# Moduli spaces on the Kuznetsov component of Fano threefolds of index 2

Matteo Altavilla  MA:Department of Mathematics
University of Utah
Salt Lake City, UT 84102, USA<EMAIL_ADDRESS>,  Marin Petković  MP:
Department of Mathematics
University of Utah
Salt Lake City, UT 84102, USA<EMAIL_ADDRESS>and  Franco Rota  FR:
Department of Mathematics
Rutgers University
Piscataway, NJ 08854, USA<EMAIL_ADDRESS>
###### Abstract.

General hyperplane sections of a Fano threefold $Y$ of index 2 and Picard rank
1 are del Pezzo surfaces, and their Picard group is a root system. To the
corresponding roots, we associate objects in the Kuznetsov component of $Y$
and investigate their moduli spaces, using the stability condition constructed
by Bayer, Lahoz, Macrì, and Stellari, and the Abel–Jacobi map. We identify a
subvariety of the moduli space isomorphic to $Y$ itself, and as an application
we prove a (refined) categorical Torelli theorem for general quartic double
solids.

###### Key words and phrases:

Derived Categories, Bridgeland stability conditions, Fano threefolds, moduli
spaces, Abel–Jacobi map

###### 2020 Mathematics Subject Classification:

Primary 14F08; secondary 14J45, 14D20

All three authors were partially supported by NSF-FRG grant DMS 1663813, PI:
Aaron Bertram.

##  1\. Introduction

Stability conditions on triangulated categories were introduced by Bridgeland
in [Bri07]. A remarkable feature of Bridgeland’s construction is that the set
$\operatorname{Stab}(\mathcal{D})$ of all stability conditions admits a
natural topology, and is in fact endowed with the structure of a complex
manifold [Bri07, Theor. 1.2]. Even when
$\mathcal{D}=D^{b}(\operatorname{Coh}(X))$ is the bounded derived category of
a smooth projective variety $X$, it is challenging to study
$\operatorname{Stab}(\mathcal{D})$, and questions about its non-emptiness,
connectedness or simple-connectedness are difficult problems. The stability
manifold is completely understood only in the case of curves (see [Oka06,
BMW15] for $\mathbb{P}^{1}$, [Bri07] for elliptic curves and [Mac07] for the
other cases). A connected component is described in [Bri08] for K3 and abelian
surfaces (and in [HMS08] for their twisted counterparts. See also [BB17] for a
more detailed description of K3 surfaces of Picard rank 1). A general
construction - recently extended to the positive characteristic case [Kos20]
\- of stability conditions on surfaces is given in [AB13], and alternative
constructions in the presence of curves of negative self-intersection appear
in [Tod13b, TX17] (these can be interpreted as stability conditions on
orbifold surfaces as in [LR20]). In dimension three, in a series of remarkable
papers, stability conditions have been constructed on Fano threefolds (see for
example [Mac14, BMT14] for $\mathbb{P}^{3}$, [Li19b] for Picard rank 1, and
[Piy17, BMSZ17] for the general case), abelian threefolds [BMS16, MP15, MP16],
some resolutions of finite quotients thereof [BMS16], quintic threefolds
[Li19a].

On the other hand, stability conditions allow for the construction of moduli
spaces of stable objects of $\mathcal{D}$. It is interesting to investigate
properties of these moduli spaces, like non-emptiness, irreducibility,
smoothness, or projectivity, or to relate them to other classical moduli
spaces: for example, Bridgeland stability regulates the birational geometry of
the Hilbert scheme of points of surfaces (see [ABCH13] for the case of
$\mathbb{P}^{2}$, and [BC13] for other surfaces), or of moduli spaces of
sheaves on a K3 surface [BM14a].

If $\mathcal{D}$ admits an exceptional collection, Bayer, Macrì, Lahoz and
Stellari give a sufficient condition to induce a stability condition on the
right orthogonal of the collection, starting from a (weak) stability condition
on $\mathcal{D}$ [BLMS17]. The criterion applies for example to cubic
fourfolds and Fano threefolds of Picard rank one [BLMS17] and to Gushel-Mukai
varieties [PPZ19].

In this paper, we focus on Fano manifolds $Y$ of Picard rank 1 and index 2
(i.e., $H\coloneqq-K_{Y}/2$ is an ample generator of
$\operatorname{Pic\,}(Y)$). They are classified by Iskovskih [Isk77] and
belong to one of the following 5 families, indexed by thir degree $d\coloneqq
H^{3}\in\\{1,..,5\\}$:

  * •

$Y_{5}=\operatorname{Gr}(2,5)\cap\mathbb{P}^{6}\subset\mathbb{P}^{9}$ is a
linear section of codimension 3 of the Grassmannian $\operatorname{Gr}(2,5)$
in the Plücker embedding;

  * •

$Y_{4}=Q\cap Q^{\prime}\subset\mathbb{P}^{5}$ is the intersection of two
quadric hypersurfaces;

  * •

$Y_{3}\subset\mathbb{P}^{4}$ is a cubic hypersurface;

  * •

$Y_{2}\xrightarrow{\pi}\mathbb{P}^{3}$ is a double cover ramified over a
quartic surface, or equivalently a hypersurface of degree 4 in the weighted
projective space $\mathbb{P}(1,1,1,1,2)$;

  * •

$Y_{1}$ is a hypersurface of degree 6 in the weighted projective space
$\mathbb{P}(1,1,1,2,3)$, or equivalently a double cover branched over a cubic
of the cone over the Veronese surface in $\mathbb{P}^{5}$.

If $Y$ is one of the above, Kuznetsov shows in a series of papers that the
derived category of $Y$ admits a semi-orthogonal decomposition

(1) $D(Y)=\langle\mathsf{Ku}(Y),\mathcal{O}_{Y},\mathcal{O}_{Y}(1)\rangle$

whose non-trivial part

(2) $\mathsf{Ku}(Y)\coloneqq\\{E\in
D(Y)\,|\,\operatorname{Hom}(\mathcal{O}_{Y},E[i])=\operatorname{Hom}(\mathcal{O}_{Y}(1),E[i])=0\mbox{
for all }i\in\mathbb{Z}\\}$

is called the Kuznetsov component of $Y$ [Kuz09, Kuz14]. We identify the
numerical Grothendieck group $K_{\text{num}}(\mathsf{Ku}(Y))$ with the image
of the chern character map

$\operatorname{ch}:K(\mathsf{Ku}(Y))\to H^{\ast}(Y,\mathbb{Q}).$

It is a rank 2 lattice spanned by the classes

(3) $v\coloneqq\left[1-\dfrac{1}{d}H^{2}\right]\qquad\mbox{and}\qquad
w\coloneqq\left[H-\dfrac{1}{2}H^{2}+\left(\dfrac{1}{6}-\dfrac{1}{d}\right)H^{3}\right]$

with a bilinear form given by the Euler pairing

$\begin{bmatrix}-1&-1\\\ 1-d&-d\end{bmatrix}.$

In this paper, we focus on moduli spaces of complexes of class $w$, with
particular attention to the degree 2 and 1 cases. A more detailed study of
moduli of class $w$ for cubic threefolds appears in [BBF+20]; moduli of
objects of class $v$ are studied in [PY20] (for degrees $d\geq 2$) and in
[PR20] for the case of degree 1.

### Summary of the results

Throughout the paper, we work under some mild generality assumptions for $Y$,
these control the singularities that may appear in the hyperplane sections of
$Y$ (see Sec. 1.1.5) and allow us to rely on previous results of [Wel81,
Tih82].

The main result of our work is a description of the moduli space
$\mathcal{M}_{\sigma}(w)$, which parametrizes complexes in $\mathsf{Ku}(Y)$ of
class $w$, semistable with respect to one of the stability conditions $\sigma$
constructed in [BLMS17].

First, we study the moduli spaces of Gieseker-stable sheaves on $Y$ of class
$w$. For a hyperplane section $S\in\lvert\mathcal{O}_{Y}(1)\rvert$ of $Y$, a
root of $S$ is a divisor $D\in\operatorname{Pic\,}(S)$ satisfying $D^{2}=-2$
and $D.H_{|S}=0$. By a root of $Y$ we mean a sheaf of the form
$\iota_{*}(\mathcal{O}_{S}(D))$, with $S\in\lvert\mathcal{O}_{Y}(1)\rvert$ and
$D$ a root of $S$ (see Def. 1.14).

######  Proposition 1.1 (= (2.1)).

Let $Y$ be a general Fano threefold of Picard rank 1 and index 2. The moduli
$\mathcal{M}_{G}(w)$ of Gieseker-semistable objects of class $w$ on $Y$ has
two irreducible components. One, denoted $\mathcal{P}$, has dimension $d+3$
and parametrizes ideal sheaves $I_{p|S}$ of a point in a hyperplane section
$S$ of $Y$. A point $[I_{p|S}]\in\mathcal{P}$ is smooth in
$\mathcal{M}_{G}(w)$ iff $p$ is a smooth point of $S$.

The second component has dimension $d+1$ and its smooth points parametrize
roots of $Y$.

Both components are smooth outside their intersection, which is the locus
parametrizing $I_{p|S}$, with $S$ singular at $p$.

We relate $\mathcal{M}_{G}(w)$ and $\mathcal{M}_{\sigma}(w)$ via wall-
crossing: there is a two-parameter family $\sigma_{(\alpha,\beta)}$ of weak
stability conditions on $D^{b}(Y)$. We show on the one hand that the moduli
space of $\sigma_{(\alpha,\beta)}$-stable complexes of class $w$ is isomorphic
to $\mathcal{M}_{G}(w)$ for $\beta=-\frac{1}{2}$ and $\alpha\gg 0$ (Prop.
2.7), and on the other hand that $\mathcal{M}_{\sigma}(w)$ is isomorphic to
the moduli space of $\sigma_{\alpha,\beta}$-stable objects with
$\beta=-\frac{1}{2}$, $0<\alpha\ll 1$ (Prop. 2.10). Then, we deform the
parameter $\alpha$: there is a unique wall between $\mathcal{M}_{G}(w)$ and
$\mathcal{M}_{\sigma}(w)$ and the universal family of $\mathcal{M}_{G}(w)$
induces a wall-crossing morphism, defined at the level of sets by replacing
objects $I_{p/S}$, fitting in a triangle

$I_{p}\to I_{p/S}\to\mathcal{O}_{Y}(-1)[1],$

with extensions of the form

(4) $\mathcal{O}(-1)[1]\to E_{p}\to I_{p}$

(Proposition 2.9). In this way we obtain a general description for all $d$:

######  Theorem 1.2 (= (2.16) + (4.4) + (4.11)).

Let $Y$ be a general Fano threefold of Picard rank 1 and index 2. Then, the
moduli space $\mathcal{M}_{\sigma}(w)$ is proper. It contains two smooth
subvarieties $\mathcal{Y}$ and $\mathcal{C}$. The locus $\mathcal{Y}$ is
isomorphic to $Y$ and it parametrizes objects $E_{p}$ of the form (4), while
the locus $\mathcal{C}$ has dimension $d+1$ and parametrizes roots of $Y$.

For $d\geq 3$, $\mathcal{M}_{\sigma}(w)$ is projective and irreducible and
$\mathcal{Y}$ is contained in the closure of $\mathcal{C}$; $\mathcal{Y}$ and
$\mathcal{C}$ are distinct irreducible components for degree $d\leq 2$.

For $d\leq 2$, we relate $\mathcal{M}_{\sigma}(w)$ to the intermediate
Jacobian $J(Y)$ of $Y$ (a similar approach has been considered before for
higher degrees, results are recollected in Section 3). We do this by defining
an Abel–Jacobi mapping, starting from the usual one

$\Phi\colon\mathcal{Z}_{1}(Y)^{\text{hom}}\to J(Y)\coloneqq
H^{1}(Y,\Omega^{2}_{Y})^{\vee}\Big{/}H^{3}(Y,\mathbb{Z}),$

defined on the space $\mathcal{Z}_{1}(Y)^{\text{hom}}$ of one-dimensional,
homologically trivial algebraic cycles on $Y$. The map $\Phi$ sends one such
$Z$ to the integral over a 3-chain $\Gamma$ such that $\partial\Gamma=Z$ (see
[Wel81, Sec. 2],[Voi03, Sec. 12.1]). One can then use the second Chern class
of objects of $\mathcal{M}_{\sigma}(w)$ to construct an analogous map

$\Psi\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
16.78633pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{M}_{\sigma}(w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
48.4925pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
48.4925pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{J(Y)}$}}}}}}}{\hbox{\kern-6.90973pt\raise-16.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.90973pt\raise-16.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
40.78633pt\raise-16.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.78633pt\raise-16.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Phi(c_{2}(F))}$}}}}}}}\ignorespaces}}}}\ignorespaces$

(see Section 2.4). In degree 2, $Y$ is a double cover of $\mathbb{P}^{3}$
ramified over a quartic K3 surface $R$. The intersection
$\mathcal{Y}\cap\mathcal{C}$ is isomorphic to $R$ (see Theorem 4.4), $\Psi$
contracts the component $\mathcal{Y}$ to a point, and it is a generic
embedding on the component $\mathcal{C}$ (Cor. 4.8). The strict transform of
$\Psi(\mathcal{C})$ in the blow-up of $J(Y)$ at $\Psi(\mathcal{Y})$ intersects
the exceptional divisor in a surface isomorphic to $R$ (Cor. 4.10).

For degree 1, we show that $\mathcal{C}\simeq F(Y)$, the Fano surface of lines
of $Y$, and the intersection $\mathcal{Y}\cap F(Y)$ in
$\mathcal{M}_{\sigma}(w)$ is a curve $C$. The map $\Psi$ is an embedding
outside $C$, and its image suffices to determine $Y$ [Tih82].

Finally, in the case of degree 2, we apply our result to show a (refined)
categorical Torelli theorem:

######  Theorem 1.3 (=5.1).

Let $Y$ and $Y^{\prime}$ be two general quartic double solids. There exists an
equivalence $u\colon\mathsf{Ku}(Y^{\prime})\simeq\mathsf{Ku}(Y)$ if and only
if $Y^{\prime}\simeq Y$.

Our proof technique is inspired by that of [BMMS12]; we use the equivalence
$u$ to construct an isomorphism between moduli spaces and argue that this is
sufficient to conclude.

### Related work and further questions

Fano threefolds of one of the five families listed above, and their related
moduli spaces, have been considered in the past. In Section 3 we give a more
detailed account of these results, which are only outlined here.

In the case of degree 5, Theorem 1.2 recovers the description of $Y_{5}$
within the Hilbert scheme of three points on $\mathbb{P}^{2}$ [Muk92]. If the
degree is 4, $Y$ is the intersection of 2 quadrics. Work of Miles Reid [Rei72]
relates $Y$ to a certain genus 2 curve $C$ (proving for example that the Fano
surface of lines on $Y$, the intermediate Jacobian of $Y$, and the Jacobian
variety of $C$ are isomorphic). As it turns out, there is an equivalence
$D^{b}(C)\simeq\mathsf{Ku}(Y)$ [BO95], so we can reinterpret Reid’s result as
an occurrence of Theorem 1.2.

The use of the Abel–Jacobi map in our work owes to the notion of the
intermediate Jacobian introduced by Clemens and Griffiths [CG72]. The
categorical techniques we use in the paper rely on the stability condition
constructed in [BLMS17], and on the base-change results of [Kuz11]. Bridgeland
moduli spaces on $Y$ have been studied in other recent papers, such as
[BBF+20] (where moduli of objects of class $w$ are used to give a new
parametrization of the Theta divisor of a cubic threefold), [PY20] (where the
authors study objects of class $v$ for degree $\geq 2$ and describe a
component of the stability manifold), and [PR20] (which considers the case of
moduli of class $v$ for degree 1).

The general expectation is that the Kuznetsov component contains sufficient
information to recover $Y$ (results in this spirit are usually called
(refined) derived Torelli theorems). Such reconstruction theorems are now
known to hold in a few cases: for example, $\mathsf{Ku}(X)$ determines $X$ if
$X$ is a cubic fourfold (see [BLMS17] for general $X$, and [LPZ18] for all
$X$) of if $X$ is a general Enriques surface [LNSZ19].

In the case of Fano threefolds, four of the five families considered satisfy a
refined Torelli theorem: $Y_{5}$ is rigid in moduli, so the statement is
vacuously true. Degrees 4 and 3 are showed in [BO95] and [BMMS12] (recently,
[PY20] gives an alternative proof of the cubic case). For degree 2, our result
(Theorem 1.3) strengthens the statement of Bernardara–Tabuada [BT16, Cor. 3.1
(iii)], who show that the same holds under the additional assumption that the
equivalence be of Fourier–Mukai type.

There are some natural questions which remain unanswered by this work.

  1. (i)

(Torelli theorem for $d=1$) It is expected that an analogue of Theorem 1.3
holds in degree 1 as well. Unfortunately, the heart of the stability condition
$\sigma$ has homological dimension 3 (instead of 2) in degree 1 [PR20, Remark
2.6], so a different technique will be needed.

  2. (ii)

(Equivalences of Fourier–Mukai type) A conjecture of Kuznetsov [Kuz07, Conj.
3.7] implies, in the case of quartic double solids, that every equivalence
$u\colon\mathsf{Ku}(Y^{\prime})\xrightarrow{\sim}\mathsf{Ku}(Y)$ is of
Fourier–Mukai type (i.e. that the composition
$D(Y^{\prime})\to\mathsf{Ku}(Y^{\prime})\xrightarrow{u}\mathsf{Ku}(Y)\to D(Y)$
is a Fourier–Mukai functor). Combining Theorem 1.3 with [BT16, Prop. 3.5]
shows that $\mathsf{Ku}(Y^{\prime})$ and $\mathsf{Ku}(Y)$ are equivalent if
and only if they are Fourier–Mukai equivalent. However, this is not sufficient
to prove the statement of [Kuz07, Conj. 3.7] for degree 2. Interestingly, this
conjecture is incompatible with Kuznetsov’s conjecture on Fano threefolds (see
[Kuz09, Conj. 3.7] for the conjecture, and [BT16, Theorem 4.2] for a proof of
incompatibility). The conjecture on Fano threefolds has been recently
disproved in [Zha20].

### Structure of the paper

Section 1.1 contains preliminary notions on (weak) stability conditions
(1.1.1, 1.1.2), moduli spaces (1.1.3), Fano threefolds of Picard rank 1 and
index 2 and the geometry of their hyperplane sections (1.1.5). Section 1.1.4
recalls results of base change of semiorthogonal decompositions from [Kuz11],
they represent our main technical tool to work with universal families.

In Section 2 we describe the moduli spaces of Gieseker-stable sheaves of class
$w$ (2.1), of $\sigma_{(\alpha,\beta)}$-stable complexes in $D^{b}(Y)$ (2.2)
and of $\sigma$-stable complexes in $\mathsf{Ku}(Y)$ (2.3). We conclude with
the definition, and some properties, of the Abel–Jacobi map $\Psi$ (2.4).

Section 3 summarizes and illustrates our results for degrees $d=3,4,5$.
Analogs of our results for degree $d=4,5$ have been known in the literature,
and the $d=3$ case recently appeared in [BBF+20].

Section 4 provides more details to the description of
$\mathcal{M}_{\sigma}(w)$ in the remaining cases of degree $d=2,1$, with
Section 4.4 dedicated to $d=1$.

Finally, Section 5 contains the proof of the Torelli theorem for quartic
double solids (Theorem 5.1).

### Acknowledgements

We are grateful to our doctoral advisor, Aaron Bertram, for his guidance and
his enthusiasm. We wish to thank Arend Bayer for encouraging us in pursuing
this problem. We also thank Paolo Stellari and Huachen Chen, for discussing
this problem with us on various occasions. This project benefited from the
participation of the second and third author to the workshop _“Semiorthogonal
decompositions, stability conditions and sheaves of categories”_ held in
Toulouse in 2018 and of the first and third author to the workshop on
_“Derived Categories, Moduli Spaces and Deformation Theory”_ held in Cetraro
in 2019. We thank the organizers and the participants of these events. We are
grateful to an anonymous referee for the invaluable feedback on a first
version of this work. We thank to Laura Pertusi, Song Yang, and Shizhuo Zhang
for our conversations on these topics.

###  1.1. Preliminaries

####  1.1.1. Weak stability conditions

In this section we refer to [BLMS17] and briefly summarize definitions and
results on weak (and Bridgeland) stability conditions.

Let $\mathcal{D}$ be a triangulated category and $K(\mathcal{D})$ its
Grothendieck group; fix a lattice $\Lambda\subset K(\mathcal{D})$ and a
surjective homomorphism $v\colon K(\mathcal{D})\twoheadrightarrow\Lambda$.

######  Definition 1.4.

The _heart of a bounded t-structure_ is a full abelian subcategory
$\mathcal{A}\subset\mathcal{D}$ with the following properties:

  1. (a)

For all $E$,$F\in\mathcal{A}$ and $n<0$ we have
$\operatorname{Hom}(E,F[n])=0$;

  2. (b)

For every $E\in\mathcal{D}$ there exist a filtration, i.e. objects
$E_{i}\in\mathcal{D}$, integers $k_{1}>\dots>k_{m}$ and triangles

$\textstyle{0=E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{m-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{m}=E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}[k_{1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}[k_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{m-1}[k_{m-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{m}[k_{m}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

such that $A_{i}\in\mathcal{A}$; these objects are called the _cohomologies_
of $E$ and are often denoted by $\mathcal{H}^{i}_{\mathcal{A}}(E)$.

######  Definition 1.5.

Let $\mathcal{A}$ be an abelian category; a group homomorphism $Z\colon
K(\mathcal{A})\to\mathbb{C}$ is called a _weak stability function_ if for all
$E\in\mathcal{A}$ we have $\Im Z(E)\geq 0$, and $\Im Z(E)=0$ implies $\Re
Z(E)\leq 0$. If moreover $\Im Z(E)=0$ and $E\neq 0$ implies $\Re Z(E)<0$, then
$Z$ is called a _stability function_.

A (weak) stability function has an associated slope function

$\mu_{Z}(E)=\begin{cases}-\dfrac{\Re Z(E)}{\Im Z(E)}\quad\text{if }\ \Im
Z(E)>0\\\ +\infty\quad\text{otherwise}\end{cases},$

######  Definition 1.6.

An object $E\in\mathcal{A}$ is called _semistable (resp. stable)_ with respect
to $Z$ if for every quotient $E\twoheadrightarrow Q$ in $\mathcal{A}$ we have
$\mu_{Z}(E)\leq\mu_{Z}(Q)$ (resp. $\mu_{Z}(E)<\mu_{Z}(Q)$).

######  Definition 1.7.

Let $\mathcal{D}$ be a triangulated category, and let $\Lambda$ and $v$ be as
above; a _weak stability condition_ on $\mathcal{D}$ is a pair
$\sigma=(\mathcal{A},Z)$ where $\mathcal{A}$ is the heart of a bounded
t-structure and $Z$ is a group homomorphism $Z\colon\Lambda\to\mathbb{C}$ ,
satisfying the following properties:

  1. (a)

The composition $Z\circ v\colon
K(\mathcal{A})=K(\mathcal{D})\to\Lambda\to\mathbb{C}$ is a weak stability
function on $\mathcal{A}$;

  2. (b)

We require all $E\in\mathcal{A}$ to have Harder–Narasimhan filtration with
factors $F_{i}\in\mathcal{A}$ semistable with respect to $Z$, with strictly
decreasing slopes;

  3. (c)

There exists a quadratic form $Q$ on $\Lambda\otimes\mathbb{R}$ that is
negative definite on $\ker Z$, such that $Q(v(E))\geq 0$ for $E$ semistable.

If moreover $Z$ is a stability function, then the pair $(\mathcal{A},Z)$ is a
_Bridgeland stability condition_.

######  Definition 1.8.

An object $E\in\mathcal{D}$ is called (semi)stable if there exists
$m\in\mathbb{Z}$ such that $E[m]\in\mathcal{A}$ and $E[m]$ is (semi)stable; in
that case we also say that $E$ is (semi)stable of phase $-m+\frac{1}{\pi}\arg
Z(E)$.

Then any $E\in\mathcal{D}$ has a Harder–Narasimhan filtration

$0=E_{0}\subset E_{1}\subset E_{2}\subset...\subset E_{m-1}\subset E_{m}=E$

such that each factor $F_{i}\coloneqq E_{i}/E_{i-1}$ is semistable of phase
$\phi_{i}$, with $\phi_{1}>\cdots>\phi_{m}$.

Given a weak stability condition, we can produce a new heart of a bounded
t-structure by _tilting_ ; indeed, given $\sigma=(\mathcal{A},Z)$ and a choice
$\mu\in\mathbb{R}$, define

$\displaystyle\mathcal{T}^{\mu,\sigma}=\\{E\in\mathcal{A}\ |\text{ All HN
factors $F$ of $E$ have slope $\mu_{Z}(F)>\mu$}\\}$
$\displaystyle\mathcal{F}^{\mu,\sigma}=\\{E\in\mathcal{A}\ |\text{ All HN
factors $F$ of $E$ have slope $\mu_{Z}(F)\leq\mu$}\\}.$

We use the notation $\langle-\rangle$ to denote the extension closure, i.e.
the smallest subcategory of $\mathcal{D}$ containing the objects in the
brackets and closed under extensions (note that this is not the same as
$\langle-\rangle$ used in the semiorthogonal decomposition, but context will
make the meaning clear). Then:

######  Proposition 1.9 ([HRS96], [BLMS17]).

The category
$\mathcal{A}^{\mu,\sigma}\coloneqq\langle\mathcal{T}^{\mu,\sigma},\mathcal{F}^{\mu,\sigma}[1]\rangle$
is the heart of a bounded t-structure.

####  1.1.2. Weak stability conditions on $D^{b}(X)$

Let $X$ be a smooth projective variety of dimension $n$, and let $H$ be an
ample divisor on $X$. We follow the setup of [BLMS17, Sec. 2] and recall the
construction of weak stability conditions on $D^{b}(X)$.

For $j\in\\{0,...,n\\}$ define lattices
$\Lambda_{H}^{j}\simeq\mathbb{Z}^{j+1}$ generated by

$\left(H^{n}\operatorname{ch}_{0},H^{n-1}\operatorname{ch}_{1},...,H^{n-j}\operatorname{ch}_{j}\right)\subset\mathbb{Q}^{j+1}$

with the sujective map $v_{H}^{j}\colon K(X)\to\Lambda_{H}^{j}$ given by the
Chern character. Then, Mumford slope gives rise to a weak stability condition
$\sigma_{H}\coloneqq(\operatorname{Coh}(X),Z_{\mu})$ where
$Z_{H}\colon\Lambda_{H}^{1}\to\mathbb{C}$ is defined by

$Z_{H}(E)\coloneqq-H^{n-1}\operatorname{ch}_{1}(E)+iH^{n}\operatorname{ch}_{0}(E).$

We can define a new weak stability condition from the one above via tilting:
for $\beta\in\mathbb{R}$, define $\operatorname{Coh}^{\beta}(X)$ to be the
heart obtained from $\sigma_{H}$ via tilting at slope $\mu_{H}=\beta$. Before
introducing the central charge, recall the notation
$\operatorname{ch}^{\beta}(E)\coloneqq\operatorname{ch}(E)\cdot e^{-\beta H}$.
We make the first three terms explicit for convenience:

(5) $\begin{split}\operatorname{ch}_{0}^{\beta}=\operatorname{ch}_{0}\\\
\operatorname{ch}_{1}^{\beta}=\operatorname{ch}_{1}-\beta
H\operatorname{ch}_{0}\\\
\operatorname{ch}_{2}^{\beta}=\operatorname{ch}_{2}-\beta
H\operatorname{ch}_{1}+\frac{\beta^{2}H^{2}}{2}\operatorname{ch}_{0}.\end{split}$

We have the following:

######  Proposition 1.10 ([BLMS17, Prop. 2.12]).

For all $(\alpha,\beta)\in\mathbb{R}_{>0}\times\mathbb{R}$, the pair given by
$\sigma_{\alpha,\beta}=(\operatorname{Coh}^{\beta}(X),Z_{\alpha,\beta})$ with

$Z_{\alpha,\beta}(E)=-H^{n-2}\operatorname{ch}_{2}^{\beta}(E)+\dfrac{\alpha^{2}}{2}H^{n}\operatorname{ch}_{0}^{\beta}(E)+iH^{n-1}\operatorname{ch}_{1}^{\beta}(E)$

defines a weak stability condition on $D^{b}(X)$. Moreover, these weak
stability conditions vary continuously with $(\alpha,\beta)$.

In virtue of the proposition above, we will consider the upper half plane
$\\{(\alpha,\beta)\in\mathbb{R}^{2}\,|\,\alpha>0\\}$ as a space parametrizing
the weak stability conditions $\sigma_{\alpha,\beta}$. We write
$\mu_{\alpha,\beta}\coloneqq\mu_{Z_{\alpha,\beta}}$ for the slope function
associated to $\sigma_{\alpha,\beta}$. We can now define walls and chambers:

######  Definition 1.11.

Fix a vector $v\in\Lambda^{2}_{H}$.

A numerical wall for $v$ is the set of pairs
$(\alpha,\beta)\in\mathbb{R}_{>0}\times\mathbb{R}$ such that there exists a
vector $w\in\Lambda^{2}_{H}$ for which
$\mu_{\alpha,\beta}(v)=\mu_{\alpha,\beta}(w)$.

A wall for $F\in\operatorname{Coh}^{\beta}(X)$ is a numerical wall for
$v=\operatorname{ch}_{\leq 2}(F)$ such that for every $(\alpha,\beta)$ in the
numerical wall there exists a short exact sequence of semistable objects $0\to
E\to F\to G\to 0$ satisfying
$\mu_{\alpha,\beta}(E)=\mu_{\alpha,\beta}(F)=\mu_{\alpha,\beta}(G)$.

A chamber is a connected component of the complement of the union of walls in
$\mathbb{R}_{>0}\times\mathbb{R}$.

The key features of this wall and chamber decomposition is that walls with
respect to a fixed $v\in\Lambda^{2}_{H}$ are locally finite in the upper half
plane. In particular, if $v=\operatorname{ch}_{\leq 2}(E)$ for some
$E\in\operatorname{Coh}^{\beta}(X)$, stability of $E$ is constant as
$(\alpha,\beta)$ varies within a chamber (see [BMS16, Prop. B.5]).

We conclude recalling another construction of a weak stability condition. In
the notation of Proposition 1.9, fix $\mu\in\mathbb{R}$ and define
$\operatorname{Coh}^{\mu}_{\alpha,\beta}(X):=\operatorname{Coh}^{\beta}(X)^{\mu,\sigma_{\alpha,\beta}}$
endowed with a stability function

$Z^{\mu}_{\alpha,\beta}:=\dfrac{1}{u}Z_{\alpha,\beta},$

where $u\in\mathbb{C}$ is the unique unit vector such that $\mu=-\dfrac{\Re
u}{\Im u}$.

######  Proposition 1.12 ([BLMS17, Prop. 2.15]).

The pair
$\sigma^{\mu}_{\alpha,\beta}=(\operatorname{Coh}^{\mu}_{\alpha,\beta}(X),Z^{\mu}_{\alpha,\beta})$
is a weak stability condition on $D(X)$.

Finally, again in [BLMS17], the authors induce a stability condition on the
Kuznetsov component of $Y$ using weak stability conditions:

######  Proposition 1.13 ([BLMS17, Theor. 1.1]).

Let $Y$ be a Fano threefold of index 2 with
$\operatorname{Pic\,}(Y)=\mathbb{Z}$, and let
$\sigma^{0}_{\alpha,-\frac{1}{2}}$ be the weak stability condition constructed
in Proposition 1.12 for $\mu=0$, $\beta=-\dfrac{1}{2}$ and $0<\alpha\ll 1$;
let moreover
$\mathcal{A}=\mathsf{Ku}(Y)\cap\operatorname{Coh}^{0}_{\alpha,\beta}(X)$ and
$Z=Z^{0}_{\alpha,-\frac{1}{2}}$. Then the pair $\sigma=(\mathcal{A},Z)$ is a
Bridgeland stability condition on $\mathsf{Ku}(Y)$.

####  1.1.3. Moduli spaces

Let $Y$ be a Fano threefold of Picard rank 1 and index 2, and fix $H$ to be
the ample class generating $\operatorname{Pic\,}(Y)$. Recall that $Y$ admits a
semiorthogonal decomposition (1) and a subcategory $\mathsf{Ku}(Y)$ whose
numerical Grothendieck group $K_{num}(\mathsf{Ku}(Y))\subset K_{num}(Y)$ is
generated by classes $v,w$ (see (3)).

Here we recall the definitions and some properties of the moduli spaces of
stable objects, focusing on those of class $w\in K_{num}(\mathsf{Ku}(Y))$.

Our starting point is the moduli space $\mathcal{M}_{G}(w)$ of sheaves of
class $w$ which are stable in the sense of Gieseker (we refer the reader to
[HL10, Chapter 4] for the definition and additional details). The space
$\mathcal{M}_{G}(w)$ is a projective scheme of finite type, whose Zariski
tangent space at a point $[F]$ is canonically isomorphic to

(6) $T_{[F]}\mathcal{M}_{G}(w)\simeq\operatorname{Ext}^{1}(F,F).$

Moreover, $\mathcal{M}_{G}(w)$ is a fine moduli space. In other words, there
is a universal family on $Y\times\mathcal{M}_{G}(w)$: this is a consequence of
[HL10, Theorem 4.6.5], combined with the observation that, if $L$ in $Y$ is
any smooth line (i.e., a smooth one dimensional subscheme of Hilbert
polynomial $1+t$, with respect to the polarization $H$), we have
$\chi(w\otimes\mathcal{O}_{L})=\chi(\mathbb{C}_{p})=1$.

Given a (weak) stability condition $\sigma_{\alpha,\beta}$ on $D^{b}(Y)$ as in
Proposition 1.10, one can define the moduli functor parametrizing
$\sigma_{\alpha,\beta}$-semistable objects of fixed class $w$. This functor is
corepresented by an algebraic stack of finite type [Tod13a, Prop. 3.7],
denoted $\mathcal{M}_{\alpha,\beta}(w)$, which is proper if every
$\sigma_{\alpha,\beta}$-semistable objects is $\sigma_{\alpha,\beta}$-stable
(quasi-properness is [TP15, Theor. 1.2], and properness is a consequence of
[AP06], with a standard argument as in [BM14b, Lemma 6.6]).

####  1.1.4. $T$-linear categories, projections, and base change

We recall some aspects of [Kuz11], which represents the main technical tool of
some parts of this paper. For any quasi-projective variety $V$, we will use
$D(V)$ to denote the category of perfect complexes on $V$. In particular, for
a smooth $V$, we will have $D(V)\simeq D^{b}(V)$ the derived category of
bounded coherent sheaves.

Let $T$ be a quasi-projective variety, and $Y$ a smooth projective variety. We
say that a subcategory $\mathcal{T}\subset D(Y\times T)$ is $T$-linear if it
is stable under tensoring by pull-backs of objects of $D(T)$ under the
projection $Y\times T\to T$. A functor
$\Phi\colon\mathcal{T}\to\mathcal{T}^{\prime}$ between two $T$-linear
categories is $T$-linear if $\Phi(F\boxtimes G)\simeq\Phi(F)\boxtimes G$ for
all $F\in\mathcal{T}$, $G\in D(T)$.

Analogously, a semiorthogonal decomposition $D(Y\times
T)=\langle\mathcal{T}_{1},...,\mathcal{T}_{n}\rangle$ is $T$-linear if the
$\mathcal{T}_{i}$ are. If $\langle\mathcal{T}_{1},...,\mathcal{T}_{n}\rangle$
is a $T$-linear semiorthogonal decomposition, then the associated projection
functors are $T$-linear.

We summarize the results of [Kuz11] which will be used in our work. Here, we
write Kuznetsov’s results in the case of a $T$ linear category arising from an
exceptional object on $Y$, which allows us to be more explicit.

Now let $E$ be an exceptional object on $Y$. Then, the category

$\langle E\rangle_{T}\coloneqq\langle E\boxtimes G\,|\,G\in D(T)\rangle\subset
D(Y\times T)$

is $T$-linear ([Kuz11, Cor. 5.9]).

There are $T$-linear projection functors $\mathbb{R}_{E}^{T}$ and
$\mathbb{L}_{E}^{T}$ on $\\!{}^{\perp}\langle E\rangle_{T}$ and $\langle
E\rangle_{T}^{\perp}$ respectively ([Kuz11, Sec. 2.3]). They are defined as
the cones of the (co)evaluation morphisms, as described by the triangles:

$\displaystyle E\boxtimes
q_{*}R\mathcal{H}om(p^{*}E,X)\xrightarrow{ev}X\to\mathbb{L}_{E}^{T}(X)$
$\displaystyle\mathbb{R}_{E}^{T}(X)\to X\xrightarrow{coev}E\boxtimes
q_{*}R\mathcal{H}om(p^{*}E,X\otimes\omega_{q}),$

where $p$ (resp. $q$) denotes the projections from $Y\times T$ onto the first
(resp. second) factor.

By [Kuz11, Sec. 5], the projection functors commute with base change, in the
following sense. For any $f\colon S\to T$, denote by $\bar{f}\colon Y\times
S\to Y\times T$ the corresponding base-change morphism. Then we have
$\mathbb{R}_{E}^{S}(\bar{f}^{*}(X))\simeq\bar{f}^{*}(\mathbb{R}_{E}^{T}(X))$
for all $X\in D(Y\times T)$, and similarly for $\mathbb{L}$. In particular, at
every closed point $t\in T$ the projection functors $\mathbb{R}_{E}^{t}$ and
$\mathbb{L}_{E}^{t}$ are the right and left mutation of objects on
$D(Y\times\\{t\\})\simeq D(Y)$ across the exceptional object $E$:

$\displaystyle
E\otimes\operatorname{Hom}^{\bullet}(E,X)\xrightarrow{ev}X\to\mathbb{L}_{E}^{t}(X)$
$\displaystyle\mathbb{R}_{E}^{t}(X)\to
X\xrightarrow{coev}E\otimes\operatorname{Hom}^{\bullet}(X,E)^{\vee}.$

####  1.1.5. Hyperplane sections of $Y$

Let $Y$ denote a smooth Fano threefold of Picard Rank 1 and index 2. As
anticipated in the Introduction, $Y$ belongs to one of five families, indexed
by degree. The assumption of smoothness suffices to prove our results if
$d=3,4,5$, but we introduce (mild) additional assumptions for lower degrees:
we make them explicit here, and fix some notation.

If $d=2$, we assume that $Y_{2}\xrightarrow{\pi}\mathbb{P}^{3}$ is a general
double cover ramified over a quartic surface $R$ (which will be often
identified with the branching locus). In this case, $\mathcal{O}_{Y}(1)$ is
the pullback of $\mathcal{O}_{\mathbb{P}^{3}}(1)$. The generality assumption
here is that of [Wel81, TM03], whose results we will need. Precisely, we
assume $R$ to be smooth and to not contain lines [Wel81, §1], consequently,
$Y$ and its Fano surface of lines are smooth [KPS18, Remark 2.2.9]. See
Section 4 for more details.

Finally, $Y_{1}$ is a sextic hypersurface in the weighted projective space
$\mathbb{P}\coloneqq\mathbb{P}(1,1,1,2,3)$ and $\operatorname{Pic\,}(Y_{1})$
is generated by restriction of $\mathcal{O}_{\mathbb{P}}(1)$. Projection onto
the first four variables induces a two-to-one cover $Y_{1}\to K$, where $K$ is
a cone over a Veronese surface (for this reason, $Y$ is often referred to as
to a double Veronese cone). We use $\pi$ to denote the projection $\pi\colon
K\dashrightarrow\mathbb{P}^{2}$, defined away from the vertex of $K$. The
cover branches over a surface $D\subset K$, whose projection to the Veronese
surface ramifies at a curve $C$ (see Section 4.4 for a more detailed
description). Throughout, we make the same generality assumption as [Tih82]:
we assume that $C$ is smooth, irreducible, and general in moduli, and that the
dual curve $\pi(C)^{\vee}\subset\check{\mathbb{P}}^{2}$ only has ordinary
double points or simple inflection points (see [Tih82, Prop. 2.3]).

In all cases, general hyperplane sections of $Y$ are del Pezzo surfaces, we
then recall here some of the related notions and some aspects of degenerations
of del Pezzo surfaces. We define:

######  Definition 1.14.

Let $S$ be a hyperplane section of $Y$, i.e.
$S\in\lvert\mathcal{O}_{Y}(1)\rvert$. A root of $S$ is a (possibly Weil)
divisor $D$ such that $D.K_{S}=0$ and $D^{2}=-2$ (see [Dol12, §8.2.3]).

By a root of $Y$ we mean a sheaf of the form $\iota_{*}(\mathcal{O}_{S}(D))$,
where $S\in\lvert\mathcal{O}_{Y}(1)\rvert$, $D$ is a root of $S$, and
$\iota\colon S\to Y$ is the inclusion map.

We’ll use the following definition (see for example [Dol12, §8.1]) to describe
degenerations of del Pezzo surfaces (which arise as hyperplane sections of
$Y$, by Lemma 1.16). Let $\Sigma=\\{x_{1},...,x_{9-d}\\}$ be a set of $9-d$
points in $\mathbb{P}^{2}$. Denote by $\Sigma_{j}$ the set
$\\{x_{1},...,x_{j}\\}$ for $1\leq j\leq 9-d$, and by $V_{j}\to\mathbb{P}^{2}$
the blow-up of $\mathbb{P}^{2}$ with center $\Sigma_{j}$. Then there is a
chain of morphisms

$V_{9-d}\to V_{9-d-1}\to...\to V_{1}\to\mathbb{P}^{2}$

where the exceptional set of the $j$-th step, $E_{j}\subset V_{j}$, is
contracted to $x_{j}\in V_{j-1}$. Every smooth del Pezzo surface arises as
$V_{9-d}$ where the $x_{j}$ are chosen to be in general position [Dol12,
§8.1.25].

######  Definition 1.15.

The points of $\Sigma$ are said to be in almost general position if:

  * •

$d\geq 1$;

  * •

the dual graph of (the strict transforms of) $\cup_{j}E_{j}\subset V_{9-d}$ is
a disjoint union of chains;

  * •

no line of $\mathbb{P}^{2}$ passes through 4 points of $\Sigma$;

  * •

no conic of $\mathbb{P}^{2}$ passes through 7 points of $\Sigma$.

We now describe hyperplane sections of $Y$:

######  Lemma 1.16.

Let $Y$ be a rank 1 index 2 Fano threefold, and let
$S\in\lvert\mathcal{O}_{Y}(1)\rvert$ be a hyperplane section of $Y$. Suppose
moreover that if $d\leq 2$ $Y$ is general in the above sense. Then:

  1. (1)

$S$ is an integral, normal, Gorenstein surface with anti-canonical bundle
$-K_{S}\simeq\mathcal{O}_{Y}(1)_{|S}$;

  2. (2)

The general $S$ is a smooth del Pezzo surface of degree $d$. If $S$ is
singular, then either

     * •

it has ordinary double points , or

     * •

it is a cone over an elliptic curve.

The degenerations above are realized by specializing $9-d$ on $\mathbb{P}^{2}$
to an almost general position (in which case they obtain rational double
points) or by degenerating a del Pezzo surface to a cone over a section of
$\lvert-K_{S}\rvert$;

  3. (3)

The set of roots of $S$ is finite in $\mathrm{Cl}(S)$. In any of the cases
above, $S$ contains no effective roots.

###### Proof.

Since $Y$ has Picard rank 1, all $S$ are irreducible and reduced.

Normality is a consequence of Zak’s theorem on tangencies: [Laz04, Cor.
3.4.19] gives normality immediately if $d=3,4,5$. For $d=1,2$ we again apply
Zak’s theorem, but not to $S$: for example, in the degree 2 case, $S$ is the
pullback of a plane $P\subset\mathbb{P}^{3}$, and is also a double cover,
ramified over the quartic $R\cap P$. Then, $S$ is only singular if the
branching locus $R\cap P$ is, i.e. if $P$ is tangent to $R$. But the
finiteness of the Gauss map of $R$ ([Laz04, Cor. 3.4.18]) implies that
singularities of $S$ must be isolated. Similarly, a hyperplane section of
$Y_{1}$ is pulled back from a line $L$ in $\mathbb{P}^{2}$, and is singular
iff $L$ is tangent to the curve $\pi(C)$. Then, the finiteness of the Gauss
mapping for $\pi(C)$ suffices to conclude (this follows from the generality
assumption as in [Tih82, Prop. 2.3]).

All surfaces $S$ are Gorenstein, their anti-canonical bundle being the
restriction $\mathcal{O}_{Y}(1)_{|S}$ (the adjunction formula [Kol13, §4.1]
applies to this case and the different vanishes [Kol13, Prop. 4.5(1)]).
Moreover, for a root $D$ we have $D.K_{S}=-D.H_{|S}=0$ and $H_{|S}$ is ample,
so $D$ cannot be effective.

Normal Gorenstein surfaces with ample anti-canonical bundle are classified in
[HW81]: if they are smooth, then they are del Pezzo surfaces of degree
$d=K_{S}^{2}$. If they have rational singularities, they are obtained from a
smooth del Pezzo surface by specializing $9-d$ points in $\mathbb{P}^{2}$ to
almost general position, and contracting all $(-2)$ curves (see [HW81, §3]).
The only non-rational singularity that may occur is obtained by degenerating a
del Pezzo surface to a cone over an elliptic curve $C\in\lvert-K_{S}\rvert$
[HW81, §4].

Roots are finite for a smooth del Pezzo surface $S$ of degree $d\leq 5$
[Dol12, §8.2.3]. Therefore, they are also finite for surfaces obtained from
$S$ by contracting $(-2)$ curves. If, instead, $S$ is a cone over an elliptic
curve, then its divisor group contains no roots [Kol13, Prop. 3.14]. ∎

##  2\. Construction of the moduli spaces

In this section, we describe three moduli spaces of objects of class $w$. Our
main interest lies in the moduli space of $\sigma$-stable objects in
$\mathsf{Ku}(Y)$, denoted $\mathcal{M}_{\sigma}(w)$, where $\sigma$ is one of
the stability conditions of Proposition 1.13. To describe it, we will make use
of the moduli space of Gieseker stable sheaves of class $w$, denoted
$\mathcal{M}_{G}(w)$, and of moduli $\mathcal{M}_{\alpha,\beta}(w)$ of
$\sigma_{\alpha,\beta}$-stable objects (see 1.1.3). We begin our study from
$\mathcal{M}_{G}(w)$ (Proposition 2.1), then we perform wall-crossing and
compare $\mathcal{M}_{G}(w)$ with one of the spaces
$\mathcal{M}_{\alpha,\beta}(w)$ (Proposition 2.7). Finally, we show that
$\mathcal{M}_{\alpha,\beta}(w)$ and $\mathcal{M}_{\sigma}(w)$ coincide
(Proposition 2.10).

###  2.1. The Gieseker moduli space $\mathcal{M}_{G}(w)$

######  Proposition 2.1.

The moduli $\mathcal{M}_{G}(w)$ of Gieseker-semistable objects of class $w$ on
$Y$ has two irreducible components. One, denoted $\mathcal{P}$, has dimension
$d+3$ and parametrizes ideal sheaves $I_{p|S}$ of a point in a hyperplane
section $S$ of $Y$. A point $[I_{p|S}]\in\mathcal{P}$ is smooth in
$\mathcal{M}_{G}(w)$ iff $p$ is a smooth point of $S$.

The second component has dimension $d+1$, and its smooth points parametrize
roots of $Y$ (see Def. 1.14).

The components intersect in the locus parametrizing $I_{p|S}$ with $S$
singular at $p$.

###### Proof.

Since the class $w$ is torsion, a Gieseker-stable sheaf $E$ of class $w$ must
be pure. This implies that $E=\iota_{*}(F)$ for some sheaf $F$ supported on a
hyperplane section $\iota\colon S\hookrightarrow Y$, otherwise the kernel of
the map $E\to\iota_{*}\iota^{*}E$ would give a destabilizing subsheaf of
smaller dimension. Stability of $E$ implies that $F$ is a torsion-free rank-
one stable sheaf on $S$.

Now we claim that $F$ must have the form $F=I_{Z}\otimes\mathcal{O}_{S}(D)$,
with $Z$ a zero-dimensional subscheme and $\mathcal{O}_{S}(D)$ a reflexive
sheaf of rank 1 associated to a Weil divisor $D$ on $S$. By Lemma 1.16, $S$ is
integral and normal. Since $S$ is integral, the map $F\to F^{\vee\vee}$ is
injective [Sch10, Theor. 2.8]. Since $S$ is normal, $F^{\vee}$ is reflexive
and therefore $F^{\vee}\simeq\mathcal{O}_{S}(-D)$ for some Weil divisor $D$
[Sch10, Sec. 3]. Then, $F\otimes\mathcal{O}(-D)\subset
F^{\vee\vee}\otimes\mathcal{O}(-D)\simeq\mathcal{O}_{S}$ and
$F\otimes\mathcal{O}(-D)$ must be an ideal sheaf of a subscheme supported in
codimension 2, which shows the claim. In particular, we have
$\operatorname{ch}(F)=\left([S],D,\operatorname{ch}_{2}(\mathcal{O}_{S}(D))-z[\mathrm{pt}]\right)$.

Next, we apply the Grothendieck-Riemann-Roch theorem to compute
$\operatorname{ch}(F)\in H^{*}(S)$. The computation can be done assuming that
$S$ is smooth: a Riemann-Roch theorem continues to hold for local complete
intersections [Suw03, §4], and the Chern class of
$N_{S/Y}\simeq\mathcal{O}_{Y}(H)_{|H}$ does not depend on the choice of $S$ in
its linear series. Then we have

$\operatorname{ch}(w)=\iota_{*}(\operatorname{ch}(F)\cdot\operatorname{td}(N_{S/Y}))$

which implies that $\operatorname{ch}(F)=([S],D,-[\mathrm{pt}])\in H^{*}(S)$,
where $D\in H^{2}(S)$ satisfies $D.(H_{|S})=0$. Then, there are two
possibilities: $z=1$ and $\operatorname{ch}_{2}(F)=0$, or $z=0$ and
$\operatorname{ch}_{2}(F)=-1$.

To proceed, we need to be more precise about the second Chern class for
reflexive rank-one sheaves. The Riemann-Roch formula on normal surfaces
presents a correction term $\delta_{S}(D)\in\mathbb{Q}$, only depending on $D$
and the singularities of $S$:

$\chi(\mathcal{O}_{S}(D))=\chi(\mathcal{O}_{S})+\dfrac{1}{2}(D^{2}-K_{S}.D)+\delta_{S}(D).$

For $S$ as in Lemma 1.16, $\delta_{S}(D)\leq 0$ (see [Rei87, Theor. 9.1] for
rational singularities, and [Lan00, §7.7] for the elliptic ones). Consistently
with the Riemann-Roch formulae, we define the second Chern character

$\operatorname{ch}_{2}(\mathcal{O}_{S}(D))=\dfrac{D^{2}}{2}+\delta_{S}(D).$

Let $p\colon\tilde{S}\to S$ denote the minimal resolution of $S$. If $z=1$,
then $0=D^{2}=(p^{*}(D))^{2}$. Then, by the Hodge index theorem [BHPVdV04,
Cor. 2.16], $p^{*}D=0$ and therefore $D=0$ in $H^{2}(S)$. If $z=0$, then $D$
is a root unless $(p^{*}D)^{2}=\frac{D^{2}}{2}=-1$ and
$\delta_{S}(D)=-\frac{1}{2}$. If $S$ is a cone, this cannot happen since
$\delta_{S}(D)\leq-1$ [Lan00, §7.7]. If $S$ has ADE singularities, then
$\tilde{S}$ is a crepant resolution, hence
$K_{\tilde{S}}=p^{*}K_{S}=p^{*}(-H_{|S})$ and therefore
$(p^{*}D).K_{\tilde{S}}=D.(H_{|S})=0$. But then, the Riemann Roch theorem on
$\tilde{S}$ implies that $(p^{*}D)^{2}$ is even, a contradiction.

For the claims about smoothness, observe the following: the locus
parametrizing sheaves of the form $I_{p|S}$ has dimension
$d+3=\dim\mathbb{P}(H^{0}(\mathcal{O}_{Y_{d}}(1))+2$, which coincides with the
rank of the Zariski tangent space of $\mathcal{M}_{G}(w)$ at $[I_{p|S}]$ iff
$p$ is a smooth point of $S$ by Lemma 2.2 and the isomorphism (6).

On the other hand, the locus of sheaves $\iota_{*}\mathcal{O}_{S}(D)$ has
dimension $d+1=\dim\mathbb{P}(H^{0}(\mathcal{O}_{Y_{d}}(1))$, since $S$ has
finitely many roots by Lemma 1.16. Then, each of these points is smooth in
$\mathcal{M}_{G}(w)$ by Lemma 2.4 and the isomorphism (6).

Now we claim that the two components intersect exactly at those points
parametrizing sheaves $I_{p|S_{0}}$, with $S_{0}$ a hyperplane section of $Y$
singular at $p$. Indeed, if $S_{0}$ has rational singularities, then one can
exhibit a flat family of sheaves of the form $\iota_{*}\mathcal{O}_{S}(D)$
degenerating to $I_{p|S}$: we do this for one explicit degeneration in Example
2.5, the other degenerations are completely analogous. If $S_{0}$ is a cone,
by the properness of $\mathcal{M}_{G}(w)$ any family of
$\iota_{*}\mathcal{O}_{S}(D)$ supported on a smoothing of $S_{0}$ has limit
$I_{p|S_{0}}$, since $\mathrm{Cl}(S_{0})$ contains no roots (Lemma 1.16).

∎

######  Lemma 2.2.

Let $S\subset Y_{d}$ be a hyperplane section, let $p$ a smooth point of $S$.
Then we have

$\dim\operatorname{Ext}^{i}(I_{p|S},I_{p|S})=\begin{cases}0&\text{ if }i=3\\\
2&\text{ if }i=2\\\ d+3&\text{ if }i=1\\\ 1&\text{ if }i=0.\end{cases}$

###### Proof.

Since $I_{p/S}\simeq[\mathcal{O}_{S}\rightarrow\mathbb{C}_{p}]$ in $D^{b}(Y)$,
then $R\operatorname{Hom}(I_{p/S},I_{p/S})$ is the totalization of a double
complex $K^{\bullet,\bullet}$ isomorphic to

$K^{\bullet}\coloneqq\left[R\operatorname{Hom}(\mathbb{C}_{p},\mathcal{O}_{S})\rightarrow
R\operatorname{Hom}(\mathcal{O}_{S},\mathcal{O}_{S})\oplus
R\operatorname{Hom}(\mathbb{C}_{p},\mathbb{C}_{p})\rightarrow
R\operatorname{Hom}(\mathcal{O}_{S},\mathbb{C}_{p})\right],$

so we may compute $\operatorname{Ext}^{i}(I_{p/S},I_{p/S})$ using a spectral
sequence

$E_{1}^{p,q}=H^{q}(K^{\bullet,p})\Rightarrow H^{p+q}(K^{\bullet}).$

Then the first page $E^{p,q}_{1}$ of the spectral sequence is

$\vdots$ | $\vdots$ | $\vdots$
---|---|---
$\operatorname{Ext}^{1}(\mathbb{C}_{p},\mathcal{O}_{S})$ | $\operatorname{Ext}^{1}(\mathcal{O}_{S},\mathcal{O}_{S})\oplus\operatorname{Ext}^{1}(\mathbb{C}_{p},\mathbb{C}_{p})$ | $\operatorname{Ext}^{1}(\mathcal{O}_{S},\mathbb{C}_{p})$
$\operatorname{Hom}(\mathbb{C}_{p},\mathcal{O}_{S})$ | $\operatorname{Hom}(\mathcal{O}_{S},\mathcal{O}_{S})\oplus\operatorname{Hom}(\mathbb{C}_{p},\mathbb{C}_{p})$ | $\operatorname{Hom}(\mathcal{O}_{S},\mathbb{C}_{p})$
($p=-1$) | ($p=0$) | ($p=1$)

with arrows pointing to the right. The dimensions of the vector spaces above
are given in the table below.

1 | 1 | 0
---|---|---
1 | 3 | 0
0 | $d+4$ | 1
0 | 2 | 1

The map in the top row is an isomorphism
$\operatorname{Ext}^{3}(\mathbb{C}_{p},\mathcal{O}_{S})\rightarrow\operatorname{Ext}^{3}(\mathbb{C}_{p},\mathbb{C}_{p})$.
The map in the bottom row is surjective because
$\operatorname{Hom}(\mathcal{O}_{S},\mathcal{O}_{S})\rightarrow\operatorname{Hom}(\mathcal{O}_{S},\mathbb{C}_{p})$
is.

Consider the maps in the third and second row. We claim that if $p$ is smooth
in $S$, then these two are both non-zero, and if $p$ is singular, then they
are both $0$. For example, the map in the third row is
$\operatorname{Ext}^{1}(\mathbb{C}_{p},\mathbb{C}_{p})\rightarrow\operatorname{Ext}^{1}(\mathcal{O}_{S},\mathbb{C}_{p})$.
By applying the functor $\operatorname{Hom}(-,\mathbb{C}_{p})$ to the sequence
$I_{p|S}\to\mathcal{O}_{S}\to\mathbb{C}_{p}$, one sees that its kernel is
$\operatorname{Hom}(I_{p|S},\mathbb{C}_{p})$. On the other hand we have

$\operatorname{Hom}(I_{p|S},\mathbb{C}_{p})\simeq\operatorname{Hom}_{S}(I_{p|S},\mathbb{C}_{p})\simeq\operatorname{Ext}^{1}_{S}(\mathbb{C}_{p},\mathbb{C}_{p})\simeq
T_{p}S,$

and the latter has rank 2, resp. 3, if $p$ is smooth, resp. singular, in $S$.
The map in the second row is
$\operatorname{Ext}^{2}(\mathbb{C}_{p},\mathcal{O}_{S})\to\operatorname{Ext}^{2}(\mathbb{C}_{p},\mathbb{C}_{p})$,
and the argument for the claim is similar.

This shows that the second page of the spectral sequence is supported on the
central column of the table, and hence $E_{2}^{p,q}$ abuts to the claimed
dimensions. ∎

######  Lemma 2.3.

If $p\in S$ is singular, then

$\dim\operatorname{Ext}^{i}(I_{p|S},I_{p|S})=\begin{cases}0&\text{ if }i=3\\\
3&\text{ if }i=2\\\ d+4&\text{ if }i=1\\\ 1&\text{ if }i=0.\end{cases}$

###### Proof.

From the above proof, we see that the dimensions of the objects on the second
page are

0 | 0 | 0
---|---|---
1 | 3 | 0
0 | $d+4$ | 1
0 | 1 | 0

and conclude that $\operatorname{Ext}^{1}(I_{p|S},I_{p|S})$ is either
$\mathbb{C}^{d+4}$ or $\mathbb{C}^{d+5}$. Using the same spectral sequence as
above, only this time starting with the short exact sequence

(7)
$\mathcal{O}_{Y}(-1)\rightarrow\mathcal{I}_{p}\rightarrow\mathcal{I}_{p|S}$

it is not hard to see that it can only be $\mathbb{C}^{d+4}$. ∎

######  Lemma 2.4.

Suppose that $E=\iota_{\ast}F$, where $F=\mathcal{O}_{S}(D)$ is a reflexive
sheaf on a hyperplane section $\iota:S\rightarrow Y$. Then

$R^{i}\mathcal{H}\text{om}(E,E)=\begin{cases}\iota_{\ast}\mathcal{O}_{S}&\text{
if }i=0\\\ \iota_{\ast}\mathcal{O}_{S}(1)&\text{ if }i=1\\\ 0&\text{
otherwise.}\end{cases}$

Therefore $\operatorname{Ext}^{1}(E,E)=\mathbb{C}^{d+1}$, and
$\operatorname{Ext}^{2}(E,E)=\operatorname{Ext}^{3}(E,E)=0$.

###### Proof.

By adjunction,

(8) $R\mathcal{H}om_{Y}(E,E)\simeq R\mathcal{H}om_{S}(Li^{*}E,F).$

We can compute $E\otimes^{L}\mathcal{O}_{S}$ by tensoring the short exact
sequence $\mathcal{O}_{Y}(-1)\rightarrow\mathcal{O}_{Y}\to\mathcal{O}_{S}$
with $E$. Then, since $\iota^{\ast}=\iota^{-1}(\underline{\hskip
10.00002pt}\otimes^{L}\mathcal{O}_{S})$, we get $L^{0}\iota^{\ast}E=F$ and
$L^{1}\iota^{\ast}E=F(-1)$. Since $F$ is reflexive, we have
$\mathcal{H}om_{S}(F,F)=\mathcal{O}_{S}$, so the statement follows from the
spectral sequence for hypercohomology of the complex in (8). ∎

In the following example we exhibit a flat family of sheaves whose general
element has the form $\iota_{*}\mathcal{O}_{S}(D)$, for $\iota\colon S\to Y$ a
hyperplane section and $D$ a root of $S$, which specializes to $I_{p|S_{0}}$
(where $p$ is a singular point in a hyperplane section $S_{0}$).

######  Example 2.5 ($I_{p/S}$ as a flat limit of
$\iota_{*}(\mathcal{O}_{S}(D))$’s).

Let $f:Y=Y_{d}\rightarrow\mathbb{P}^{d+1}$ be the map induced by $-K_{Y}/2$
(assume $d\geq 2$). Denote with $\mathcal{S}$ the universal hyperplane
section. Let $h_{0}\in(\mathbb{P}^{d+1})^{\vee}$ cut out a hyperplane section
$S_{0}\subset Y$ that is singular at $p$. Let
$\Delta\subset(\mathbb{P}^{d+1})^{\vee}$ be a 1-dimensional analytic disk
containing $h_{0}$. Hyperplane sections with rational singularities arise by
specializing the $9-d$ points in $\mathbb{P}^{2}$ (Lemma 1.16). Therefore,
after possibly replacing it with a finite cover, $\Delta$ supports a family of
hyperplane sections of $Y$ constructed by blowing up $9-d$ sections $s_{i}$ of
the projection $p:\Delta\times\mathbb{P}^{2}\rightarrow\Delta$.

For $h_{t}\in\Delta$, we denote by $S_{t}\subset Y$ the corresponding
hyperplane section, and by $f_{t}:S_{t}\rightarrow\mathbb{P}^{d}$ the
restriction of $f$ to $S_{t}$. The sections $s_{i}$ are in general position
for $t\neq 0$, and they are in almost general position (see Lemma 1.16) over
$t=0$. We now assume that $s_{1},s_{2},s_{3}$ are collinear over $t=0$, but
one can carry over the same argument for the other degenerations. Denote the
blowup with $g:\mathcal{S}^{\prime}\rightarrow\Delta\times\mathbb{P}^{2}$. The
map induced by the dual of the relative dualizing sheaf $\omega_{p\circ g}$
factors through the universal hyperplane section pulled back to $\Delta$:

${\mathcal{S}^{\prime}}$${\Delta\times\mathbb{P}^{d}}$${\mathcal{S}_{\Delta}}$${\Delta\times
Y}$$\scriptstyle{-\omega_{p\circ g}}$

Here the vertical map is induced by the dual of the relative dualizing sheaf
of $\mathcal{S}_{\Delta}\rightarrow\Delta$. Its fibers are
$f_{t}:S_{t}\rightarrow\mathbb{P}^{d}$.

Let $\mathcal{E}_{i}$ be the exceptional divisors of $g$. Define
$D=-g^{\ast}(\mathcal{O}_{\mathbb{P}^{2}}(1))+\mathcal{E}_{1}+\mathcal{E}_{2}+\mathcal{E}_{3}$.
Then the fiber of $D$ over $t\neq 0$ is a Cartier divisor $D_{t}$ of degree
$0$ and $D_{t}^{2}=-2$, and when $t=0$, $D$ restricts to $-E$, where $E$ is
the $(-2)$ curve in the central fiber of the blowup $g$. If we pushforward to
$\Delta\times Y$, we get a flat family of divisors
$\iota_{*}(\mathcal{O}_{S_{t}}(D_{t}))$ over a general fiber, and
$I_{p/S_{0}}$ in the central fiber.

###  2.2. Moduli spaces and wall crossing in the $(\alpha,\beta)$-plane

Now we turn to the study of the wall crossing; we investigate moduli spaces of
stable objects of class $w$ with respect to a stability condition
$\sigma_{\alpha,\beta}$ of Proposition 1.10. In particular, we fix
$\beta=-1/2$, $\alpha^{+}\gg 1$ and $0<\alpha^{-}\ll 1$. We denote the
corresponding moduli spaces by $\mathcal{M}_{+}(w)$, $\mathcal{M}_{-}(w)$
respectively. In Proposition 2.7 we describe them and show that the former
coincides with $\mathcal{M}_{G}(w)$ and is separated from the latter by a
single wall. We need a preparatory Lemma:

######  Lemma 2.6.

The only tilt-semistable sheaves
$A\in\operatorname{Coh}_{\alpha,-\frac{1}{2}}(Y)$ having truncated Chern class
$\operatorname{ch}^{-\frac{1}{2}}_{\leq
2}(A)=\left(1,\dfrac{H}{2},\dfrac{H^{2}}{8}\right)$ are $\mathcal{O}_{Y}$ and
$I_{Z}$, with $Z$ a zero-dimensional subscheme.

###### Proof.

Such an object $A$ has
$H^{2}\operatorname{ch}_{1}^{-\frac{1}{2}}(A)=\frac{H^{3}}{2}$, which is the
minimal positive value of $H^{2}\operatorname{ch}_{1}^{-\frac{1}{2}}$ on
objects of $\operatorname{Coh}_{\alpha,-\frac{1}{2}}(Y)$. By [BMT14, Lemma
7.2.2], $A$ must be a slope-stable sheaf, so in particular torsion-free. Since
the untwisted Chern character of $A$ is $\operatorname{ch}_{\leq
2}(A)=\left(1,0,0\right)$, we must have $A\simeq\mathcal{O}_{Y}$ or $A\simeq
I_{Z}$ for some zero-dimensional subscheme $Z$. ∎

We will also need the following definition: let $p$ be a point on $Y$ and
$I_{p}$ its ideal sheaf; applying the functor
$\operatorname{Hom}(\mathcal{O}_{Y},-)$ to the short exact sequence

$0\to I_{p}(-1)\to\mathcal{O}_{Y}(-1)\to\mathbb{C}_{p}\to 0$

and since $\mathcal{O}_{Y}(-1)$ has no cohomology, one computes that
$\operatorname{Ext}^{1}(\mathcal{O}_{Y},I_{p}(-1))=\mathbb{C}$. Serre duality
with $K_{Y}=\mathcal{O}_{Y}(-2)$ gives
$\operatorname{Ext}^{1}(\mathcal{O}_{Y},I_{p}(-1))=\operatorname{Ext}^{1}(I_{p},\mathcal{O}_{Y}(-1)[1])^{\vee}$:
let then $E_{p}$ be the unique non-trivial extension (4) so that $[E_{p}]=w$.

######  Proposition 2.7.

There are only two chambers in the $(\alpha,\beta)$-plane for class $w$, and
the line $\beta=-\frac{1}{2}$ intersects both. The moduli space of
$\sigma_{\alpha,-\frac{1}{2}}$-semistable objects for $\alpha\gg 1$ is
isomorphic to $\mathcal{M}_{G}(w)$. If $0<\alpha\ll 1$,
$\sigma_{\alpha,-\frac{1}{2}}$-semistable objects of class $w$ have the form
$\iota_{*}\mathcal{O}_{S}(D)$ (where, as usual, $D$ is a root of a hyperplane
section $S$), or $E_{p}$ as in (4).

###### Proof.

The isomorphism $\mathcal{M}_{+}(w)\simeq\mathcal{M}_{G}(w)$ is a standard
argument: every object in $\mathcal{M}_{+}(w)$ is Gieseker-stable [BMT14,
Lemma 7.2.1]. Conversely, assume $E$ is a Gieseker-stable sheaf of class $w$
on $Y$. Then $H^{2}\operatorname{ch}^{-\frac{1}{2}}_{1}(E)=H^{3}$ is twice the
minimal positive value of $H^{2}\operatorname{ch}^{-\frac{1}{2}}_{1}$ on
$\operatorname{Coh}^{-\frac{1}{2}}(Y)$. Then, the only possibility for a
destabilizing sequence $A\to E\to B$ is that $A$ satisfies
$H^{2}\operatorname{ch}^{-\frac{1}{2}}_{1}(A)\leq\frac{H^{3}}{2}$, and hence
$A$ must be a slope stable sheaf of dimension $\geq 2$ (by applying [BMT14,
Lemma 7.2.2] to $A$, and observing that $H^{-1}(A)=0$, since $H^{-1}(E)=0$).
If $A$ has dimension 2, we must have $H^{-1}(B)=0$, but $E$ is pure of
dimension 2 so $A$ cannot destabilize it. If, instead,
$\operatorname{rk}(A)>0$, we must have $\mu_{H}(A)>-\frac{1}{2}$ and therefore
$H^{2}\operatorname{ch}^{\frac{1}{2}}_{1}(A)>0$. Then, one has

$\mu_{\alpha,-\frac{1}{2}}(A)=\frac{H\operatorname{ch}^{\frac{1}{2}}_{2}(A)-\frac{\alpha^{2}}{2}\operatorname{ch}_{0}(A)}{H^{2}\operatorname{ch}^{\frac{1}{2}}_{1}(A)}\xrightarrow{\alpha\to+\infty}-\infty,$

so $A$ cannot destabilize for $\alpha$ sufficiently large. Equivalently, $E$
is $\sigma_{\alpha^{+},-\frac{1}{2}}$-stable if and only if it is Gieseker
stable, and the two moduli spaces are isomorphic.

Since $w$ is a torsion class, walls in the $(\alpha,\beta)$-plane are
concentric semicircles centered at $\alpha=0$ and $\beta=-\dfrac{1}{2}$
[BMSZ17, Theorem 2.2]. In particular, the line $\beta=-\dfrac{1}{2}$
intersects all the possible walls for the class $w$. The twisted Chern
character of $E$ satisfies

$\operatorname{ch}^{-\frac{1}{2}}_{\leq 2}(E)=(0,H,0),$

hence if $A$ gives a numerical wall for $E$ and
$\operatorname{ch}^{-\frac{1}{2}}_{\leq 2}(A)=(x,y,z)$, the equation of the
wall at the point with coordinate $\beta=-\dfrac{1}{2}$ yields

$z-\dfrac{\alpha^{2}}{2}x=0$

meaning that $z$ and $x$ must have the same sign and cannot be equal to 0
(otherwise they would not give a wall).

An actual wall in tilt-stability intersecting the vertical line $\beta=-1/2$
must be given by a short exact sequence $0\to A\to E\to B\to 0$ with
$0<\operatorname{ch}_{1}^{-1/2}(A)<\operatorname{ch}_{1}^{-1/2}(E)=1$; since
$\operatorname{ch}_{1}^{-1/2}(A)\in\frac{1}{2}\mathbb{Z}$, we get
$\operatorname{ch}_{1}^{-1/2}(A)=\dfrac{1}{2}$.

Moreover, by the support property one has $0\leq\Delta(A)\leq\Delta(E)=1$,
which can be rearranged to

$-1\leq-8xz\leq 3;$

since $\beta=-\dfrac{1}{2}$, it follows that $8z\in\mathbb{Z}$, and since $x$
and $z$ have the same sign and can’t be 0 we must have $x=1$ and
$z=\dfrac{1}{8}$.

This means there’s only one possible actual wall, and it would be given by a
subobject with $\operatorname{ch}^{-\frac{1}{2}}_{\leq
2}(A)=\left(1,\dfrac{H}{2},\dfrac{H^{2}}{8}\right)$. As before, $A$ must be a
sheaf and we can assume $A$ to be tilt-semistable. By Lemma 2.6, either
$A\simeq\mathcal{O}_{Y}$ or $A\simeq I_{Z}$ for $Z$ a zero-dimensional
subscheme.

Now if $E=\iota_{*}\mathcal{O}_{S}(D)$, with $D$ as in Prop. 2.1, neither of
the $A$’s can map to $E$: in both cases, a map $A\to E$ would produce a
section of $\mathcal{O}_{S}(D)$, by restricting to $S$ and then dualizing
$A_{|S}\to\mathcal{O}_{S}(D)$ twice. This does not happen since $D$ is not
effective by Lemma 1.16.

On the other hand, there are always maps $I_{p}\to I_{p/S}$ fitting in the
triangle

$0\to I_{p}\to I_{p/S}\to\mathcal{O}(-1)[1]\to 0.$

This means that the numerical wall is an actual wall: all objects in
$\mathcal{M}_{G}(w)\setminus\mathcal{P}_{d}$ are stable on both sides of the
wall, while the $I_{p/S}$’s are destabilized. In turn, the unique extensions

$0\to\mathcal{O}(-1)[1]\to E_{p}\to I_{p}\to 0,$

are also unstable at the wall and become stable below it, since both
$\mathcal{O}(-1)[1]$ and $I_{p}$ are stable, and stay such in the whole
chamber because there are no other walls. ∎

######  Lemma 2.8.

There is an isomorphism of functors
$\left(\operatorname{Sch}/\mathbb{C}\right)^{o}\to\operatorname{Sets}$ between

$\displaystyle\mathtt{I}\colon T\longmapsto\left\\{\text{$\mathcal{I}\in
D(Y\times T)$ a flat family of ideal sheaves of points of $Y$}\right\\}$
$\displaystyle\mathtt{E}\colon T\longmapsto\left\\{\text{$\mathcal{E}\in
D(Y\times T)$ a flat family of complexes $E_{p}$ for some $p\in Y$}\right\\}.$

Consequently, any moduli space for the second functor is a fine moduli space
isomorphic to $Y$, since the scheme $Y$ represents the first functor.

###### Proof.

To prove the lemma, we will show that a scheme $S$ parametrizes a flat family
of sheaves of the form $I_{p}$, for $p\in Y$, if and only if $S$ parametrizes
a flat family of sheaves of the form $E_{p}$, for $p\in Y$, and that moreover
this correspondence is natural, i.e. it commutes with the pull back of
families. Let $\mathcal{I}\in D^{b}(Y\times S)$ be a family of ideal sheaves
of points of $Y$. Then, using the notation of 1.1.4, we consider the
projection $\mathbb{R}_{\mathcal{O}_{Y}(-1)}^{S}\coloneqq D^{b}(Y\times
S)\to\\!^{\perp}\langle\mathcal{O}_{Y}(-1)\rangle_{S}$, and the fibers of
$\mathbb{R}_{\mathcal{O}_{Y}(-1)}^{S}(\mathcal{I})$ over
$Y\times\\{[I_{p}]\\}$ fit in triangles

$\mathbb{R}_{\mathcal{O}_{Y}(-1)}^{S}(\mathcal{I})_{p}\to
I_{p}\to\mathcal{O}_{Y}(-1)[2].$

Hence $\mathbb{R}_{\mathcal{O}_{Y}(-1)}^{S}(\mathcal{I})_{p}\simeq E_{p}$ by
(4). Conversely, if $\mathcal{E}$ is a family of $E_{p}$ over $S$, one can
apply the left mutation functor (see Sec. 1.1.4) and obtain
$\mathbb{L}_{S}(\mathcal{E})_{p}\simeq I_{p}$. The statement about naturality
follows from the base-change properties of the projection functors, that about
moduli from the universal properties of moduli spaces. ∎

######  Proposition 2.9.

There is a locus $\mathcal{Y}\subset\mathcal{M}_{-}(w)$, isomorphic to $Y$,
parametrizing objects defined as in (4), and a wall-crossing morphism
$s:\mathcal{M}_{G}(w)\to\mathcal{M}_{-}(w)$. The map $s$ is surjective on
closed points. It is an isomorphism on
$\mathcal{M}_{G}(w)\setminus\mathcal{P}$, so that the complement of
$\mathcal{Y}$ parametrizes roots of $Y$. Its fiber over a point $p$ of $Y$ is
the projective space $\mathbb{P}(I_{p}(1))$.

###### Proof.

Lemma 2.8 shows that $Y$ supports a family of objects $E_{p}$, which gives a
morphism $\phi\colon Y\to\mathcal{M}_{-}(Y)$. Denote by $\mathcal{Y}$ its
schematic image. We claim $\mathcal{Y}$ corepresents the functor $\mathtt{E}$:
every scheme $T$ parametrizing a family $\mathcal{T}$ of
$\sigma_{\alpha,\beta}$-semistable objects has a map $\psi\colon
T\to\mathcal{M}_{-}(w)$, and by Lemma 2.8 $\psi$ factors through $\phi$ if and
only if $\mathcal{T}_{t}$ is isomorphic to some $E_{p}$ for all $t\in T$
closed. Then, $Y$ and $\mathcal{Y}$ corepresent isomorphic functors. This
implies that they are isomorphic (see e.g. [HL10, Sec. 2.2]) and that $\phi$
is an isomorphism between them.

To define $p$, we construct a family of $\sigma_{\alpha,\beta}$-stable objects
on $\mathcal{M}_{G}(w)$, starting from the universal family $\mathcal{G}$ of
Gieseker stable sheaves on $Y\times\mathcal{M}_{G}(w)$.

Then, let
$\mathcal{J}\coloneqq\mathbb{R}_{\mathcal{O}_{Y}(-1)}^{\mathcal{M}_{G}(w)}(\mathcal{G})\in
D(Y\times\mathcal{M}_{G}(w))$ the projection of $\mathcal{G}$ onto the
category
$\\!{}^{\perp}\langle\mathcal{O}_{Y}(-1)\rangle_{\mathcal{M}_{G}(w)}$. We
claim that $\mathcal{J}$ parametrizes $\sigma_{\alpha,\beta}$-stable object,
and therefore induces a morphism
$p\colon\mathcal{M}_{G}(w)\to\mathcal{M}_{-}(w)$.

This can be checked after restricting to fibers $Y\times\\{x\\}$ for
$x\in\mathcal{M}_{G}(w)$. If $x$ represents a sheaf $I_{p|S}$, then one
computes
$\operatorname{Hom}^{\bullet}(I_{p|S},\mathcal{O}_{Y}(-1))\simeq\mathbb{C}[1]\oplus\mathbb{C}[2]$
and sees that $\mathcal{J}_{x}$ fits in the right mutation triangle

$\mathcal{J}_{x}\to
I_{p|S}\xrightarrow{coev}\mathcal{O}_{Y}(-1)[1]\oplus\mathcal{O}_{Y}(-1)[2].$

Applying the octahedral axiom to the composition of $coev$ with the projection
onto the summand $\mathcal{O}_{Y}(-1)[1]$, we find a distinguished triangle

$\mathcal{J}_{x}\to I_{p}\to\mathcal{O}_{Y}(-1)[2].$

It follows that $\mathcal{J}_{x}$ is isomorphic to $E_{p}$ (see the triangle
(4)) and is $\sigma_{\alpha,\beta}$-stable. On the other hand, if
$x=[\iota_{*}\mathcal{O}_{S}(D)]$, we have

$\operatorname{Hom}^{*}(\iota_{*}\mathcal{O}_{S}(D),\mathcal{O}_{Y}(-1))\simeq\operatorname{Hom}^{3-*}(\mathcal{O}_{Y}(1),\iota_{*}\mathcal{O}_{S}(D))^{\vee}=0,$

by Serre duality and Lemma 2.15 below, so
$\mathcal{J}_{x}\simeq\iota_{*}\mathcal{O}_{S}(D)$ is
$\sigma_{\alpha,\beta}$-stable.

This defines the map $p$. The statements about its properties also follow from
the description of the objects $\mathcal{J}_{x}$ just given: for example, the
map $p_{|\mathcal{P}}\colon\mathcal{P}\to\mathcal{Y}$ coincides with the
assignment $[I_{p|S}]\mapsto p$, and the fiber over a point $p\in\mathcal{Y}$
is the projective space
$\mathbb{P}(\operatorname{Hom}(\mathcal{O}_{Y}(-1),I_{p}))\simeq\mathbb{P}(H^{0}(I_{p}(1)))\simeq\mathbb{P}^{d}$.
∎

###  2.3. The moduli space $\mathcal{M}_{\sigma}(w)$

Fix $0<\alpha\ll 1$ and $\beta=-\frac{1}{2}$. Consider the weak stability
condition
$\sigma^{0}_{\alpha,\beta}=(\operatorname{Coh}^{0}_{\alpha,\beta}(Y),Z_{\alpha,\beta}^{0})$
on $D(Y)$ defined in Proposition 1.12. Recall that
$\operatorname{Coh}^{0}_{\alpha,\beta}(Y)$ is a tilt of
$\operatorname{Coh}^{\beta}(Y)$, and

$Z_{\alpha,\beta}^{0}(E)=H^{2}\operatorname{ch}^{\beta}_{1}(E)+i\left(H\operatorname{ch}_{2}^{\beta}(E)-\dfrac{\alpha^{2}}{2}H^{3}\operatorname{ch}_{0}(E)\right).$

The condition $\sigma^{0}_{\alpha,\beta}$ gives rise to a stability condition
$\sigma=(\mathsf{Ku}(Y)\cap\operatorname{Coh}(X)^{0}_{\alpha,\beta},Z^{0}_{\alpha,-\frac{1}{2}})$
on $\mathsf{Ku}(Y)$ by Proposition 1.13. In this section, we focus on the
moduli space $\mathcal{M}_{\sigma}(w)$ of $\sigma$-semistable objects of
$\mathsf{Ku}(Y)$ of class $w$, and we prove:

######  Proposition 2.10.

$\mathcal{M}_{\sigma}(w)$ is isomorphic to $\mathcal{M}_{-}(w)$.

We start by relating stability with respect to $\sigma$,
$\sigma^{0}_{\alpha,-\frac{1}{2}}$, and $\sigma_{\alpha,\beta}$.

######  Remark 2.11.

Notice that with this choice of stability condition,
$Z^{0}_{\alpha,\beta}(w)=1$, so an object $E$ of class $-w$ is stable if and
only if an even shift $E[i]$ of it belongs to the heart
$\operatorname{Coh}^{0}_{\alpha,\beta}(Y)$. Indeed, such an object $E[i]$ has
infinite slope, so it is semistable; moreover, the class $w$ is primitive in
$\mathsf{Ku}(Y)$, so that there are no strictly semistable objects of that
class.

######  Lemma 2.12.

Let $E\in\mathsf{Ku}(Y)$ of class $-w$. Then $E$ is stable with respect to
$\sigma$ if and only if $E$ is $\sigma^{0}_{\alpha,-\frac{1}{2}}$-semistable
in $D(Y)$.

###### Proof.

If $E\in\mathsf{Ku}(Y)$ is $\sigma$-stable, up to a shift we have
$E\in\mathcal{A}=\operatorname{Coh}^{0}_{\alpha,-\frac{1}{2}}(Y)\cap\mathsf{Ku}(Y)$,
so in particular $E\in\operatorname{Coh}^{0}_{\alpha,-\frac{1}{2}}(Y)$. Since
$E$ is in the heart of a (weak) stability condition and it has maximal phase
(because
$\operatorname{Im}Z^{0}_{\alpha,\beta}(E)=\operatorname{Im}Z^{0}_{\alpha,\beta}(w)=0$),
by definition $E$ must be semistable.

Conversely, if $E\in\mathsf{Ku}(Y)$ is semistable in
$\operatorname{Coh}^{0}_{\alpha,-\frac{1}{2}}(Y)$ then in particular $E$
belongs to the heart $\mathcal{A}$, hence it is stable by the above Remark
2.11. ∎

######  Lemma 2.13.

If $E\in\operatorname{Coh}^{0}_{\alpha,\beta}(Y)$ be
$\sigma^{0}_{\alpha,\beta}$-semistable object of class $-w$, then
$E[-1]\in\operatorname{Coh}^{\beta}(Y)$ and it is
$\sigma_{\alpha,\beta}$-semistable.

###### Proof.

Since $E$ is in $\operatorname{Coh}^{0}_{\alpha,\beta}(Y)$, there is a
triangle

$E^{\prime}[1]\to E\to T$

with $E^{\prime}\in\operatorname{Coh}^{\beta}(Y)_{\mu_{\alpha,\beta}\leq 0}$,
$T\in\operatorname{Coh}^{\beta}(Y)_{\mu_{\alpha,\beta}>0}$. Since $E$ is
semistable with respect to $\mu^{0}_{\alpha,\beta}$, $Z_{\alpha,\beta}(T)$ has
to be $0$, so $T$ is supported on points, that is, $T$ has finite length.
Also, $E^{\prime}$ needs to be $\sigma_{\alpha,\beta}$-semistable, because
otherwise the destabilizing subobject of $E^{\prime}$ would make $E$
$\sigma_{\alpha,\beta}^{0}$-unstable.

We now prove that $T$ must vanish, and conclude. By a result of Li [Li15], we
have $\operatorname{ch}^{\beta}_{3}(E^{\prime})\leq 0$. Now,
$\operatorname{ch}(E^{\prime})=w+t\frac{H^{3}}{d}$, where $t$ denotes the
length of $T$. The inequality then reads

$\operatorname{ch}^{\beta}_{3}(E^{\prime})=H^{3}\left[\frac{t}{d}+\frac{(d-6)}{6d}-\frac{1}{8}\right]=H^{3}\left[\frac{1}{24}+\frac{t-1}{d}\right]\leq
0$

which shows $t=0$, and therefore $T=0$. ∎

Lemma 2.12 and Lemma 2.13, together, show that if an object
$E\in\mathsf{Ku}(Y)$ is $\sigma$-semistable then it is
$\sigma_{\alpha,\beta}$-semistable. Next, we work towards a converse: we
classified all $\sigma_{\alpha,\beta}$-semistable objects of class $w$ in
Prop. 2.7, we check that they belong to $\mathsf{Ku}(Y)$ in Lemma 2.14 and
Lemma 2.15 below.

######  Lemma 2.14.

Let $E_{p}$ be defined as in (4). Then
$\operatorname{Hom}^{\bullet}(\mathcal{O}_{Y}(1),E_{p})=\operatorname{Hom}^{\bullet}(\mathcal{O}_{Y},E_{p})=~{}0$.
In other words, the objects $E_{p}$ belong to $\mathsf{Ku}(Y)$.

###### Proof.

From the short exact sequence

$0\to I_{p}\to\mathcal{O}_{Y}\to\mathbb{C}_{p}\to 0,$

it is straightforward to see that
$\operatorname{Hom}^{\bullet}(\mathcal{O}_{Y},I_{p})=0$, and we already
pointed out that $\mathcal{O}_{Y}(-1)$ has no cohomology, so that
$\operatorname{Hom}^{\bullet}(\mathcal{O}_{Y},E_{p})=0$ using the defining
sequence (4).

On the other hand, from the sequence above we have
$\operatorname{Hom}^{\bullet}(\mathcal{O}(1),I_{p})\simeq\mathbb{C}[2]$, and
again by applying $\operatorname{Hom}(\mathcal{O}_{Y}(1),-)$ to the
distinguished triangle (4) we see that
$\operatorname{Hom}^{\bullet}(\mathcal{O}_{Y}(1),E_{p})=0$. ∎

######  Lemma 2.15.

Let $S$ be a hyperplane section of $Y$, and $D$ a root of $S$. Then
$\operatorname{Hom}^{\bullet}(\mathcal{O}_{Y}(1),\iota_{*}(\mathcal{O}_{S}(D))=\operatorname{Hom}^{\bullet}(\mathcal{O}_{Y},\iota_{*}(\mathcal{O}_{S}(D))=0$.
In other words, $\iota_{*}\mathcal{O}_{S}(D)\in\mathsf{Ku}(Y)$.

###### Proof.

By adjunction of the functors $\iota_{*}$ and $\iota^{*}$, it’s enough to
check the required vanishing on $S$. Let $H$ be the pullback of the ample
divisor on $Y$ to $S$: we get
$\operatorname{Hom}^{\bullet}(\mathcal{O}_{Y}(1),\iota_{*}(\mathcal{O}_{S}(D))\simeq
H^{\bullet}(S,\mathcal{O}_{S}(D-H))$ and
$\operatorname{Hom}^{\bullet}(\mathcal{O}_{Y},\iota_{*}(\mathcal{O}_{S}(D))\simeq
H^{\bullet}(S,\mathcal{O}_{S}(D))$. The adjunction isomorphisms above, and the
condition $\chi([\mathcal{O}_{Y}],w)=\chi([\mathcal{O}_{Y}(1)],w)=0$, imply
that $\chi(S,\mathcal{O}_{S}(D))=\chi(S,\mathcal{O}_{S}(D-H))=0$. Since $D$
and $D-H$ are not effective (see Lemma 1.16), we have $H^{0}(D)=H^{0}(D-H)=0$,
and by Serre duality it follows that $H^{2}(D)=H^{2}(D-H)=0$. Then
$H^{1}(D)=H^{1}(D-H)=0$ must be $0$ as well, and the claim is proven. ∎

We’re now ready to prove Proposition 2.10 and lastly Theorem 1.2:

###### Proof of Proposition 2.10.

The proposition follows from the fact that the two moduli spaces
$\mathcal{M}_{-}(w)$ and $\mathcal{M}_{\sigma}(w)$ corepresent isomorphic
functors. Observe that stability and semistability coincide for both $\sigma$
and $\sigma_{\alpha,\beta}$ and class $w$.

Then, for all $E\in D(Y)$, we have that $E$ is a $\sigma$-stable object of
$\mathcal{A}$ if and only if $E[-1]\in\operatorname{Coh}^{\beta}(Y)$ and $E$
is $\sigma_{\alpha,\beta}$-stable: the forward direction follows from Lemma
2.12 and Lemma 2.13. For the converse, observe that
$\sigma_{\alpha,\beta}$-stable objects have been classified in Proposition
2.7, and all belong to $\mathcal{A}$ (after a shift) by Lemma 2.14 and Lemma
2.15. Therefore, they must be stable by Remark 2.11.

∎

We recollect here the results of Section 2:

######  Theorem 2.16.

The moduli space $\mathcal{M}_{\sigma}(w)$ is proper. There is a subvariety
$\mathcal{Y}\subset\mathcal{M}_{\sigma}(w)$, isomorphic to $Y$, parametrizing
objects $E_{p}$ fitting in a distinguished triangle (4). The complement
$\mathcal{M}_{\sigma}(w)\setminus\mathcal{Y}$ is smooth of dimension $d+1$ and
parametrizes roots of $Y$.

###### Proof.

The statement is a combination of the description of $\mathcal{M}_{-}(w)$
(Proposition 2.7, Proposition 2.9), and the isomorphism
$\mathcal{M}_{\sigma}(w)\simeq\mathcal{M}_{-}(w)$ (Proposition 2.10).
Properness follows from properness of $\mathcal{M}_{-}(w)$ (see Section
1.1.3). ∎

###  2.4. The Abel–Jacobi map on $\mathcal{M}_{\sigma}(w)$

Here we recall some facts about the intermediate Jacobian $J(Y)$ of a Fano
threefold $Y$ and its associated Abel–Jacobi map (we refer the reader to
[Voi03, Sec. 12.1] for more details), with particular focus to the case of
curves on threefolds. We also recall the tangent bundle sequence (TBS)
technique developed in [Wel81, Sec. 2] to study the infinitesimal behavior of
the AJ map.

The _intermediate Jacobian_ of $Y$ is the complex torus

$J(Y)\coloneqq H^{1}(Y,\Omega^{2}_{Y})^{\vee}\Big{/}H^{3}(Y,\mathbb{Z});$

and it comes equipped with an _Abel–Jacobi_ map
$\Phi\colon\mathcal{Z}_{1}(Y)^{\text{hom}}\longrightarrow J(Y)$, where
$\mathcal{Z}_{1}(Y)^{\text{hom}}$ is the space of 1-dimensional algebraic
cycles on $Y$ which are homologous to 0, defined as follows. Every
$Z\in\mathcal{Z}_{1}(Y)^{\text{hom}}$ admits a 3-chain $\Gamma$ in $Y$ such
that $\partial\Gamma=Z$. Then one has the linear map on $H^{3,0}(Y)\oplus
H^{2,1}(Y)$ given by integration against $\Gamma$, whose image in $J(Y)$
(projecting onto the second factor) is well-defined and defines the map

$\Phi(Z)\coloneqq\int_{\Gamma}.$

Now suppose $T$ is a smooth variety parametrizing algebraic 1-cycles
$\\{Z_{t}\\}_{t\in T}$ on $Y$. Picking a base point $0\in T$ yields a map
$T\to\mathcal{Z}_{1}(Y)^{\text{hom}}$, which one composes with $\Phi$ to
define an analogous map (see [Wel81, Sec. 2]):

$\Psi\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.78195pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
40.85623pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.85623pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{J(Y)}$}}}}}}}{\hbox{\kern-7.78195pt\raise-16.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
7.78195pt\raise-16.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
31.78195pt\raise-16.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.78195pt\raise-16.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Phi(Z_{t}-Z_{0})}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

We will apply this construction to families of complexes of sheaves, using the
(Poincaré dual of) the second Chern class $c_{2}$: to every point in
$\mathcal{M}_{\sigma}(w)$, parametrizing an object $F\in\mathsf{Ku}(Y)$, we
associate an algebraic 1-cycle $c_{2}(F)$. Abusing notation, we identify
points of $\mathcal{M}_{\sigma}(w)$ with the objects they parametrize, and
pick a base point $F_{0}$. Then, we get a map from $\mathcal{M}_{\sigma}(w)$
to the intermediate Jacobian:

$\Psi\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
16.78633pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{M}_{\sigma}(w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
63.68639pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
63.68639pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{J(Y_{d})}$}}}}}}}{\hbox{\kern-6.90973pt\raise-16.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.90973pt\raise-16.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
40.78633pt\raise-16.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.78633pt\raise-16.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Phi(c_{2}(F)-c_{2}(F_{0}))}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Finally, we recall the TBS technique of [Wel81, Sec. 2] for the infinitesimal
study of $\Psi$. Suppose $Y$ is embedded in a quasi-projective fourfold $W$.
Let $Z$ be a smooth curve on $Y$, and $0\in T$ a smooth family of deformations
of $Z=Z_{0}$, so that $\Psi$ is defined on $T$. Then the dual map of the
differential of $\Psi$ at $0$, denoted $\psi_{Z}^{\vee}$, appears in the
following diagram:

(9) ${H^{0}(N_{Y/W}\otimes
K_{Y})}$${H^{1}(\Omega^{2}_{Y})}$${0}$${H^{0}((N_{Y/W}\otimes
K_{Y})_{|Z})}$${H^{1}(N_{Z/Y}\otimes K_{Y})}$${H^{1}(N_{Z/W}\otimes
K_{Y}).}$$\scriptstyle{R}$$\scriptstyle{res}$$\scriptstyle{\psi_{Z}^{\vee}}$$\scriptstyle{\beta_{Z}}$

Here, $res$ is the restriction map, and the rows are part of long exact
sequences associated, respectively, with the short exact sequence

$\Omega_{Y}^{2}\otimes
N_{Y/W}^{\vee}\to\Omega_{W}^{3}\otimes\mathcal{O}_{Y}\to\Omega_{Y}^{3},$

([Wel81, 2.6]) and with the normal bundle sequence of $Z\subset Y\subset W$.
We use the isomorphism $H^{1}(N_{Z}Y\otimes K_{Y})\simeq H^{0}(N_{Z}Y)^{\vee}$
and the Kodaira-Spencer isomorphism $H^{0}(N_{Z}Y)^{\vee}\simeq\Omega_{T,0}$.

######  Remark 2.17.

The map $\Psi$ contracts the locus
$\mathcal{Y}\subset\mathcal{M}_{\sigma}(w)$, for every $d$. One can show this
in two different ways: on the one hand, the objects $E_{p}$ all have the same
second Chern class, so $\Psi$ is constant on the set $\mathcal{Y}$
parametrizing them. Alternatively, the Abel–Jacobi map contracts rational
curves since it maps to an abelian variety, and $Y_{d}$ (and hence
$\mathcal{Y}$) is rationally connected for all degrees.

##  3\. A summary of the cases $d=3,4,5$

Before focusing more on the low degree case, we recollect here some results
about the $d=3,4,5$ case. In these cases, the moduli spaces
$\mathcal{M}_{\sigma}(w)$ have been considered in the literature, and there is
often a nice interpretation for them, and for their Abel–Jacobi map. We will
follow the notation given in the Introduction.

###  3.1. Degree 5

The Fano threefold of Picard rank 1 and degree 5 is often denoted $V_{5}$ and
called the quintic del Pezzo threefold (it is unique up to isomorphism
[Isk77]). It arises as the intersection of a Grassmannian
$\operatorname{Gr}(2,5)\subset\mathbb{P}^{9}$ with three hyperplanes of
$\mathbb{P}^{9}$ [Muk92, Prop. 10]. The universal sub-bundle and quotient on
$\operatorname{Gr}(2,5)$ restrict to exceptional vector bundles $S,Q$ on
$V_{5}$, moreover, it is shown in [Orl91] that $V_{5}$ has a full strong
exceptional collection $D^{b}(V_{5})=\langle
S,Q^{\vee},\mathcal{O},\mathcal{O}(1)\rangle$, which in turn gives rise to a
semiorthogonal decomposition

$\mathsf{Ku}(V_{5})\simeq\langle S,Q^{\vee}\rangle.$

The space $\operatorname{Hom}(S,Q^{\vee})$ is three dimensional. Therefore,
the category $\mathsf{Ku}(V_{5})$ is equivalent to the derived category of
representations of the Kronecker quiver with three arrows.

To illustrate Theorem 2.16 in this case, we recall the alternative description
of $V_{5}$ from [Fae05] and the properties of the quiver moduli space from
[Dre88].

First of all, consider a three dimensional vector space $B$ and the projective
spaces $\mathbb{P}^{2}=\mathbb{P}(B)$ and
$\check{\mathbb{P}}^{2}=\mathbb{P}(B^{*})$. Fix a smooth conic
$F\in\operatorname{Sym}^{2}(B)$. Then, considering $F$ as a function
$F\colon\operatorname{Sym}^{2}(B^{*})\to\mathbb{C}$, one sees as in [Fae05,
Lemma 2.3] that the variety $V_{5}$ is isomorphic to the following subset of
$\operatorname{Hilb}_{3}(\check{\mathbb{P}}^{2})$:

(10) $\left\\{Z\in\operatorname{Hilb}_{3}(\check{\mathbb{P}}^{2})\mid
H^{0}(I_{Z}(2))\subset\ker F\right\\},$

and that there is a natural isomorphism
$B\simeq\operatorname{Hom}(S,Q^{\vee})$.

On the other hand, the universal bundles have classes

$[S]=2-H+\frac{H^{2}}{10}+\frac{H^{3}}{30}\qquad\qquad[Q^{\vee}]=3-H-\frac{H^{2}}{10}+\frac{H^{3}}{30}$

in $K_{num}(\mathsf{Ku}(Y))$ (see [Fae05, Sec. 3]), so that
$w=2[Q^{\vee}]-3[S]$. In other words, $\mathcal{M}_{\sigma}(w)$ is identified
with the moduli space of quiver representations of the form $S^{\oplus
3}\rightarrow(Q^{\vee})^{\oplus 2}$.

In turn, such a map is a point of the variety $\mathbb{G}(2\times 3,B^{*})$ of
$2\times 3$ matrices on the space $B^{*}$, i.e.

$\mathcal{M}_{\sigma}(w)\simeq\mathbb{G}(2\times
3,B^{*})=\mathbb{P}(M_{2,3}(B^{*}))//\operatorname{SL}(2)\times\operatorname{SL}(3).$

The variety $\mathbb{G}(2\times 3,B^{*})$ has been studied in detail in
[Dre88]. The morphism

$\displaystyle\operatorname{Hilb}_{3}(\check{\mathbb{P}}^{2})$
$\displaystyle\rightarrow\operatorname{Gr}(3,6)=\operatorname{Gr}(3,\operatorname{Sym}^{2}(B))$
$\displaystyle\phantom{aaa}Z\phantom{aaa}$ $\displaystyle\mapsto
H^{0}(I_{Z}(2))$

factors through

$q\colon\operatorname{Hilb}_{3}(\check{\mathbb{P}}^{2})\to\mathbb{G}(2\times
3,B^{*}),$

defined by sending a subscheme $Z\subset\check{\mathbb{P}}^{2}$ to the map
between the first order syzygies and the ideal generators of $I_{Z}(2)$. The
variety $\mathbb{G}(2\times 3,B^{*})$ is a birational model of
$\operatorname{Hilb}_{3}(\check{\mathbb{P}}^{2})$, it compactifies the locus
of reduced subschemes, and $q$ contracts the locus of collinear schemes to a
plane in $\mathbb{G}(2\times 3,B^{*})$ [Dre88, Theor. 4]. Moreover, $q$ is an
isomorphism on the set (10) [Fae05, Lemma 2.3].

In summary, $V_{5}$ is realized in
$\operatorname{Gr}(3,\operatorname{Sym}^{2}(B))$ as the intersection of the
moduli space $\mathcal{M}_{\sigma}(w)$ with the cell
$\operatorname{Gr}(2,\operatorname{Sym}^{2}(B)/\langle F\rangle)$ of subspaces
containing $F$:

$Y_{5}=\mathcal{M}_{\sigma}(w)\cap\operatorname{Gr}(2,5)\subset\operatorname{Gr}(3,6).$

The intermediate Jacobian of $V_{5}$ is a point, so there is not much
interesting to say about the Abel–Jacobi map [Sha99, Table 12.2].

###  3.2. Degree 4

In this case, $Y$ is the complete intersection of two quadric hypersurfaces in
$\mathbb{P}^{5}$. The corresponding pencil of quadrics degenerates at 6
points, and determines a genus 2 curve $C$ defined as a double cover of
$\mathbb{P}^{1}$, ramified over the six singular quadrics. Every smooth even
dimensional quadric admits two spinor bundles $S_{+}$ and $S_{-}$. At the six
singular points, the quadrics degenerate to a cone over a smooth quadric,
which has only one spinor bundle. Restricting these spinor bundles to $Y$ and
varying the quadric in the pencil exhibits $C$ as a moduli space on $Y$. As it
turns out (see [BO95, Theor. 2.9]), $C$ is a fine moduli space, and the
universal spinor bundle induces a fully faithful Fourier-Mukai functor
$\phi^{\prime}\colon D^{b}(C)\to D^{b}(Y)$ (depending on a choice of a line
$\ell_{0}\subset Y$) giving rise to a decomposition

$D^{b}(Y)=\langle\mathcal{O}_{Y}(-1),\mathcal{O}_{Y},\phi(D^{b}(C))\rangle.$

For consistency with Definition 2, we let $\phi\colon
D^{b}(C)\to\mathsf{Ku}(Y)$ be the equivalence
$\phi(M)\coloneqq\mathcal{O}_{Y}(-1)\otimes\phi^{\prime}(M)$ for all $M\in
D^{b}(C)$.

By construction, the functor $\phi$ sends points of $C$ to twisted spinor
bundles on $X$. The numerical class in $K_{num}(Y)$ of both spinor bundles is

$[S_{\pm}]=2+H-\frac{H^{3}}{12}$

(see [Ott88, Remark 2.9]). This implies $[S(-1)]=2v-w$.

On the other hand, ideal sheaves of lines of $Y$ are images of line bundles on
$C$: suppose in fact that $\phi(M)=I_{\ell}$ for some $M\in\mathsf{Ku}(Y)$ and
some line $\ell\subset Y$. Then we have
$\operatorname{Hom}(M,M)\simeq\operatorname{Hom}(I_{\ell},I_{\ell})=\mathbb{C}$,
and by Riemann-Roch

$-1=\chi_{Y}(v,v)=\chi_{C}(M,M)=-\operatorname{rk}(M)^{2},$

therefore $M$ is a line bundle. Indeed, it is a known fact that $\phi$ induces
an isomorphism $J(C)\simeq F(Y)$ between the Jacobian of $C$ and the Fano
surface of $Y$ [Rei72, Theor. 4.8].

These remark imply that the class $w$ corresponds to the class of a rank 2
vector bundle on $C$ of fixed odd degree (we can assume that degree is 1 by
tensoring the universal family with a line bundle). Since there is a unique
notion of stability on a curve, $\sigma$ must pull-back to slope-stability,
and the functor $\phi$ induces an isomorphism between
$\mathcal{M}_{\sigma}(w)$ and $M_{C}(2,1)$, the moduli space of slope-stable
rank 2 vector bundles of degree 1 on $C$. This shows that $M_{\sigma}(w)$ is
an irreducible smooth projective variety of dimension five. As a consequence
of [Rei72, Theorem 4.14(c’)], the intermediate Jacobian $J(Y)$ is isomorphic
to the Jacobian of $C$, and under this isomorphism the Abel–jacobi map
coincides with the determinant:

${M_{C}(2,1)}$${M_{\sigma}(w)}$${\operatorname{Pic\,}^{1}(C)}$${J(Y)}$$\scriptstyle{\det}$$\scriptstyle{\sim}$$\scriptstyle{\Psi}$$\scriptstyle{\sim}$

Moreover, fibers of $\det$ (and hence of $\Psi$) are isomorphic to $Y_{4}$
[New68].

###  3.3. Degree 3

The case of a cubic threefold $Y_{3}$ is treated in the recent [BBF+20]. Here,
the authors show that $\mathcal{M}_{\sigma}(w)$ is isomorphic to the moduli
space $M_{G}(\kappa)$ of Gieseker-stable sheaves with
$\kappa=\left(3,-H,-\frac{1}{2}H^{2},-\frac{1}{6}H^{3}\right)$. The space
$M_{G}(\kappa)$ is smooth and irreducible of dimension 4. $M_{G}(\kappa)$ maps
birationally to the intermediate Jacobian of $Y_{3}$ and its image is a theta
divisor $\Theta$. More precisely, $M_{G}(\kappa)\to\Theta$ is the blow-up of
$\Theta$ in its unique singular point, and the exceptional divisor is the
variety $Y_{3}\subset\mathcal{M}_{\sigma}(w)\simeq M_{G}(\kappa)$ [BBF+20,
Theorem 7.1].

######  Remark 3.1 (Projectivity of $\mathcal{M}_{\sigma}(w)$ for $d\geq 3$).

We point out that $\mathcal{M}_{\sigma}(w)$ is projective for degrees $d\geq
3$. this follows from the explicit descriptions given in these sections: In
the case $d=5$, $\mathcal{M}_{\sigma}(w)$ is projective since it is a moduli
space of quiver representation, and for $d=3,4$ it is projective since it
coincides with a space of Gieseker-stable sheaves.

##  4\. Low degrees

In this section we detail the study of the moduli spaces
$\mathcal{M}_{\sigma}(w)$ and the corresponding Abel–Jacobi maps when $d\leq
2$. We start by focusing on the degree 2 case.

###  4.1. Quartic double solids ($d=2$): generalities

We follow the notation of Section 1.1.5. The manifold $Y=Y_{2}$ (from here on,
we drop the subindex everywhere) is a general double cover $\pi\colon
Y\to\mathbb{P}^{3}$ of the projective space ramified over a smooth quartic K3
surface $R$ (we will identify the branching and ramification locus, and denote
both by $R$) which does not contain lines.

A hyperplane section $\iota\colon S\hookrightarrow Y$ is given by the pullback
$\pi^{*}(P)$ of a plane in $\mathbb{P}^{3}$. It is a double cover of
$\mathbb{P}^{2}$ branched over $R\cap P$, and it is singular if and only if
$P$ is tangent to $R$. The singularities that may appear are of rational type
(by Lemma 1.16) (the elliptic type can only appear if $R\cap P$ is four lines
meeting at a point [HW81, Prop. 4.6]).

Finally, $Y$ is equipped with an involution $\tau$ that swaps the two sheets
of the cover, acting on objects in the derived category via pull-back; since
$\mathcal{O}_{Y}(1)$ is pulled-back from $\mathbb{P}^{3}$, the involution
preserves hyperplane sections, and therefore descends to an involution on each
surface $S$. Following [Kuz15], we can moreover compute the Serre functor for
$\mathsf{Ku}(Y)$ to be

$S_{\mathsf{Ku}(Y)}(E)=\tau E[2].$

Next, we focus on properties of the heart
$\mathcal{A}=\mathsf{Ku}(Y)\cap\operatorname{Coh}^{0}_{\alpha,-\frac{1}{2}}(Y)$
of $\mathsf{Ku}(Y)$ associated to the stability condition $\sigma$. We say
that the heart of a bounded t-structure $\mathcal{B}\subset\mathsf{Ku}(Y)$ is
$\tau$-invariant if $\tau(\mathcal{B})=\mathcal{B}$.

######  Lemma 4.1.

The heart $\mathcal{A}$ is $\tau$-invariant.

###### Proof.

By definition,
$\mathcal{A}=\mathsf{Ku}(Y)\cap\operatorname{Coh}^{0}_{\alpha,-\frac{1}{2}}(Y)$.
Clearly $\tau(\mathsf{Ku}(Y))\subseteq\mathsf{Ku}(Y)$ since up to shift $\tau$
is the Serre functor of $\mathsf{Ku}(Y)$. Then, it suffices to show that
$\tau$ preserves $\operatorname{Coh}^{0}_{\alpha,-\frac{1}{2}}(Y)$ to
conclude.

Observe that $\tau$ acts trivially on numerical classes of objects in $D(Y)$,
since it preserves the class $H$. Therefore, $\tau$ preserves slopes.
Moreover, it preserves exact sequences of sheaves and hence slope stability.
This implies that $\tau$ preserves $\operatorname{Coh}^{\beta}(Y)$ for all
$\beta\in\mathbb{R}$. Similarly one sees that $\tau$ preserves
$\sigma_{\alpha,\beta}$-semistable objects for all
$(\alpha,\beta)\in\mathbb{R}_{>0}\times\mathbb{R}$, and hence it preserves the
double tilt. ∎

######  Lemma 4.2.

The homological dimension of a $\tau$-invariant heart
$\mathcal{B}\subset\mathsf{Ku}(Y)$ is equal to 2, i.e.
$\operatorname{Ext}^{i}(E,F)=0$ for $i\neq 0,1,2$ for $E,F\in\mathcal{B}$.

###### Proof.

Since $\mathcal{B}$ is the heart of a bounded t-structure, we have by
definition that $\operatorname{Ext}^{i}(E,F)=0$ for any $E,F\in\mathcal{B}$,
$i<0$. Now, using the above Serre functor we get:

$\operatorname{Ext}^{i}(E,F)=\operatorname{Hom}(E,F[i])=\operatorname{Hom}(F[i],S_{\mathsf{Ku}(Y)}(E))=\operatorname{Ext}^{2-i}(F,\tau
E),$

which concludes the proof, since $\mathcal{B}$ is $\tau$-invariant and hence
$\tau E\in\mathcal{B}$. ∎

###  4.2. The moduli space $\mathcal{M}_{\sigma}(w)$

In this section we refine the description of $\mathcal{M}_{\sigma}(w)$ for
general quartic double solids. The locus $\mathcal{Y}$ turns out to be an
irreducible component, and the following Lemma is instrumental in describing
the intersection of $\mathcal{Y}$ with the rest of $\mathcal{M}_{\sigma}(w)$.
Recall that for a complex $E\in D^{b}(Y)$, the set of first-order
infinitesimal deformations of $E$ is a torsor under the space
$\operatorname{Ext}^{1}(E,E)$ [Lie06, §3].

######  Lemma 4.3.

Let $d\leq 2$ and $Y$ of degree $d$. Consider $p\in Y$ and $E_{p}$ be one of
the objects as in (4); then there is a natural identification

(11)
$\operatorname{Ext}^{1}(E_{p},E_{p})=\operatorname{Ext}^{1}(I_{p},I_{p})\oplus
V_{p}\simeq\mathbb{C}^{3}\oplus V_{p},$

where $V_{p}$ is non-zero only if there exists a hyperplane section $S$
singular at $p$, in which case it is one dimensional, generated by the element
$f_{p}\in\operatorname{Ext}^{1}(\mathcal{O}(-1)[1],I_{p})\simeq
H^{0}(I_{p}(1))$ such that $S=\\{f_{p}=0\\}$.

###### Proof.

To describe the splitting, first observe that
$\operatorname{Ext}^{1}(E_{p},E_{p})\simeq\operatorname{Ext}^{1}(E_{p},I_{p})$
(e.g. by applying $\operatorname{Hom}(E_{p},-)$ to the sequence (4)). Applying
$\operatorname{Hom}(-,I_{p})$ to (4) one obtains

$\begin{split}&\operatorname{Ext}^{3}(E_{p},E_{p})=0;\\\
0\to\operatorname{Ext}^{1}(I_{p},I_{p})\to\operatorname{Ext}^{1}(E_{p},I_{p})&\to\ker\left(\operatorname{Ext}^{1}(\mathcal{O}(-1)[1],I_{p})\xrightarrow{\alpha}\operatorname{Ext}^{2}(I_{p},I_{p})\right)\to
0,\end{split}$

where $\alpha$ is pre-composition with the unique map
$\epsilon_{p}:I_{p}\to\mathcal{O}(-1)[2]$ of the triangle (4). Let
$V_{p}\coloneqq\ker\alpha$. Since
$\operatorname{Ext}^{1}(\mathcal{O}(-1)[1],I_{p})\simeq H^{0}(I_{p}(1))$,
every element $f\in\operatorname{Ext}^{1}(\mathcal{O}(-1)[1],I_{p})$ can be
identified with a hyperplane section $S_{f}$ containing $p$.

We now prove that a nonzero
$f\in\operatorname{Ext}^{1}(\mathcal{O}(-1)[1],I_{p})$ is in $\ker\alpha$ if
and only if $S_{f}$ is singular at $p$. We have the following diagram:

${I_{p}}$${I_{p/S}[1]}$${\mathcal{O}(-1)[2]}$${I_{p}[2]}$$\scriptstyle{\epsilon_{p}}$$\scriptstyle{f}$

Then $\alpha(f)=f\epsilon_{p}=0$ if and only if $\epsilon_{p}$ lifts to a map
$I_{p}\rightarrow I_{p/S}[1]$. Next, we check that $\epsilon_{p}$ is in the
image of
$\operatorname{Hom}(I_{p},I_{p/S}[1])\rightarrow\operatorname{Hom}(I_{p},\mathcal{O}(-1)[2])\simeq\mathbb{C}$
if and only if $p\in S$ is a singular point (this also shows that $V_{p}$ is
at most one-dimensional). Indeed, applying $\operatorname{Hom}(I_{p},-)$ to
the triangle

$\mathcal{O}(-1)\to I_{p}\to I_{p|S}$

we get an exact sequence

$0\to\mathbb{C}^{3}\to\operatorname{Ext}^{1}(I_{p},I_{p|S})\to\operatorname{Hom}(I_{p},\mathcal{O}(-1)[2]).$

This sequence shows that it suffices to argue that
$\operatorname{Ext}^{1}(I_{p},I_{p|S})$ is $3$-dimensional if $p\in S$ i
smooth, and $4$-dimensional if $p\in S$ is singular. To see this, apply the
$\operatorname{Hom}(-,I_{p|S})$ to the sequence
$I_{p}\to\mathcal{O}_{Y}\to\mathbb{C}_{p}$.

∎

Recall from Proposition 2.1 that $\mathcal{M}_{G}(w)$ has two irreducible
components; one, $\mathcal{P}$, is a rank 2 projective bundle over $Y$
parametrizing $I_{p|S}$. The complement of $\mathcal{P}$ is smooth and
parametrizes sheaves $\iota_{*}\mathcal{O}_{S}(D)$ associated to roots on
hyperplane sections. Points in the intersection are $I_{p|S}$ with $p$
singular in $S$. As illustrated in Section 4.1, $p$ is the singular point of
some hyperplane section if and only if $p\in R$, the ramification locus of the
double cover $\pi\colon Y\to\mathbb{P}^{3}$.

We restate Theorem 1.2 in the case of a general quartic double solid:

######  Theorem 4.4.

Suppose $Y$ is a general quartic double solid. The space
$\mathcal{M}_{\sigma}(w)$ on $\mathsf{Ku}(Y)$ has two irreducible components
$\mathcal{Y}$ and $\mathcal{C}$, both of dimension 3. The component
$\mathcal{Y}$ is isomorphic to $Y$, and intersects $\mathcal{C}$ exactly at
the ramification locus $R$. Both components are smooth outside $R$.

###### Proof of Theorem 4.4.

The statements about the component $\mathcal{C}$ are already in Theorem 2.16.
The addition is in Lemma 4.3, which implies that the inclusion $Y\setminus
R\hookrightarrow\mathcal{M}_{\sigma}(w)$ is an open immersion into a component
of $\mathcal{M}_{\sigma}(w)$ (since the tangent space at a point $p\in
Y\setminus R$ is naturally isomorphic to the tangent space of its image).

As for the intersection, recall that the components of $\mathcal{M}_{G}(w)$
intersect at the points $[I_{p|S}]$, with $p\in S\cap R$ singular in $S$. The
wall-crossing morphism
$p\colon\mathcal{M}_{G}(w)\to\mathcal{M}_{-}(w)\simeq\mathcal{M}_{\sigma}(w)$
corresponds to the assignment $[I_{p|S}]\to p$, and hence it maps the
intersection of the components onto the image of $R$ via $R\subset
Y\simeq\mathcal{Y}$ (see Proposition 2.9, Proposition 2.10).

∎

###  4.3. The Abel-Jacobi map

The following construction expresses the relation between $\mathcal{C}$ and a
family of conics on $Y$. First, we recollect some facts about the geometry of
hyperplane sections of $Y$, following [Dol12, Chapter 8].

####  4.3.1. Del Pezzo surfaces of degree 2: roots and lines

Smooth hyperplane sections of $Y_{2}$ are del Pezzo surfaces of degree 2.
Every such surface $S$ can be constructed from a blowup of $\mathbb{P}^{2}$ in
7 points in general position, its Picard group is free of rank 8, generated by
the hyperplane class $\mathbf{e}_{0}$ of $\mathbb{P}^{2}$ and the exceptional
curves $\mathbf{e}_{i}$, $i=1,...,7$. The line bundle $\mathcal{O}_{Y}(H)$
restricts on $S$ to

$H_{|S}=-K_{S}=3\mathbf{e}_{0}-\mathbf{e_{1}}-...-\mathbf{e_{7}}.$

The map $\phi_{\lvert-K_{S}\rvert}$ realizes $S$ as a double cover of
$\mathbb{P}^{2}$ ramified over a quartic curve, and the orthogonal lattice
$(K_{S})^{\perp}$, equipped with the intersection product, is a root lattice
$E_{7}$. The following lemma is well-known.

######  Lemma 4.5.

Let $S$ be a hyperplane section of $Y$. Suppose $S$ is a del Pezzo surface.
Define vectors in $(K_{S})^{\perp}$:

$\displaystyle\alpha_{i}\coloneqq
2\mathbf{e}_{0}-\mathbf{e}_{1}-...-\mathbf{e}_{7}+\mathbf{e}_{i}$
$\displaystyle\quad 1\leq i\leq 7$
$\displaystyle\alpha_{ij}\coloneqq\mathbf{e}_{i}-\mathbf{e}_{j},$
$\displaystyle\quad 1\leq i<j\leq 7$
$\displaystyle\alpha_{ijk}\coloneqq\mathbf{e}_{0}-\mathbf{e}_{i}-\mathbf{e}_{j}-\mathbf{e}_{k},$
$\displaystyle\quad 1\leq i<j<k\leq 7$

  * •

$S$ admits 126 roots of the form $\alpha_{i},\alpha_{ij},\alpha_{ijk}$.

  * •

$S$ contains 56 lines. These come in pairs $(L,-K_{S}-L)$ (i.e., there is a
unique divisor in $\lvert-K_{S}-L\rvert$ and it is a line).

  * •

every root can be expressed (non-uniquely) as the difference of two disjoint
lines. Conversely, the difference of two disjoint lines is a root in
$\operatorname{Pic\,}(S)$.

If $S$ has rational singularities, then its minimal resolution $\tilde{S}\to
S$ is obtained by the blowup of $\mathbb{P}^{2}$ at 7 points in almost general
position, and the morphism contracts all effective roots on $\tilde{S}$.
Moreover, every singular point of $S$ lies on a line.

###### Proof.

The roots of $S$ are computed in [Dol12, Prop. 8.2.7]. Consider the double
cover $\phi_{\lvert-K_{S}\rvert}\colon S\to\mathbb{P}^{2}$, let $C$ denote the
ramification locus (a smooth plane quartic). The curve $C$ has 28 bitangent
lines ([Dol12, §6.1.1]), the preimage of which has two irreducible components,
each of which is a line in $S$. Two lines obtained this way add up to
$\phi^{*}_{\lvert-K_{S}\rvert}\mathcal{O}_{\mathbb{P}^{2}}(1)=-K_{S}$ in
$\operatorname{Pic\,}(S)$.

One can easily list the classes of lines in $\operatorname{Pic\,}(S)$ by
solving the system of equations $L.K_{S}=L^{2}=-1$, and check directly that
every root can be expressed as a difference of disjoint lines. Conversely, it
is straightforward to see that the difference of disjoint lines is a root.

The statement about the minimal resolutions is [HW81, Theorem 3.4]. It is
straightforward to check that one can always find a line through a singular
point of $S$: equivalently, every effective root of $\tilde{S}$ intersects a
line. This can be checked on a case by case basis using the explicit
description of effective roots in [Dol12, §8.2.7], together with the above
description of lines. ∎

######  Lemma 4.6.

Let $S$ be a smooth del Pezzo surface of degree 2, and let $D$ be a root of
$S$. Then, $H^{1}(D-K_{S})=H^{2}(D-K_{S})=0$.

###### Proof.

We show the claim by applying the Kodaira vanishing theorem to the divisor
$D-2K_{S}$, after showing that it is ample on $S$. There are 126 roots on $S$
[Dol12, Prop. 8.2.7], and the nef cone of $S$ is described in [BC13, Theorem
2.4]. It is not hard to check that if $D$ is any of the 126 roots then
$D-2K_{S}$ belongs to the interior of the nef cone, and is therefore an ample
divisor on $S$. ∎

####  4.3.2. Conics on $Y$ and the Abel–Jacobi map

A conic on $Y$ is, by definition, a curve $Z\subset Y$ whose image $\pi_{|Z}$
is an isomorphism onto a conic of $\mathbb{P}^{3}$. Consider the Hilbert
scheme of conics on $Y$, and let $\mathcal{C}^{\prime}$ be the component of
the Hilbert scheme containing smooth conics (note that since $H$ is not very
ample, there may be unusual subschemes of $Y$ with Hilbert polynomial $1+2t$).

Let $Z\in\mathcal{C}^{\prime}$ be a conic; its image $\pi(Z)$ spans a
hyperplane of $\mathbb{P}^{3}$ and hence it determines a hyperplane section
$S\subset Y$. Suppose $S$ is smooth. Then, $Z-H$ is a root of $S$ since
$(Z-H).H=0$ and $(Z-H)^{2}=-2$, so the push forward of the line bundle
$\mathcal{O}_{S}(Z-H)$ to $Y$ is in the Gieseker moduli space
$\mathcal{M}_{G}(w)$, as in Prop. 2.1. Conversely, given a root $D$ on $S$,
the divisor class $D+H$ has the Hilbert polynomial of a conic. By Riemann-Roch
and Lemma 4.6 we see
$h^{0}(\mathcal{O}_{S}(D+H))=\chi(\mathcal{O}_{S}(D+H))=2$, and hence $|D+H|$
is in a pencil whose general element is a smooth conic.

Therefore, the assignment $Z\mapsto\iota_{*}(\mathcal{O}_{S}(Z-H))$ then
defines a rational map

$f\colon\mathcal{C}^{\prime}\dashrightarrow\mathcal{C}$

whose fibers are pencils of linearly equivalent conics (since rational
equivalence implies homological equivalence), and which is dominant, since the
open locus of $\iota_{*}\mathcal{O}_{S}(D)$ where $D$ is a root of a smooth
$S$ is in the image of $f$. Consider the subscheme of $\mathcal{C}^{\prime}$
given by $\mathcal{C}^{\prime}_{0}=\\{Z\in\mathcal{C}^{\prime}\,|\,Z\subset
S\text{ with $S$ smooth}\\}$, so that $f$ is defined on
$\mathcal{C}_{0}^{\prime}$

######  Proposition 4.7.

Let $Z$ be a smooth conic in $\mathcal{C}^{\prime}_{0}$, and consider the
Abel–Jacobi map $\mathcal{C}^{\prime}_{0}\to J(Y)$. Its codifferential at $Z$
has rank 3, and the Abel–Jacobi map factors through
$f\colon\mathcal{C}^{\prime}_{0}\to\mathcal{C}$.

###### Proof.

The restriction of the Abel–Jacobi map to $\mathcal{C}^{\prime}_{0}$ is
constant on pencils of curves, so it factors through $f$. We use diagram (9)
to study the codifferential (take $W$ to be weighted projective space
$\mathbb{P}(1,1,1,1,2)$, which implies $N_{Y/W}\simeq\mathcal{O}_{Y}(4)$).
Recall that the lower row of (9) is part of the long exact sequence associated
to

$0\to N_{Z/Y}\otimes K_{Y}\to N_{Z/W}\otimes K_{Y}\to(N_{Y/W}\otimes
K_{Y})_{|Z}\to 0.$

The normal bundle $N_{Z/Y}$ is computed in [TM03, Lemma 1.2] to be
$\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus 2}$ or
$\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(2)$; in any
case, we have $h^{0}(N_{Z/Y}\otimes K_{Y})=0$ and $h^{1}(N_{Z/Y}\otimes
K_{Y})=4$.

The bundle $N_{Z}W$ can be computed as follows: let
$\overline{Z}\subset\mathbb{P}^{3}$ be the smooth conic image of $Z$ under
$\pi$. The standard exact sequence
$\mathcal{O}_{W}(2)\to\mathcal{T}_{W}\to\pi^{*}\mathcal{T}_{\mathbb{P}^{3}}$
yields the diagram

${\mathcal{O}_{Z}(2)}$${\mathcal{O}_{Z}(2)}$${\mathcal{T}_{Z}}$${\mathcal{T}_{W|Z}}$${N_{Z/W}}$${\mathcal{T}_{\overline{Z}}}$${\pi^{*}\mathcal{T}_{\mathbb{P}^{3}|\overline{Z}}}$${\pi^{*}N_{\overline{Z}/\mathbb{P}^{3}}}$$\scriptstyle{\cong}$

where all rows and columns are exact. Since $\overline{Z}$ is a complete
intersection we have
$N_{\overline{Z}/\mathbb{P}^{3}}\simeq\mathcal{O}_{\overline{Z}}(2)\oplus\mathcal{O}_{\overline{Z}}(1)$,
hence from the third column $N_{Z/W}\otimes K_{Y}\simeq\mathcal{O}_{Z}^{\oplus
2}\oplus\mathcal{O}_{Z}(-1)\simeq\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
2}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-2)$.

Finally, we have $(N_{Y/W}\otimes
K_{Y})_{|Z}\simeq\mathcal{O}_{\mathbb{P}^{1}}(4)$. Therefore the connecting
homomorphism $\beta_{Z}\colon H^{0}((N_{Y/W}\otimes K_{Y})_{|Z})\to
H^{1}(N_{Z/Y}\otimes K_{Y})$ has rank 3.

To conclude that $\psi_{Z}$ has rank 3, it is enough to show that the
restriction map $H^{0}(K_{Y}(4))\to H^{0}(K_{Y}(4)_{|Z})$ is surjective. Since
$Z$ lies on a hyperplane section $S$ of $Y$, this can be checked in two steps:
$H^{0}(K_{Y}(4))\to H^{0}(K_{Y}(4)_{|S})$ is surjective because
$H^{1}(\mathcal{O}_{Y}(1))=0$, and the surjectivity of
$H^{0}(K_{Y}(4)_{|S})\to H^{0}(K_{Y}(4)_{|Z})$ follows from the sequence

$\mathcal{O}_{S}(2H-Z)\to\mathcal{O}_{S}(2H)\to\mathcal{O}_{Z}(2),$

since $H-Z$ is a root of $S$, and applying Lemma 4.6 to $2H-Z=(H-Z)+H$. ∎

We summarize the above Proposition 4.7 and Remark 2.17 in the following
Corollary.

######  Corollary 4.8.

The Abel–Jacobi map $\Psi\colon\mathcal{M}_{\sigma}(w)\to J(Y)$ contracts the
irreducible component $\mathcal{Y}$ to a point $y\in J(Y)$ and is a generic
embedding on $\mathcal{C}$.

We conclude this section with a description of the Abel-Jacobi differential on
the intersection $R$ of the two components.

######  Proposition 4.9.

The differential $\psi_{p}\colon
T_{p}\mathcal{M}\simeq\operatorname{Ext}^{1}(E_{p},E_{p})\to T_{y}J(Y)$ of
$\Psi$ at a point $p\in R$ is non-zero.

###### Proof.

Let $S_{0}$ be the hyperplane section of $Y$ that is singular at $p$; our
generality assumption on $R$ implies that $S_{0}$ has rational singularities
and its minimal resolution $\tilde{S_{0}}$ is obtained by the blowup of
$\mathbb{P}^{2}$ at 7 points in almost general position [HW81]. We are in the
situation of Example 2.5, and we use the notation thereof. By Lemma 4.5, there
is a line $L\subset S_{0}$ passing through $p$. The linear series $\lvert
I_{L}(1)\rvert$ is a one-dimensional family of hyperplane sections of $Y$
containing $L$, let $\Delta$ be an analytic neighborhood of $S_{0}$ in this
pencil, and $\mathcal{S}\to\Delta$ the family of surfaces it parametrizes.

Like in Example 2.5, we may pass to a cover of $\Delta$ and assume that the
family $\mathcal{S}$ is obtained by blowing up 7 sections of the projection
$\mathbb{P}^{2}\times\Delta\to\Delta$. Again, we assume that the degeneration
is caused by three sections $s_{1},s_{2},s_{3}$ becoming collinear and that
$S_{0}$ is obtained contracting the line $E$ passing through the three
sections. We may also assume that $L=\mathcal{E}_{1}\cap S_{0}$. This helps to
make the notation explicit but the same argument can be run in the other
cases. For some $0\neq t\in\Delta$, choose a line $L_{t}\subset S_{t}$
disjoint from $E$ and $L$, for example $L_{t}=\mathcal{E}_{4}\cap S_{t}$.
Then, the sheaf $\mathcal{O}_{\mathcal{S}}(\mathcal{E}_{1}-\mathcal{E}_{4})$
restricts to $\mathcal{O}_{S_{t}}(L-L_{t})$ above $t$. The locus
$\\{L_{t}\coloneqq\mathcal{E}_{4}\cap S_{t}\\}_{t}\subset F(Y)$ in the Hilbert
scheme of lines has limit a line $L_{0}$ in $S_{0}$. This locus is contained
in the incidence divisor $D_{L^{\prime}}\subset F(Y)$, of lines intersecting
$L^{\prime}=H-L$ (see Lemma 4.5). The differential $\psi_{p}$ is equal to the
differential of the Abel Jacobi map from $F(Y)$, restricted to
$D_{L^{\prime}}$. The assumption on $R$ and [Wel81, Prop. 2.13] imply that the
Abel Jacobi map is a local embedding $F(Y)\to J(Y)$, so $\psi_{p}$ is nonzero.
∎

From this, we obtain:

######  Corollary 4.10.

There is an isomorphism $\psi\colon R\xrightarrow{\sim}T$, where $T$ is the
intersection of the strict transform of $\Psi(\mathcal{C})$ with the
exceptional divisor
$\mathbb{P}^{9}\simeq\mathbb{P}(H^{0}(\mathcal{O}_{\mathbb{P}^{3}}(2)))$ of
$\operatorname{Bl}_{y}J(Y)$.

###### Proof.

Combining Lemma 4.3 and Proposition 4.9, one sees that for a point $p\in R$
the vector space $\langle f_{p}\rangle$ coincides with the tangent space of
$\Delta^{\prime}$ in $T_{L_{0}}F\simeq H^{0}(N_{L_{0}}Y)$ ($\Delta^{\prime}$
is the image of $\Delta$ in $F(Y)$). Then, writing diagram (9) for
$\psi_{\Delta^{\prime},L_{0}}^{\vee}$, we have

${H^{0}(\mathcal{O}_{Y}(2))}$${H^{0}(\mathcal{O}_{\mathbb{P}^{3}}(2))}$${H^{0}(\mathcal{O}_{L_{0}}(2))}$${H^{0}(N_{L_{0}}Y)^{\vee}}$${\langle\psi_{p}(f_{p})\rangle^{\vee}}$$\scriptstyle{res}$$\scriptstyle{\psi_{\Delta^{\prime},L_{0}}^{\vee}}$$\scriptstyle{\psi_{p}^{\vee}}$

which shows that $\psi$ induces a rational map on $R$ via quadrics to
$\mathbb{P}^{9}\simeq\mathbb{P}\left(H^{0}(\mathcal{O}_{\mathbb{P}^{3}}(2))\right)$,
whose image is dense in $T$ and is generically one-to-one. ∎

###  4.4. Double Veronese cones

In this section, we describe $\mathcal{M}_{\sigma}(w)$ and its Abel–Jacobi map
for a Fano threefold $Y$ of degree 1. In this case, $Y$ is cut out by a sextic
equation in the weighted projective space
$\mathbb{P}\coloneqq\mathbb{P}(1,1,1,2,3)$ (we use $x_{1},x_{2},x_{3}$ for
coordinates of degree 1, and $u,v$ for those of degree 2,3 respectively). We
recall the description of [Isk77, §6.10]: completing a cube and a square, we
may assume that the equation for $Y$ has the form

$v^{2}=u^{3}-f(x_{1},x_{2},x_{3}),$

where $f$ is a degree 6 polynomial. The linear series
$H\coloneqq\mathcal{O}_{\mathbb{P}}(1)_{|Y}$ has three sections and a unique
base point $y_{0}\coloneqq\\{x_{1}=x_{2}=x_{3}=0\\}$, hence it induces a
rational map $\phi_{H}\colon Y\dashrightarrow\mathbb{P}^{2}$. On the other
hand, $2H\sim-K_{Y}$ is base point free, and induces a morphism
$\phi_{2H}\colon(Y)\to\mathbb{P}^{6}$ whose image $K\simeq\mathbb{P}(1,1,1,2)$
is the cone over a Veronese surface with vertex $k\coloneqq\phi(y_{0})$. The
morphism $\phi_{2H}$ is smooth of degree 2 outside the vertex $k$ and the
divisor $D\coloneqq\\{u^{3}-f=0\\}\in\lvert\mathcal{O}_{K}(3)\rvert$ (for this
reason, $Y$ is often referred to as to a double Veronese cone). There is a
commutative diagram

(12)
${{Y}}$${{K}}$${{\mathbb{P}^{2}}}$$\scriptstyle{\phi_{H}}$$\scriptstyle{\phi_{2H}}$$\scriptstyle{\pi}$

where $\pi$ is the projection from $k$. Let $\sigma_{K}$ be the blowup
$\sigma_{K}\colon\widetilde{K}\to K$ of the vertex $k$ with exceptional
divisor $E$. Then, the pull-back $\widetilde{Y}\coloneqq
Y\times_{K}\widetilde{K}$ resolves the indeterminacy of diagram (12):

${\widetilde{Y}}$${\widetilde{K}}$${{Y}}$${{K}}$${{\mathbb{P}^{2}}}$$\scriptstyle{\widetilde{\tau}}$$\scriptstyle{\phi_{2H}}$$\scriptstyle{\sigma_{Y}}$$\scriptstyle{\sigma_{K}}$$\scriptstyle{\widetilde{\pi}}$$\scriptstyle{\pi}$

where $\widetilde{\tau}\colon\widetilde{Y}\to\widetilde{K}$ is a degree 2
cover ramified over the divisor $E\cup\sigma_{K}^{-1}(D)$.

The map $\pi$ restricted to $D$ is a 3-to-1 cover of $\mathbb{P}^{2}$, and it
ramifies at a curve $C$. Throughout this section, we assume that $Y$ is
smooth, equivalently that $C$ is irreducible and general in moduli (this is
the generality assumption used in [Tih82], whose results we will use).

We now give a brief description of $F(Y)$, the Fano surface of lines in $Y$: a
_line_ in $Y$ is defined as a closed one-dimensional subscheme $L\subset Y$
with Hilbert polynomial $p_{H}(t)=t+1$. Under our generality assumption, the
scheme $F(Y)$ parametrizing lines in $Y$ is given by a smooth projective
surface with an embedded copy of $C$ ([Tih82, Thm 4]).

The curve $C\subset F(Y)$ parametrizes _singular_ lines ([Tih82, §3, §8]):
indeed, a line according to the above definition can either be a smooth
rational curve $L$ with $L\cdot H=1$, or a rational curve with a single node
at a point $p\in C$, with non-reduced scheme structure at the point.

We reformulate Theorem 1.2 in the case of a general double Veronese cone. The
arguments are similar to the proof of Theorem 4.4.

######  Theorem 4.11.

Suppose $Y=Y_{1}$ is a general double Veronese cone. The space
$\mathcal{M}_{\sigma}(w)$ has two irreducible components $\mathcal{Y}$ and
$\mathcal{F}$, isomorphic respectively to $Y$ and $F(Y)$. The two intersect
exactly at the curve $C$, and they are smooth outside of $C$.

The Abel–Jacobi map $\Psi\colon\mathcal{M}_{\sigma}(w)\to J(Y)$ contracts
$\mathcal{Y}$ to a singular point $y$ and it is an embedding elsewhere.
Moreover, the image $\Psi(\mathcal{M}_{\sigma}(w))$ determines $Y$ uniquely.

###### Proof.

The statements about the component
$\mathcal{F}\coloneqq\overline{\mathcal{M}_{\sigma}(w)\setminus\mathcal{Y}}$
are already in Theorem 2.16. We can argue as in the proof of Theorem 4.4 (with
the variation that now $p$ is a singular point of some hyperplane section if
and only if $p\in C$) and conclude that $\mathcal{Y}$ and $\mathcal{F}$ are
irreducible components of the desired dimensions, intersecting at $C$ and
smooth outside the intersection.

We’re then only left with proving that $\mathcal{F}$ is isomorphic to $F(Y)$.
To this end, we define a morphism $F(Y)\to\mathcal{F}$ by constructing a
family of objects of $\mathcal{F}$ with base $F(Y)$. Let $\mathcal{H}\subset
Y\times\mathbb{P}(H^{0}(\mathcal{O}_{Y}(1)))$ be the universal hyperplane
section. Since any line in $Y$ is contained in a unique hyperplane section of
$Y$, there is a finite map $F(Y)\to\mathbb{P}(H^{0}(\mathcal{O}_{Y}(1)))$, so
we can define $\mathcal{S}\subset Y\times F(Y)$ as the pullback of
$\mathcal{H}$.

Let $\mathcal{L}\subset\mathcal{S}\subset Y\times F(Y)$ be the universal line,
and consider the sheaf
$\mathcal{K}\coloneqq\mathcal{O}_{\mathcal{S}}(p_{Y}^{\ast}H-\mathcal{L})$ on
$Y\times F(Y)$. Its fiber is $\mathcal{O}_{S}(H-L)$ over a smooth line $L\in
F(Y)$. Observe that $H-L$ is a root of $S$, therefore the restriction of
$\mathcal{K}$ to $Y\times(F(Y)\setminus C)$ defines a rational map
$F(Y)\dashrightarrow\mathcal{M}_{G}(w)$. By the properness of $F(Y)$ and
$\mathcal{M}_{G}(w)$, the above map extends to a morphism
$F(Y)\to\mathcal{M}_{G}(w)$, which is an isomorphism of $F(Y)\setminus C$ onto
the smooth locus of the irreducible component of $\mathcal{M}_{G}(w)$
parametrizing roots on hyperplane sections. Let $\mathcal{K}^{\prime}$ be the
pull-back of the universal family to $Y\times F(Y)$ which is isomorphic to
$\mathcal{K}$ on $Y\times(F(Y)\setminus C)$. The fiber of
$\mathcal{K}^{\prime}$ at a singular line in $C\subset F(Y)$ is the flat limit
of a family of push forwards of roots, and therefore is of the form $I_{p|S}$
for some $p\in S$ and some hyperplane section $S$.

Then, the object
$\mathcal{K}^{\prime\prime}\coloneqq\mathbb{R}_{\mathcal{O}_{Y}(-1)}^{F(Y)}(\mathcal{K})\in\\!^{\perp}\langle\mathcal{O}_{Y}(-1)\rangle_{F(Y)}$
(notation as in Sec. 1.1.4) is a family of objects of $\mathcal{F}$
parametrized by $F(Y)$, and it induces the desired morphism $\mu\colon
F(Y)\to\mathcal{F}$.

Moreover, $\mathcal{K}^{\prime\prime}$ is isomorphic to $\mathcal{K}$ off the
curve $C$, and its fiber over a singular line $[L]\in C$ fits in a triangle

$\mathcal{\mathcal{K}}^{\prime\prime}_{[L]}\to
I_{p|S}\xrightarrow{coev}\mathcal{O}_{Y}(-1)[1]\oplus\mathcal{O}_{Y}(-1)[2]$

whence $\mathcal{K}_{[L]}\simeq E_{p}$ (as in the proof of Prop. 2.9).
Therefore, $\mathcal{K}^{\prime\prime}$ is a universal family for
$\mathcal{F}$ supported on $F(Y)$. Then, the morphism $\mu$ is an isomorphism.

Lastly, the component $\mathcal{Y}$ is contracted by $\Psi$ as in Remark 2.17,
while the Abel–Jacobi mapping is an embedding on smooth lines by [Tih82, §10],
and $Y$ is determined uniquely via $\Psi(\mathcal{M}_{\sigma}(w))=\Psi(F(Y))$
by [Tih82, Thm. 12]. ∎

##  5\. A categorical Torelli theorem for quartic double solids

As an application of the previous construction, this section is dedicated to
the proof of Theorem 1.3, we state it here for the reader’s convenience.

######  Theorem 5.1.

Let $Y$ and $Y^{\prime}$ be two general quartic double solids such that there
exists an equivalence
$u\colon\mathsf{Ku}(Y^{\prime})\xrightarrow{\sim}\mathsf{Ku}(Y)$. Then
$Y^{\prime}\simeq Y$.

This theorem improves a result of Bernardara and Tabuada [BT16, Cor. 3.1
(iii)] who show that the same holds under the additional assumption that $u$
be of Fourier–Mukai type. Our technique is inspired by that of [BMMS12]; we
use the equivalence to construct an isomorphism between moduli spaces and
argue that this is sufficient to conclude.

###  5.1. Images of stable objects

First, we prove that given an equivalence as in Theorem 1.3, we can produce
another equivalence with nice properties on the objects parametrized by
$\mathcal{M}^{\prime}$. In this section, . We start with a series of lemmas,
and denote by $\mathcal{A}$ the heart of the stability condition $\sigma$ (see
Sec. 4.1).

######  Lemma 5.2.

Let $u\colon\mathsf{Ku}(Y^{\prime})\xrightarrow{\sim}\mathsf{Ku}(Y)$ be an
equivalence of categories, and let $E\in\mathsf{Ku}(Y^{\prime})$ be an object
of class $w$; then up to a sign, we have $[u(E)]=w$ or $[u(E)]=2v-w$.

###### Proof.

Since $u$ is an equivalence of categories, it must preserve dimensions of
$\operatorname{Hom}$ spaces, so in particular $\chi(u(E),u(E))=\chi(E,E)=-2$.
From the intersection matrix of the Euler form given in section 2, if
$[u(E)]=av+bw$, then $\chi(u(E),u(E))=-(a+b)^{2}-b^{2}$, The only possible
pairs $(a,b)$ are $(0,\pm 1)$ and $(\mp 2,\pm 1)$. Finally, since an
equivalence of categories induces a homomorphism on the Grothendieck groups,
we can choose the sign uniformly for all objects. ∎

######  Lemma 5.3.

Let $0\neq C\in\mathcal{A}\subset\mathsf{Ku}(Y)$; then
$\dim\operatorname{Ext}^{1}(C,C)\geq 2$.

###### Proof.

Since $\mathcal{A}$ is the heart of a stability condition, $C\neq 0$ implies
that $[C]\neq 0$. Then, we have $\chi(C,C)\leq-1$ and
$\dim\operatorname{Hom}(C,C)\geq 1$, and $\operatorname{Ext}^{3}(C,C)=0$ from
Lemma 4.2. Therefore, $\dim\operatorname{Ext}^{1}(C,C)\geq 2$. ∎

######  Lemma 5.4.

Let $C\in\mathsf{Ku}(Y)$ with $\dim\operatorname{Ext}^{1}(C,C)=3$; then up to
shift $C\in\mathcal{A}$, and if additionally $[C]=w$ then $C$ is stable.

###### Proof.

Consider the spectral sequence for objects in $\mathsf{Ku}(Y)$ whose second
page is given by

(13)
$E^{p,q}_{2}=\bigoplus_{i}\operatorname{Hom}^{p}(\mathcal{H}^{i}(C),\mathcal{H}^{i+q}(C))\Longrightarrow\operatorname{Hom}^{p+q}(C,C)$

(see [BMMS12, Lemma 4.5]), where the cohomology is taken with respect to the
heart $\mathcal{A}$. Since by Lemma 4.2 the Ext-dimension of $\mathcal{A}$ is
2, it follows that $E^{1,q}_{2}=E^{1,q}_{\infty}$, so that if we take $q=0$,
by Lemma 5.3 we get

$3=\hom^{1}(C,C)\geq\bigoplus_{i}\hom^{1}(\mathcal{H}^{i}(C),\mathcal{H}^{i}(C))\geq
2r,$

where $r>0$ is the number of non-zero cohomology objects of $C$. Then $r=1$
and $C\in\mathcal{A}$ up to shift; if we also assume $[C]=w$, then $C$ must be
stable since $w$ has maximal slope and is primitive. ∎

######  Lemma 5.5.

Let $C\in\mathsf{Ku}(Y)$ of class $[C]=w$ with the following numerics:

$\hom^{i}(C,C)=\begin{cases}1\qquad i=0\\\ 4\qquad i=1\\\ 1\qquad i=2\\\
0\qquad i=3\end{cases}$

then up to shift $C\in\mathcal{A}$, and in particular $C$ is stable.

###### Proof.

Similarly to the proof of the previous Lemma, we get that

$4=\hom^{1}(C,C)\geq\bigoplus_{i}\hom^{1}(\mathcal{H}^{i}(C),\mathcal{H}^{i}(C))\geq
2r,$

which in this case yields $r=1$ or $r=2$. If $r=1$, we’re done. Assume by
contradiction that $r=2$, and call $M$ and $N$ the two non-zero cohomology
objects of $C$; one necessarily has $\hom^{1}(M,M)=\hom^{1}(N,N)=2$, and since
$\chi(F,F)\leq-1$ for all non-zero objects of $\mathcal{A}$, we also get
$\hom^{0}(M,M)=\hom^{0}(N,N)=1$ and $\hom^{2}(M,M)=\hom^{2}(N,N)=0$, yielding
$\chi(M,M)=\chi(N,N)=-1$ in particular. This means that $[M]$ and $[N]$ can
only be equal to $\pm v$ or $\pm(v-w)$. Moreover, the $M$ and $N$ must be
adjacent cohomologies, otherwise $C$ must splits as a sum of their shifts
since $\operatorname{Ext}^{i}(M,N)=0$ for $i>2$, contradicting
$\hom^{0}(C,C)=1$.

In order for their classes (with appropriate shift) to add up to $w$, for $M$
and $N$ to be in the heart at the same time and for the spectral sequence to
abut to the correct numerics of $C$, we see that the only possibility is the
following:

$\quad M=\mathcal{H}^{j}(C),\;[M]=\pm v\qquad\text{and}\qquad
N=\mathcal{H}^{j\pm 1}(C),\;[N]=\mp(w-v),$

where the signs are determined by the parity of $j$; notice in particular that
in all of these cases $\chi(M,N)=0$.

i.e. $M$ and $N$ are adjacent cohomologies, and exactly one of them has class
$\pm v$ and the other has class $\mp(w-v)$, where the signs are determined by
the parity of $j$; notice in particular that in all of these cases
$\chi(M,N)=0$. Now, if we gather all this information together, the second
page of the spectral sequence (13) has zeroes except for the spaces:

$E_{2}^{p,q}$ =  0 0 0 $\operatorname{Hom}(N,M)$ $\operatorname{Hom}^{1}(N,M)$
$\operatorname{Hom}^{2}(N,M)$
$\operatorname{Hom}(M,M)\oplus\operatorname{Hom}(N,N)$
$\operatorname{Hom}^{1}(M,M)\oplus\operatorname{Hom}^{1}(N,N)$ 0
$\operatorname{Hom}(M,N)$ $\operatorname{Hom}^{1}(M,N)$
$\operatorname{Hom}^{2}(M,N)$

Therefore, the whole sequence degenerates at page 3, and the central column
survives in the limit. We denote with $a,b,c,d,e,f$ the dimensions of the
spaces in the first and third row, the table of dimensions for page 2 is:

$\dim E_{2}^{p,q}$ =  0 0 0 $a$ $b$ $c$ 2 4 0 $d$ $e$ $f$

Next, we impose the numerics of $C$, which constrain the dimensions on the
diagonals of page 2. One sees right away that we must have $a=0$, $b=1$, $c=0$
and $d=0$, and a surjection
$\operatorname{Hom}(M,M)\oplus\operatorname{Hom}(N,N)\twoheadrightarrow\operatorname{Hom}^{2}(M,N)$
with kernel of dimension $g\coloneqq 2-f$. On the other hand, we also have
$e=f$ since the alternate sum of dimension on the row $q=-1$ is $\chi(M,N)=0$.
This implies the contradiction that $1=\hom(C,C)=g+e=g+f=2$. ∎

The category $\mathsf{Ku}(Y)$ admits an autoequivalence called the rotation
functor

$\mathsf{R}(-)\coloneqq\mathbb{L}_{\mathcal{O}_{Y}}(-\otimes\mathcal{O}_{Y}(1))$

(see [Kuz15, Sec. 3.3], in particular [Kuz15, Cor. 3.18] for the proof of the
fact that $\mathsf{R}$ is an autoequivalence).

######  Lemma 5.6.

The map induced on $K_{num}(\mathsf{Ku}(Y))$ by the rotation functor maps $w$
to $w-2v$ and $v$ to $w-v$.

###### Proof.

To prove the first statement, we compute $\mathsf{R}(E_{p})$ since
$[E_{p}]=w$. Twisting the sequence (4) and mutating across $\mathcal{O}_{Y}$,
we see that $\mathsf{R}(E_{p})=\mathbb{L}_{\mathcal{O}_{Y}}(I_{p}(1))$. The
latter is computed by the triangle

(14) $\mathcal{O}_{Y}^{3}\to
I_{p}(1)\to\mathbb{L}_{\mathcal{O}_{Y}}(I_{p}(1)),$

and it has class $[I_{p}(1)]-3[\mathcal{O}_{Y}]=2v-w$ in
$K_{num}(\mathsf{Ku}(Y))$. Arguing similarly, one considers the ideal sheaf of
a line $l$, which has class $v$, and checks that
$[L_{\mathcal{O}_{Y}}(I_{p}(1))]=[I_{l}(1)]-2[\mathcal{O}_{Y}]=w-v$. ∎

Now if we take an object $E\in\mathcal{M}^{\prime}$, then $u(E)$ has class
$\pm w$ or $\pm(2v-w)$. In the latter case we can replace $u$ with
$u\circ\mathsf{R}$ so that $u(E)$ has class $\pm w$. Lemmas 5.4 and 5.5 show
that there exists an integer $n$ such that $u(E)[n]\in\mathcal{A}$ and is
stable of class $w$ (in particular, it has phase 1).

We want to prove that the shift can be taken uniformly, i.e. that, for the
same $n$ and any other object $F$ in $\mathcal{M}^{\prime}$, we have
$u(F)[n]\in\mathcal{A}$. For all such $F$, we must have
$\operatorname{Hom}(E,F[1])=\operatorname{Ext}^{1}(E,F)\neq 0$, since
$\operatorname{Hom}(E,F)=0$ by stability,
$\operatorname{Ext}^{3}(E,F)=\operatorname{Ext}^{-1}(F,E)=0$ by use of the
Serre functor on $\mathsf{Ku}(Y^{\prime})$, and $\chi(E,F)=-2$ because of
their numerical class. Likewise, $\operatorname{Hom}(F,E[1])\neq 0$. Applying
$u(-)[n]$ we get $\operatorname{Hom}(u(E)[n],u(F)[n+1])\neq 0$ and
$\operatorname{Hom}(u(F)[n],u(E)[n+1])\neq 0$. Now, $u(E)[n]$ has phase 1, and
let $\phi\in\mathbb{Z}$ be the phase of $u(F)[n]$: from the above we get
$1<\phi+1$ and $\phi<2$, hence $\phi=1$ and $u(F)[n]\in\mathcal{A}$.

We can thus summarize the results of this section with the following
Proposition:

######  Proposition 5.7.

The equivalence
$u\colon\mathsf{Ku}(Y^{\prime})\xrightarrow{\sim}\mathsf{Ku}(Y)$ induces a
bijection on closed points $u\colon\mathcal{M}^{\prime}(k)\to\mathcal{M}(k)$.

###### Proof.

We have shown that the assignment $E\mapsto u(E)$ sends points of
$\mathcal{M}^{\prime}$ to points of $\mathcal{M}$ (and similarly, $u^{-1}$
takes $\mathcal{M}$ to $\mathcal{M}^{\prime}$). Since $u$ is an equivalence,
the induced map is a bijection on closed points. ∎

###  5.2. Universal families and convolutions

Let $\mathcal{M}$ and $\mathcal{M}^{\prime}$ denote the moduli spaces of
objects of class $w$ in $\mathsf{Ku}(Y)$, resp. $\mathsf{Ku}(Y^{\prime})$. As
usual, we will denote by $\mathcal{Y}$ the component isomorphic to $Y$ in
$\mathcal{M}$ (we use the same convention in the case of $Y^{\prime}$). The
component $\mathcal{Y}$ admits a universal family, whose construction we
outline here (see also the proof of Prop. 2.9). Let $\mathcal{I}\in
D^{b}(Y\times\mathcal{Y})$ be the pull back of the ideal sheaf of the diagonal
via the isomorphism $Y\times\mathcal{Y}\xrightarrow{\sim}Y\times Y$. Then,
define
$\mathcal{E}\coloneqq\mathbb{R}_{\mathcal{O}_{Y}(-1)}^{\mathcal{Y}}(\mathcal{I})[-1]$
the projection to
$\\!{}^{\perp}\langle\mathcal{O}_{Y}(-1)\rangle_{\mathcal{Y}}$ (notation as in
1.1.4). It is the desired universal family: for example the restriction of
$\mathcal{E}$ to a fiber $Y\simeq Y\times\\{s\\}$ above a closed point $s\in
S$ fits in a triangle

$I_{s}\to\mathcal{O}_{Y}(-1)[2]\to\mathcal{E}_{s}[1],$

whence $\mathcal{E}_{s}=E_{s}$ (this triangle is (4)). Likewise, define
$\mathcal{E}^{\prime}$ to be the universal family above
$\mathcal{Y}^{\prime}$.

Define the composite functor

$F\colon
D(Y^{\prime})\xrightarrow{\rho^{\prime}}\mathsf{Ku}(Y^{\prime})\xrightarrow{u}\mathsf{Ku}(Y)\xrightarrow{\epsilon}D(Y)$

where $\rho^{\prime}$ is the natural projection, $\epsilon$ the full
embedding, and $u$ an exact equivalence. If $F$ is a Fourier–Mukai functor,
i.e., $F\simeq\Phi_{\mathcal{G}}$ for some $\mathcal{G}\in D^{b}(Y\times
Y^{\prime})$, then one defines

$\Phi_{\mathcal{G}}\times\operatorname{id}_{\mathcal{Y}^{\prime}}\coloneqq\Phi_{\mathcal{G}\boxtimes\mathcal{O}_{\Delta_{\mathcal{Y}^{\prime}}}}\colon
D^{b}(Y^{\prime}\times\mathcal{Y}^{\prime})\to
D^{b}(Y\times\mathcal{Y}^{\prime})$

and the object

$\tilde{\mathcal{E}}\coloneqq\Phi_{\mathcal{G}}\times\operatorname{id}_{\mathcal{Y}^{\prime}}(\mathcal{E}^{\prime})\in
D^{b}(Y\times\mathcal{Y}^{\prime})$

is a family of objects of $D(Y)$ parametrized by $\mathcal{Y}^{\prime}$, which
defines a morphism $Y^{\prime}\simeq\mathcal{Y}^{\prime}\to\mathcal{M}$.

If $F$ is not a Fourier–Mukai functor, then one can construct the family
$\tilde{\mathcal{E}}$ ”by hand”, with the same argument as in [BMMS12, Sec.
5.2]. The main technical tool is that of convolutions, we will recall here
some aspects of the argument for the convenience of the reader. A _right
convolution_ of a bounded complex

$A_{m}\xrightarrow{d_{m}}A_{m-1}\xrightarrow{d_{m-1}}...\xrightarrow{d_{1}}A_{0}$

in a triangulated category $\mathbf{T}$ is an object $A$ with a morphism
$d_{0}\colon A_{0}\to A$ such that there is a diagram in $\mathbf{T}$

${A_{m}}$${A_{m-1}}$${...}$${A_{1}}$${A_{0}}$${A_{m}}$${C_{m-1}}$${...}$${C_{1}}$${A}$$\scriptstyle{d_{m}}$$\scriptstyle{\operatorname{id}}$$\circlearrowleft$$\circlearrowleft$$\circlearrowleft$$\scriptstyle{d_{0}}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$