{\mathbb{S}}\to
0,\\\
&0\to\widetilde{p}^{*}\widetilde{\mathcal{A}}^{\vee}\otimes(\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X_{2}}(-1))\to\widetilde{\mathcal{W}}\to\mathbb{I}\otimes\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X_{2}}(m)\to
0.\end{split}$

Note that the extensions (24) and (25) are specified by elements
$\xi\in\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\\\
\mathcal{O}_{X}(-n)^{\oplus 2})$ and
$\widetilde{\xi}\in\operatorname{Ext}\nolimits^{1}(\mathcal{I}(m),\mathcal{O}_{X_{2}}(-1)^{\oplus
2})$ respectively, considered as
homomorphisms

(42) $\xi:\
\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n))^{\vee}\to\mathbf{k}^{2},\
\ \ \widetilde{\xi}:\
\operatorname{Ext}\nolimits^{1}(\mathcal{I}(m),\mathcal{O}_{X_{2}}(-1))^{\vee}\to\mathbf{k}^{2}.$

We will carry out further argument in this subsection for the upper exact
triple (41). (For the lower triple (41) the argument is completely similar —
see subsection 4.2.) By the universality, this triple, restricted to
$\\{S\\}\times X$, in view of the base change (39) is included, together with
the extension (24), where $E=E_{\xi}$, into a commutative pushout diagram

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Note that the homomorphism $\xi$ in (42) is non-zero, since the extension (24)
is non-trivial. Moreover, if $E_{\xi}$ is stable, then $\xi$ is an
epimorphism. Indeed, if $\xi$ has rank 1, then the diagram (43) goes through a
subextension $0\to\mathcal{O}_{X}(-n)\to E^{\prime}\to\mathcal{O}_{S}(m)\to 0$
of the extension (24), which implies the existence of a homomorphism
$E_{\xi}\twoheadrightarrow\mathcal{O}_{X}(-n)$, contrary to the stability of
$E_{\xi}$.

Suppose that $\xi$ in (43) is an epimorphism. The class $[\xi]$ of the
epimorphism $\xi$ modulo automorphisms of the space $\mathbf{k}^{2}$ is a
point of the grassmannian
$\mathrm{Gr}(\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n))^{\vee},2)$.
Therefore, let us consider the grassmannization

(44) $\pi:\ {\mathcal{G}}r(\mathcal{A}^{\vee},2)\to\mathbb{P}^{N}$

of two-dimensional quotient spaces of the bundle $\mathcal{A}^{\vee}$ on
$\mathbb{P}^{N}$ and its open subsets

(45) $\mathcal{Y}_{k,m,n}:=\\{[\xi]\in{\mathcal{G}}r(\mathcal{A}^{\vee},2)\ |\
E_{\xi}\ \textit{is stable}\\},$ (46)
$\mathcal{R}_{k,m,n}:=\\{[\xi]\in{\mathcal{G}}r(\mathcal{A}^{\vee},2)\ |\
E_{\xi}\ \textit{is reflexive}\\}.$

######  Theorem 4.1.

There are embeddings of dense open sets

(47)
$\mathcal{R}_{k,m,n}\subset\mathcal{Y}_{k,m,n}\subset{\mathcal{G}}r(\mathcal{A}^{\vee},2).$

###### Proof.

(1) First, let us check the reflexivity of a general sheaf $E$ in the triple
(24). (In the remaining cases the proofs are completely similar.) Namely,
applying to the triple (24) the functor
$\mathbf{R}\mathcal{H}om(-,\mathcal{O}_{X})$ and taking into account that
$\mathcal{E}xt^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X})\\\
\cong\mathcal{O}_{S}(k-m)$, we obtain the exact sequence

(48) $0\to E^{\vee}\to\mathcal{O}_{X}(n)^{\oplus
2}\to\mathcal{O}_{S}(m)\to\tau\to 0,\ \ \ \ \ \ \ \ \
\tau:=\mathcal{E}xt^{1}(E,\mathcal{O}_{X}).$

The long exact sequence defined by the spectral sequence of local and global
Exts
$E^{pq}_{2}=H^{p}(\mathcal{E}xt_{\mathcal{O}_{X}}^{q}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n)^{\oplus
2}))\Longrightarrow\operatorname{Ext}\nolimits^{p+q}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n)^{\oplus
2})$, gives an isomorphism

(49)
$\begin{split}&\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n)^{\oplus
2}))\xrightarrow{\cong}H^{0}(\mathcal{E}xt^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n)^{\oplus
2})))\\\ &\cong H^{0}(\mathcal{O}_{S}(k-m-n)^{\oplus 2}).\end{split}$

Under this isomorphism the element
$\xi\in\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n)^{\oplus
2}))$, defining the extension (24) corresponds to a section $s_{\xi}\in
H^{0}(\mathcal{O}_{S}(k-m-n)^{\oplus 2})$, scheme of zeros
$Z_{\xi}=(s_{\xi})_{0}$ of which, due to the ampleness of the sheaf
$\mathcal{O}_{S}(k-m-n)$ for $k\geq 1,\ m+n<0$, is zero-dimensional for $\xi$
belonging to an open dense subset in
$\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n)^{\oplus
2}))$. A standard local calculation shows that $\operatorname{Sing}E=Z_{\xi}$,
so $\tau$ is an artinian sheaf and thus
$\mathcal{E}xt^{i}(\tau,\mathcal{O}_{X})=0$ for $i\leq 2$. Therefore, applying
the functor $\mathbf{R}\mathcal{H}om(-,\mathcal{O}_{X})$ to (48) and taking
into account the isomorphism
$\mathcal{E}xt^{1}(\mathcal{O}_{S}(k-m),\mathcal{O}_{X})\cong\mathcal{O}_{S}(m)$,
we obtain an exact triple $0\to\mathcal{O}_{X}(-n)^{\oplus 2}\to
E^{\vee\vee}\to\mathcal{O}_{S}(m)\to 0$ and its isomorphism with the triple
(24) via the functor $(-)^{\vee\vee}$. This implies the required isomorphism
$E\cong E^{\vee\vee}$.

(2) Let us prove the stability of an arbitrary sheaf
$E\in\mathcal{R}_{k,m,n}$. Recall that in (24) $k\geq 1$,
$n=\lceil\frac{k}{2}\rceil\geq 1$, $m<-n\leq 1$. This implies, What

(50) $-1\leq c_{1}(E)\leq 0,$ (51) $h^{0}(E)=0.$

Assume that $E$ is unstable. In view of (50) the maximal destabilizing rank 1
subsheaf $F$ of $E$ has the form $F=\mathcal{I}_{Z,\mathbb{P}^{3}}(q),\ q\geq
0$. By virtue of reflexivity of $E$, the monomorphism $F\to E$ extends to the
monomorphism $\mathcal{O}_{\mathbb{P}^{3}}(q)\cong
F^{\vee\vee}\overset{s}{\to}E$, that is, a section $0\neq s\in h^{0}(E(-q))$.
This contradicts (51), since $q\geq 0$. ∎

Consider the modular morphism of the variety $\mathcal{Y}_{k,m,n}$ into the
Gieseker-Maruyama moduli scheme $M_{X}(v)$:

(52) $\Theta:\ \mathcal{Y}_{k,m,n}\to M_{X}(v),\ \ \ [\xi]\mapsto[E_{\xi}],$

where $[E_{\xi}]$ is the isomorphism class of the stable sheaf $E_{\xi}$
obtained from the point $\xi$ as the lower extension in the diagram (43), and
$v=\mathrm{ch}(E)$ is determined by formulas (7) and (26)-(27).

Note that by virtue of the first equality (33) the sheaf
$[E=E_{[\xi]}]\in\Theta(\mathcal{Y}_{k,m,n})$ has a unique subsheaf
$\mathcal{O}_{X}(-n)^{\oplus 2}$, and hence the extension (24) is uniquely
defined, that is, the class $[\xi]$ can be reconstructed from the point $[E]$
uniquely. Therefore,

(53) $\Theta:\ \mathcal{Y}_{k,m,n}\to M_{k,m,n}:=\Theta(\mathcal{Y}_{k,m,n})\
-\ \text{bijection},$ (54) $\begin{split}&\dim
M_{k,m,n}=\dim\mathcal{Y}_{k,m,n}=\dim\mathrm{Gr}(\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n))^{\vee},2)+\\\
&+\dim|\mathcal{O}_{X}(k)|=\mathrm{ext}^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n)^{\oplus
2})-4+N.\end{split}$

Applying the bifunctor $\mathbf{R}\operatorname{Hom}(-,-)$ to the triple (24)
and taking into account the equalities (32) and (33), we obtain for
$[E]\in\dim\mathcal{Y}_{k,m,n}$ a commutative diagram

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From this diagram and the equality
$\mathrm{ext}^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{S}(m))=N$ (see (32)) we get

(56)
$\mathrm{ext}^{1}(E,E)=\mathrm{ext}^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n)^{\oplus
2})-4+N,$

hence, due to (54), we find
$\dim\Theta(\mathcal{Y}_{k,m,n})=\mathrm{ext}^{1}(E,E)=\dim_{[E]}M_{X}(v)$.
Taking into account Theorem 4.1, this means that $M_{k,m,n}$ is a smooth open
subset of an irreducible component $\overline{M}_{k,m,n}$ of the moduli scheme
$M_{X}(v)$. From here and from (53), according to [31, Ch. 2, §4.4, Thm.
2.16], it follows that

(57) $\Theta:\ \mathcal{Y}_{k,m,n}\to M_{k,m,n}\ \text{is an isomorphism},$

that is, $M_{k,m,n}$ is a smooth rational variety, isomorphic in view of
Theorem 4.1 to a dense open subset in the smooth variety
${\mathcal{G}}r(\mathcal{A}^{\vee},2)$. At the same time, it directly follows
from the diagram (43) that on $M_{k,m,n}$ there is a universal family of
sheaves. Formulas for dimensions of varieties $M_{k,m,n}$ as functions of
$k,m,n$ are obtained taking into account the Riemann–Roch Theorem for $X_{5}$,
formulas (28), (30), (54) and standard resolutions for the sheaves
$\mathcal{O}_{X_{2}}$, $\mathcal{O}_{X_{4}}$:

(58)
$\begin{split}&0\to\mathcal{O}_{\mathbb{P}^{4}}(-2)\to\mathcal{O}_{\mathbb{P}^{4}}\to\mathcal{O}_{X_{2}}\to
0,\\\
&0\to\mathcal{O}_{\mathbb{P}^{5}}(-4)\to\mathcal{O}_{\mathbb{P}^{5}}(-2)^{\oplus
2}\to\mathcal{O}_{\mathbb{P}^{5}}\to\mathcal{O}_{X_{4}}\to 0,\\\ \end{split}$

Moreover, in view of Theorem 4.1, the reflexive sheaves in $M_{k,m,n}$
constitute a dense open set

(59) $R_{k,m,n}:=\Theta(\mathcal{R}_{k,m,n})\cong\mathcal{R}_{k,m,n}.$

Thus, the following theorem holds.

######  Theorem 4.2.

Let $X$ be one of the Fano varieties $X_{1}$, $X_{2}$, $X_{4}$, $X_{5}$. For
an arbitrary natural number $k$ and integers $m$ and
$n=\lceil\frac{k}{2}\rceil$ such that $m+n<0$ let $M_{X}(v)$ be the Gieseker-
Maruyama moduli scheme of semistable sheaves $E$ on $X$ with Chern character
$v=\mathrm{ch}(E)$, determined by the formulas (7) and (26)-(27). Then the
following statements are true.
(i) The family

$M_{k,m,n}=\\{[E]\in M_{X}(v)\ |\ E\ \textit{is a stable extension
\eqref{extension}, where}\ S\in|\mathcal{O}_{X}(k)|\\}$

is a smooth dense open subset of an irreducible rational component of scheme
$M_{X}(v)$, and there is an isomorphism (57), in which $\mathcal{Y}_{k,m,n}$
is the open subset of the grassmannization defined in (45) $\pi:\
{\mathcal{G}}r(\mathcal{A}^{\vee},2)\to\mathbb{P}^{N}$, where $\mathcal{A}$ is
the locally free sheaf on $\mathbb{P}^{N}=|\mathcal{O}_{X}(k)|$, given by the
first formula (38); in this case $M_{k,m,n}$ is a fine moduli space, and
reflexive sheaves constitute a dense open subset $R_{k,m,n}$ defined in (59).
(ii) The dimension of the variety $M_{k,m,n}$ is given by the formulas:
(ii.1) $\dim
M_{k,m,n}=2\binom{k-m-n+3}{3}-2\binom{-m-n+3}{3}+\binom{k+3}{k}-5$ for
$X=X_{1}$,
(ii.2) $\dim
M_{k,m,n}=2\binom{k-m-n+4}{4}-2\binom{k-m-n+2}{4}-2\binom{-m-n+4}{4}+2\binom{-m-n+2}{4}+\binom{k+4}{4}-\binom{k+2}{4}-5$
for $X=X_{2}$,
(ii.3) $\dim
M_{k,m,n}=2\binom{k-m-n+5}{5}-4\binom{k-m-n+3}{5}+2\binom{k-m-n+1}{5}-2\binom{-m-n+5}{5}+4\binom{-m-n+3}{5}-2\binom{-m-n+1}{5}+2\binom{k+5}{5}-2\binom{k+3}{5}+\binom{k+1}{5}-5$
for $X=X_{4}$,
(ii.4) $\dim
M_{k,m,n}=\frac{5}{3}(k-m-n)^{3}+5(k-m-n)^{2}+\frac{16}{3}(k-m-n)-\frac{5}{3}(-m-n)^{3}-5(-m-n)^{2}-\frac{16}{3}(-m-n)+\frac{5}{3}k^{3}+5k^{2}+\frac{16}{3}k-4$
for $X=X_{5}$.
(iii) The dimension of the variety $M_{k,m,n}$ coincides with the virtual
dimension of the scheme $M_{X}(v)$, that is, $\operatorname{ext}^{2}(E,E)=0$
for $[E]\in M_{k,m,n}$, if and only if $k<i$ and $m>k-n-i$, where $i$ is the
index of the Fano variety $X$.

###### Proof.

Statements (i) and (ii) of the theorem were proven above. Statement (iii) is
proven by a direct calculation by analogy with the formulas (33). ∎

######  Remark 4.3.

Direct checking using formulas (26) and (27) for the Chern classes of the
sheaf $[E]\in M_{k,m,n}$ shows that
$\frac{c_{3}(E)}{H^{3}}<\frac{4}{k}(\frac{c_{2}(E)}{H^{2}})^{2}$ in case of
even $k$, and, respectively,
$\frac{c_{3}(E)}{H^{3}}<\frac{4k}{(k-1)^{2}}(\frac{c_{2}(E)}{H^{2}})^{2}$ in
case of odd $k$.

######  Remark 4.4.

On the quadric $X=X_{2}$ we consider a nontrivial extension of the form

(60) $\begin{split}&0\to\SS(-n)\to E\to\mathcal{O}_{S}(m)\to 0,\ \ \ \ \ \ \ \
\ S\in|\mathcal{O}_{X}(k)|,\\\ &k\geq 1,\ \ \ \ \
n=\lfloor\frac{k}{2}\rfloor+1,\ \ \ \ \ m\leq-n,\end{split}$

where $\SS$ is the spinor bundle on $X$. Let $M_{X}(v)$ be the Gieseker-
Maruyama moduli scheme of semistable sheaves $E$ with Chern character
$v=\mathrm{ch}(E)$, determined by the formulas (7), (10) and (60). From (60)
it follows that $E$ satisfies the formulas (32), (33), (56) and the diagram
(55), in which $\mathcal{O}_{X}(-n)^{\oplus 2}$ is everywhere replaced by
$\SS(-n)$, and the left vertical triple in (55) is replaced by the triple
$\mathbf{k}\rightarrowtail\mathbf{k}\twoheadrightarrow 0$. Respectively, in
the notation of the first formula (38), on $\mathbb{P}^{N}$ the following
locally free sheaf is defined:

(61)
$\mathcal{A}_{\SS}=\mathcal{E}xt_{p}^{1}(\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(m)|_{\mathbb{S}},\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\SS(-n)).\\\
$

In addition, replacing the sheaf $\mathcal{O}_{X}(-n)^{\oplus 2}$ in the
formulas (44)-(46) by $\SS(-n)$, we obtain the projectivization $\pi:\
\mathbb{P}(\mathcal{A}_{\SS}^{\vee})\to\mathbb{P}^{N}$ and its dense open
subsets $\mathcal{Y}_{\SS,k,m,n}$ and $\mathcal{R}_{\SS,k,m,n}$ of stable and
reflexive sheaves respectively. For them, by analogy with (47), we have
embeddings of dense open sets

(62)
$\mathcal{R}_{\SS,k,m,n}\subset\mathcal{Y}_{\SS,k,m,n}\subset{\mathcal{G}}r(\mathcal{A}_{\SS}^{\vee},2)$

and, as in (57), an isomorphism

(63) $\Theta:\
\mathcal{Y}_{\SS,k,m,n}\xrightarrow{\cong}M_{k,m,n}:=\Theta(\mathcal{Y}_{\SS,k,m,n}),\
\ \ [\xi]\mapsto[E_{\xi}].$

Repeating for the indicated varieties the corresponding stages of the proof of
Theorem 4.2, we obtain the following theorem.

######  Theorem 4.5.

Let $X=X_{2}$ be the three-dimensional quadric. Then, in the notation of
Remark 4.4, the following statements are true.
(i) The family

$M_{k,m,n}=\\{[E]\in M_{X}(v)\ |\ E\ \textit{is a stable extension
\eqref{extension Spin}, where}\S\in|\mathcal{O}_{X}(k)|\\}$

is a smooth dense open subset of an irreducible rational component of scheme
$M_{X}(v)$, and there is an isomorphism $\Theta$ in (63), in which
$\mathcal{Y}_{\SS,k,m,n}$ is an open subset (defined in Remark 4.4) of the
projectivization $\pi:\
\mathbb{P}(\mathcal{A}_{\SS}^{\vee})\to\mathbb{P}^{N}$, where
$\mathcal{A}_{\SS}$ is the locally free sheaf on
$\mathbb{P}^{N}=|\mathcal{O}_{X}(k)|$, given by the formula (61); wherein
$M_{k,m,n}$ is a fine moduli space, and reflexive sheaves constitute in
$M_{k,m,n}$ a dense open set $R_{\SS,k,m,n}=\Theta(\mathcal{R}_{\SS,k,m,n})$,
where $\mathcal{R}_{\SS,k,m,n}$ is defined in Remark 4.4.
(ii) The dimension of the variety $M_{k,m,n}$ is given by the formula $\dim
M_{k,m,n}=\\\
4\binom{k-m-n+3}{3}-4\binom{k-m-n+3}{3}-4\binom{-m-n+3}{3}+4\binom{-m-n+3}{3}+3$.

###  4.2. Second series.

In this subsection we treat in more detail the extensions (25) on the quadric
$X=X_{2}$.

Let us first find the dimension of the space
$\operatorname{Ext}\nolimits^{1}(\mathcal{I}(m),\mathcal{I}(m))$, which we
will need below. To do this, apply the functor
$\mathbf{R}\operatorname{Hom}(-,\mathcal{I}(1))$ to the triple (13):

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

From here and from (35) and (36) we find

(64)
$\operatorname{ext}^{1}(\mathcal{I}(m),\mathcal{I}(m))=\operatorname{ext}^{1}(\mathcal{I}(1),\mathcal{I}(1))=4.$

Let us now consider the lower triple (41). By universality, this triple
restricted to $\\{x\\}\times X$, $x\in\mathbb{G}$, in view of the base change
(39) can be included with the extension (25), where $\widetilde{\xi}:\
\operatorname{Ext}\nolimits^{1}(\mathcal{I}(m),\mathcal{O}_{X}(-1))^{\vee}\to\mathbf{k}^{2}$
is the second homomorphism in (42) and $E=E_{\widetilde{\xi}}$, into a
commutative diagram similar to (43):

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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
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70.02242pt\raise-43.12444pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
194.67319pt\raise-43.12444pt\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
194.67319pt\raise-43.12444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E_{\widetilde{\xi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
245.35876pt\raise-43.12444pt\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
245.35876pt\raise-43.12444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{I}(m).}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

Moreover, as in (43), if $E_{\widetilde{\xi}}$ is stable, then
$\widetilde{\xi}$ is an epimorphism. Therefore, by analogy with (45)-(46),
consider the grassmannization

$\widetilde{\pi}:\
{\mathcal{G}}r(\widetilde{\mathcal{A}}^{\vee},2)\to\mathbb{G}$

of two-dimensional quotient spaces of the bundle
$\widetilde{\mathcal{A}}^{\vee}$ on $\mathbb{G}$, defined in (38), and its
open subsets

(66)
$\widetilde{\mathcal{Y}}_{m}:=\\{[\widetilde{\xi}]\in{\mathcal{G}}r(\widetilde{\mathcal{A}}^{\vee},2)\
|\ E_{\widetilde{\xi}}\ \textit{is stable}\\},$ (67)
$\widetilde{\mathcal{R}}_{m}:=\\{[\widetilde{\xi}]\in{\mathcal{G}}r(\widetilde{\mathcal{A}}^{\vee},2)\
|\ E_{\widetilde{\xi}}\ \textit{is reflexive}\\}.$

Repeating the proof of Theorem 4.9 for $E$ from the triple (25), we obtain
embeddings of dense open subsets

(68)
$\widetilde{\mathcal{R}}_{m}\subset\widetilde{\mathcal{Y}}_{m}\subset{\mathcal{G}}r(\widetilde{\mathcal{A}}^{\vee},2).$

As in (52) we have the modular morphism $\Theta:\
\widetilde{\mathcal{Y}}_{m}\to M_{X}(v),\ [\xi]\mapsto[E_{\widetilde{\xi}}]$,
where $[E_{\widetilde{\xi}}]$ is the isomorphism class of the bundle
$E_{\widetilde{\xi}}$ obtained from the point $\widetilde{\xi}$ as the lower
extension in the diagram (65). Assuming
$\widetilde{M}_{m}=\Theta(\widetilde{\mathcal{Y}}_{m})$, as in (53) and (54),
we have a bijection

(69) $\Theta:\
\widetilde{\mathcal{Y}}_{m}\xrightarrow{\simeq}\widetilde{M}_{m}$

and an equality

(70)
$\begin{split}&\dim\widetilde{M}_{m}=\dim\widetilde{\mathcal{Y}}_{m}=\dim\mathrm{Gr}(\operatorname{Ext}\nolimits^{1}(\mathcal{I}(m),\mathcal{O}_{X}(-1))^{\vee},2)+\dim\mathbb{G}=\\\
&2\mathrm{ext}^{1}(\mathcal{I}(m),\mathcal{O}_{X}(-1))-4+4=\mathrm{ext}^{1}(\mathcal{I}(m),\mathcal{O}_{X}(-1)^{\oplus
2}).\end{split}$

Applying the bifunctor $\mathbf{R}\operatorname{Hom}(-,-)$ to the triple (25)
and taking into account the equalities (32)-(35), we obtain a commutative
diagram

(71) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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From this diagram and the equalities (64) and (70) we obtain

(72)
$\operatorname{ext}^{1}(E,E)=\operatorname{ext}^{1}(\mathcal{I}(m),\mathcal{O}_{X}(-n)^{\oplus
2})=\dim\widetilde{M}_{m}.$

From here, according to the deformation theory, in view of the stability of
$E$, it follows that $\widetilde{M}_{k,m,n}$ is a smooth open subset of an
irreducible component of the module scheme $M_{X}(v)$. Therefore from (69), as
in (57), it follows that

(73) $\Theta:\ \widetilde{\mathcal{Y}}_{m}\to\widetilde{M}_{m}\text{is an
isomorphism},$

that is, $\widetilde{M}_{m}$ is a smooth rational variety isomorphic, due to
(68), to a dense open subset of a smooth variety
${\mathcal{G}}r(\widetilde{\mathcal{A}}^{\vee},2)$. Moreover,
$\widetilde{M}_{m}$ is also a fine moduli space. Formulas for the dimensions
of the varieties $\widetilde{M}_{m}$ as functions of $m$ are obtained taking
into account (28)-(31) and (70) by analogy with formulas for dimensions of
$M_{m}$.
Moreover, due to (68), reflexive sheaves in $\widetilde{M}_{m}$ constitute a
dense open subset

(74)
$\widetilde{R}_{m}:=\Theta(\widetilde{\mathcal{R}}_{m})\cong\widetilde{\mathcal{R}}_{m}.$

So, the following theorem holds.

######  Theorem 4.6.

Let $X=X_{2}$ be the quadric, and let $M_{X}(v)$ be the Gieseker-Maruyama
moduli scheme of semistable sheaves $E$ on $X$ with Chern character
$v=\mathrm{ch}(E)$ defined by the formulas (9) and (25). Then the following
statements are true.
(i) The family

(75) $\begin{split}&\widetilde{M}_{m}=\\{[E]\in M_{X}(v)\ |\ E\ \textit{is a
stable extension \eqref{extension I}}\\},\end{split}$

is a smooth dense open subset of an irreducible rational component of the
scheme $M_{X}(v)$, and there is an isomorphism (73), in which
$\widetilde{\mathcal{Y}}_{m}$ is the defined in (66) open subset of the
grassmannization $\widetilde{\pi}:\
{\mathcal{G}}r(\widetilde{\mathcal{A}}^{\vee},2)\to\mathbb{G}$, where
$\widetilde{\mathcal{A}}$ is the locally free sheaf on $\mathbb{G}$, given by
the second formula (38); moreover, $\widetilde{M}_{m}$ is a fine moduli space,
and reflexive sheaves constitute in $\widetilde{M}_{m}$ the dense open set
$\widetilde{R}_{m}$, defined in (74).
(ii) The dimension of the variety $\widetilde{M}_{m}$ is given by the formula
$\dim\widetilde{M}_{m}=2\binom{-m+4}{4}-2\binom{-m+2}{4}-2\binom{-m+3}{4}+2\binom{-m+1}{4}-2m+2=2m^{2}-6m+4$.

######  Remark 4.7.

On the quadric $X=X_{2}$ we consider a nontrivial extension of the form

(76) $\begin{split}&0\to\SS(-1)\to E\to\mathcal{I}(m)\to 0,\ \ \ \ \ \ \ \ \ \
m\leq-1,\end{split}$

where $\SS$ is the spinor bundle on $X$, and the sheaf $\mathcal{I}$ is the
same as in (25). As in the case of sheaves $E$ from (25), we assume that the
sheaf $E$ is stable. Let $M_{X}(v)$ be the Gieseker-Maruyama moduli scheme of
semistable sheaves $E$ with Chern character $v=\mathrm{ch}(E)$ defined by the
formulas (7), (10) and (76). From (76) it follows that $E$ satisfies the
formulas (32), (35), (37), (72) and the diagram (71), in which the sheaf
$\mathcal{O}_{X}(-n)^{\oplus 2}$ is replaced everywhere by $\SS(-1)$, and the
left vertical triple in (71) is replaced by a triple
$\mathbf{k}\rightarrowtail\mathbf{k}\twoheadrightarrow 0$. Respectively, in
the notation of the second formula (38) on $\mathbb{G}$ the following locally
free sheaf is defined:

(77)
$\widetilde{\mathcal{A}}_{\SS}=\mathcal{E}xt_{\widetilde{p}}^{1}(\mathbb{I}\otimes\mathcal{O}_{\mathbb{G}}\boxtimes\mathcal{O}_{X}(m),\mathcal{O}_{\mathbb{G}}\boxtimes\SS(-1)).$

Next, as in Remark 4.4, we consider the projectivization $\widetilde{\pi}:\
\mathbb{P}(\widetilde{\mathcal{A}}_{\SS}^{\vee})\to\mathbb{G}$ and its open
subsets $\widetilde{\mathcal{Y}}_{\SS,m}$ and
$\widetilde{\mathcal{R}}_{\SS,m}$ of stable and reflective sheaves
respectively. For them, by analogy with (62) and (63) we have embeddings of
dense open sets

(78)
$\widetilde{\mathcal{R}}_{\SS,m}\subset\widetilde{\mathcal{Y}}_{\SS,m}\subset{\mathcal{G}}r(\widetilde{\mathcal{A}}_{\SS}^{\vee},2)$

and an isomorphism

(79) $\Theta:\
\widetilde{\mathcal{Y}}_{\SS,m}\xrightarrow{\cong}M_{m}:=\Theta(\widetilde{\mathcal{Y}}_{\SS,m}),\
\ \ [\widetilde{\xi}]\mapsto[E_{\widetilde{\xi}}].$

Repeating the proof of Theorem 4.6 for the indicated varieties, we obtain the
following theorem.

######  Theorem 4.8.

Let $X=X_{2}$ be the quadric. Then, in the notation of Remark 4.7, the
following statements are true.
(i) The family

$M_{m}=\\{[E]\in M_{X}(v)\ |\ E\ \textit{is a stable extension
\eqref{extension Spin I}}\\}$

is a smooth dense open subset of an irreducible rational component of the
scheme $M_{X}(v)$, and there is an isomorphism $\Theta$ in (79), in which
$\widetilde{\mathcal{Y}}_{\SS,m}$ is an open subset of the projectivization
$\widetilde{\pi}:\
\mathbb{P}(\widetilde{\mathcal{A}}_{\SS}^{\vee})\to\mathbb{G}$, defined in
Remark 4.7, where $\widetilde{\mathcal{A}}_{\SS}$ is the locally free sheaf of
rank $\operatorname{{rk}}\widetilde{\mathcal{A}}_{\SS}=2m^{2}-8m+7$ on
$\mathbb{G}$ defined by the formula (77); in this case $M_{m}$ is a fine
moduli space, and reflexive sheaves constitute a dense open set
$\widetilde{R}_{\SS,m}=\Theta(\widetilde{\mathcal{R}}_{\SS,m})$ in $M_{m}$,
where $\widetilde{\mathcal{R}}_{\SS,m}$ is defined in Remark 4.7.
(ii) The dimension of the variety $M_{m}$ is given by the formula $\dim
M_{m}=2m^{2}-8m+10$.

###  4.3. Third series.

Let $k\geq 1$, $n=\lfloor\frac{k}{2}\rfloor+1$ and $m+n<0$. Consider on $X$ a
sheaf $F$ of rank 2 and a variety $W$, defined as follows.
(I) In the case $X=X_{1}$ the sheaf $F$ is determined from the exact triple

(80)
$0\to\mathcal{O}_{\mathbb{P}^{3}}(-1)\to\mathcal{O}_{\mathbb{P}^{3}}^{\oplus
3}\to F\to 0.$

As is known (see, for example, [16, Example 4.2.1]), or [2, Remark 2]), the
moduli space $W$ of sheaves $F$ is isomorphic to ${\mathbb{P}^{3}}$, and the
isomorphism $W\xrightarrow{\simeq}{\mathbb{P}^{3}}$ is given by the formula
$[F]\mapsto\operatorname{Sing}(F)$.
(II) In the case when $X=X_{4}$ is a complete intersection of a general pencil
of hyperquadrics in ${\mathbb{P}^{5}}$, let
${\mathbb{P}^{1}}\subset|\mathcal{O}_{{\mathbb{P}^{5}}}(2)|$ be the base of
this pencil of quadrics, and let $\Gamma$ be the hyperelliptic curve of genus
2, defined as the double covering $\rho:\ \Gamma\to\mathbb{P}^{1}$, branched
at points corresponding to degenerate quadrics of the pencil. Let
$\Gamma^{*}=\rho^{-1}({{\mathbb{P}^{1}}}^{*})$, where
${{\mathbb{P}^{1}}}^{*}\subset{\mathbb{P}^{1}}$ is an open subset of
nondegenerate quadrics of the pencil, and let
$\Delta=\rho^{-1}({\mathbb{P}^{1}}\smallsetminus{{\mathbb{P}^{1}}}^{*})$. Any
point $y\in\Gamma^{*}$ corresponds to one of the two series of generating
planes on a non-degenerate 4-dimensional quadric $Q(y):=\rho(y)$, and this
series corresponds to a spinor bundle $\SS(y)$ of rank 2 on $Q(y)$ with
$\det\SS(y)=\mathcal{O}_{Q(y)}(1)$. In this case we set $F_{y}=\SS(y)|_{X}$.
It is known that $\SS(y)$ is an arithmetically Cohen–Macaulay sheaf (briefly:
an ACM sheaf), that is, it satisfies the equalities:

(81) $h^{i}(\SS(y)(\ell))=0\ \ \ \ \ell\in\mathbb{Z},\ \ \ \ i=1,2.$

From this and from the cohomology of the triple $0\to\SS(y)(-2)\to\SS(y)\to
F_{y}\to 0$ it follows that $F_{y}$ is also an ACM sheaf. Let now
$y\in\Delta$, that is, the degenerate quadric $Q(y)$ is a cone with its vertex
at the point say $z(y)$, so the projection
$\mu:Q(y)\smallsetminus\\{z(y)\\}\to Q_{y}$ is defined, where $Q_{y}$ is a
smooth three-dimensional quadric. On $Q_{y}$ the spinor bundle $\SS_{Q_{y}}$
with $\det\SS_{Q_{y}}=\mathcal{O}_{Q_{y}}(1)$ is defined, and we set
$F_{y}=\mu^{*}\SS_{Q_{y}}|_{X}$. The sheaf $F_{y}$ is also an ACM sheaf. In
this case we set $W=\\{[F_{y}]\ |\ y\in\Gamma\\}\simeq\Gamma$.
(III) In the case $X=X_{5}$, the sheaf $F$ is defined as the restriction to
$X$ of the tautological bundle on the grassmannian
$\mathop{\mathrm{Gr}}\nolimits(2,5)$, twisted by $\mathcal{O}_{X}(1)$. The
isomorphism class $[F]$ of $F$ is uniquely defined, and we assume that $W$ is
a point $\\{[F]\\}$. Note that $F$ is also an ACM sheaf.

Note that in all cases (I)-(III) the universal family $\mathbb{F}$ of sheaves
$F$ on $W\times X$ is defined.

Consider the sheaf $E$ on $X$ obtained as a nontrivial extension

(82) $0\to F(-n)\to E\to\mathcal{O}_{S}(m)\to 0,\ \ \ \ \ \
S\in|\mathcal{O}_{X}(k)|,\ \ \ \ \ k\geq 1.$

Further, in case when $E$ is stable, by analogy with (33), using the
definition and the above properties of the sheaf $F$, the exact triple
$0\to\mathcal{O}_{X}(-k)\to\mathcal{O}_{X}\to\mathcal{O}_{S}\to 0$ and the
condition $m+n<0$, we have the equalities

(83)
$\begin{split}&\hom(E,E)=\hom(F,F)=\hom(\mathcal{O}_{S}(m),\mathcal{O}_{S}(m))=\hom(E,\mathcal{O}_{S}(m))=\\\
&\hom(F(-n),E)=1,\ \ \ \
\hom(F(-n),\mathcal{O}_{S}(m))=\operatorname{ext}^{2}(\mathcal{O}_{S}(m),F(-n))=0.\end{split}$

Let, as above, $\mathbb{P}^{N}=|\mathcal{O}_{X}(k)|$, let
$\mathbb{S}\subset\mathbb{P}^{N}\times X$ be a universal family of surfaces of
degree $k$ in $X$, and let $W\times X\xleftarrow{q}W\times\mathbb{P}^{N}\times
X\xrightarrow{p}W\times\mathbb{P}^{N}$ be the projections. On
$W\times\mathbb{P}^{N}$, by analogy with (38)-(40), we have a locally free
sheaf

(84)
$\mathcal{A}=\mathcal{E}xt_{p}^{1}(\mathcal{O}_{W\times\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(m)|_{W\times\mathbb{S}},\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(-n)\otimes
q^{*}\mathbb{F}),$

the construction of which commutes with the base change:

(85) $\tau_{F,S}:\
\mathcal{A}\otimes\mathbf{k}_{\\{F,S\\}}\xrightarrow{\cong}\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),F(-n)),$

and we have an equality

(86)
$\mathcal{E}xt_{p}^{0}(\mathcal{O}_{W\times\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(m)|_{W\times\mathbb{S}},\mathcal{O}_{W\times\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(-n)\otimes
q^{*}\mathbb{F})=0.$

Consider the varieties $Y=\mathbb{P}(\mathcal{A}^{\vee})$ and $X_{Y}=Y\times
X$ with projections $Y\xrightarrow{\rho}\mathbb{P}^{N}$ and
$Y\xleftarrow{p_{Y}}X_{Y}\xrightarrow{\rho_{Y}}W\times\mathbb{P}^{N}\times X$.
From (85), (86) and [21, Cor. 4.5] it follows that on $X_{Y}$ there is the
universal family of extensions

$0\to
p_{Y}^{*}\mathcal{O}_{P}(1)\otimes(\mathcal{O}_{W\times\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(-n)\otimes
q^{*}\mathbb{F})\to\mathcal{V}\to\rho_{Y}^{*}(\mathcal{O}_{W\times\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(m)|_{W\times\mathbb{S}})\to
0.$

Applying the functor $\rho_{Y*}$ to this triple, we obtain on
$W\times\mathbb{P}^{N}\times X$ the universal extension, where
$\mathcal{U}=\rho_{Y*}\mathcal{V}$:

(87) $0\to
p^{*}\mathcal{A}^{\vee}\otimes(\mathcal{O}_{W\times\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(-n))\to\mathcal{U}\to\mathcal{O}_{\mathbb{P}^{N}}\boxtimes\mathcal{O}_{X}(m)|_{\mathbb{S}}\to
0.$

Note that the extension (82) is specified by an element
$\xi\in\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),F(-n))$, considered
as a homomorphism

(88) $\xi:\
\operatorname{Ext}\nolimits^{1}(\mathcal{O}_{S}(m),\mathcal{O}_{X}(-n))^{\vee}\to\mathbf{k}.$

By the universality, the exact triple (87), restricted to $\\{S\\}\times X$,
in view of the base change (85) can be included with the extension (82) in a
commutative diagram

(89) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern