# One-cycles on Gushel-Mukai fourfolds and the Beauville-Voisin filtration

Ruxuan Zhang  Ruxuan Zhang, Shanghai Center for Mathematical Sciences, Fudan
University, Jiangwan Campus, Shanghai, 200438, China<EMAIL_ADDRESS>
###### Abstract.

We prove that the invariant locus of the involution associated with a general
double EPW sextic is a constant cycle surface and introduce a filtration on
$\operatorname{CH}_{1}$ of a Gushel-Mukai fourfold. We verify the sheaf/cycle
correspondence for sheaves supported on low-degree rational curves, parallel
to the cubic fourfolds case of Shen-Yin’s work.

###### Key words and phrases:

Beauville-Voisin filtration, irreducible holomorphic symplectic variety,
Gushel-Mukai fourfold

###### 2020 Mathematics Subject Classification:

14J42 , 14C15

##  1\. Introduction

A Gushel-Mukai (GM) fourfold is a smooth prime Fano fourfold
$X\subset\mathbb{P}^{8}$ of degree $10$ and index $2$. Throughout the paper, a
GM fourfold is always an ordinary GM fourfold, which can be obtained as a
smooth dimensionally transverse intersection

$X=\operatorname{Gr}(2,5)\cap\mathbb{P}^{8}\cap Q\subset\mathbb{P}^{9},$

where $\operatorname{Gr}(2,5)\subset\mathbb{P}^{9}$ is the Plücker embedding,
$\mathbb{P}^{8}$ is a linear subspace and $Q$ is a quadric hypersurface.

GM fourfolds share many properties with cubic fourfolds. One of the
distinguished facts is that there is a parallel between GM and cubic fourfolds
from a categorical viewpoint. Similarly to the semiorthogonal decomposition of
the derived category of a cubic fourfold, Kuznetsov and Perry showed in [15]
that the derived category of a GM fourfold admits a semiorthogonal
decomposition:

${\rm
D}^{b}(X)=\langle\mathcal{A}_{X},\mathcal{O}_{X},\mathcal{U}_{X}^{\vee},\mathcal{O}_{X}(1),\mathcal{U}_{X}^{\vee}(1)\rangle,$

where $\mathcal{U}_{X}$ is the tautological bundle of $\operatorname{Gr}(2,5)$
restricted to $X$ and $\mathcal{A}_{X}$ is a K3 category.

Another fact is that a GM fourfold has an associated irreducible holomorphic
symplectic (IHS) variety, which is called the dual double EPW sextic, compared
with the Fano variety of lines associated with a cubic fourfold, see
[3][19][14] for details.

The Fano variety of lines of a cubic fourfold and the dual double EPW sextic
of a GM fourfold both admit constructions via moduli spaces of stable objects
and Hilbert schemes of rational curves. In [22, Proposition 5.17], it has been
shown that at least one of the double EPW sextic and the dual double EPW
sextic can be realized as a moduli space of stable objects of
$\mathcal{A}_{X}$ and conjecturally both are. In [14], Iliev and Manivel gave
a more geometrical description of the dual double EPW sextic. For a general GM
fourfold $X$, let $F(X)$ be the Hilbert scheme of conics on $X$ and
$\tilde{Y}^{\vee}$ be the associated dual double EPW sextic of $X$, there
exists a morphism

$F(X)\rightarrow\tilde{Y}^{\vee}$

such that a general fiber is $\mathbb{P}^{1}$.

Quite generally, for a $2n$-dimensional IHS variety $M$, Voisin predicted in
[32] that there exists a filtration of the Chow ring
$\operatorname{CH}^{*}(M)$ (called Beauville–Voisin filtration), which can be
viewed as an opposite to the conjectural Bloch-Beilinson filtration. For
0-cycles, the filtration ${\rm S}_{\bullet}\operatorname{CH}_{0}(M)$ is
defined by

${\rm S}_{i}\operatorname{CH}_{0}(M):=\left<x\in M|~{}\dim O_{y}\geq
n-i\right>,$

where $O_{y}$ is the orbit of $y$ under the rational equivalence (here we
follow the definition in [24], which is opposite to Voisin’s original
definition). Let $Z\subset M$ be a subvariety. If any two points on $Z$ have
the same class in $\operatorname{CH}_{0}(M)$, we call that $Z$ is a constant
cycle variety on $M$. In particular, for an IHS variety of dimension $4$, the
filtration is determined by constant cycle surfaces and uniruled divisors. Let
${\rm S}_{i}M\subset M$ be the set of points with dim $O_{y}\geq n-i$. Voisin
conjectured in [32, Conjecture 0.4] that

$\mathrm{dim}~{}{\rm S}_{i}M=n+i.$ (1.1)

It is closely related to the existence of algebraically coisotropic
subvarieties on $M$.

Let $M$ be a nonsingular projective moduli space of stable objects on a K3
category $\mathcal{A}$, then it is an IHS variety in many cases. Shen and Yin
predicted that all the Beauville–Voisin filtrations associated with different
moduli spaces of stable objects on a fixed K3 category should be controlled by
a universal filtration on the Grothendieck group of the K3 category, see [24,
Speculation 0.1].

When the K3 category is the derived category of a K3 surface $S$ and $M$ is a
$2d$-dimensional nonsingular moduli space of stable objects, the O’Grady’s
filtration on $\operatorname{CH}_{0}(S)$

${\rm S}_{i}(S):=\bigcup_{\begin{subarray}{c}\deg([z])=i\\\
z~{}\text{effective}\end{subarray}}\\{[z]+\mathbb{Z}\cdot[o_{S}]\\}$

serves as the universal filtration for all moduli space of stable objects. It
has been shown in [25] that

$c_{2}(\mathcal{E})\in{\rm S}_{d}(S)$

for any $\mathcal{E}\in M$ and that $c_{2}(\mathcal{E})\in{\rm S}_{i}(S)$
implies $\mathcal{E}\in{\rm S}_{i}\operatorname{CH}_{0}(M)$.

Another example arises when the K3 category is the Kuznetsov component
$\mathcal{A}_{X}$ of a cubic fourfold $X$. Using the Beauville-Voisin
filtration on the Fano variety of lines on $X$, Shen-Yin introduced in [25] a
filtration ${\rm S}_{\bullet}(X)$ on $\operatorname{CH}_{1}(X)$ induced by the
incidence correspondence. They conjectured that the filtration ${\rm
S}_{\bullet}(X)$ may serve as a universal filtration on
$K_{0}(\mathcal{A}_{X})$, i.e., for $\mathcal{E}\in\mathcal{A}_{X}$, there
should be

$c_{3}(\mathcal{E})\in{\rm S}_{d}(X),$

where $d=\frac{1}{2}\dim\operatorname{Ext}^{1}(\mathcal{E},\mathcal{E})$. This
implies (1.1) by a standard argument and they verified the prediction when
$\mathcal{E}$ is supported on low-degree rational curves.

It is natural to ask whether there exists a universal filtration on
$\mathcal{A}_{X}$ for a GM fourfold, similarly to the case of a cubic
fourfold. We first study the Beauville–Voisin filtration on
$\operatorname{CH}_{0}(\tilde{Y}^{\vee})$ by finding a natural constant cycle
surface on $\tilde{Y}^{\vee}$. The invariant locus $Z$ of the involution
associated with the dual double EPW sextic is a surface of general type, whose
$\operatorname{CH}_{0}$ might be huge. However, it was asked whether $Z$ is a
constant cycle surface on $\tilde{Y}^{\vee}$, see [17, Remark 2.7]. Our first
result is (Theorem 4.14):

######  Theorem 1.1.

The invariant locus $Z$ of the involution associated with a general dual
double EPW sextic $\tilde{Y}^{\vee}$ is a constant cycle surface. If
$\tilde{Y}^{\vee}$ is very general, any other constant cycle surface
$Z^{\prime}$ meets $Z$.

The $0$-piece of the Beauville–Voisin filtration of $\tilde{Y}^{\vee}$ is
represented by a point on $Z$ and the $1$-piece is represented by a point on a
uniruled divisor. Theorem 1.1 allows us to construct a filtration ${\rm
S}_{\bullet}(X)$ on $\operatorname{CH}_{1}(X)$ by the incidence correspondence
of Iliev and Manivel’s construction. For a very general GM fourfold, we
predict that the filtration should serve as a universal filtration on
$\mathcal{A}_{X}$ in the following way:

The Grothendieck group of $\mathcal{A}_{X}$ is shown to be generated by
$\operatorname{pr}[\mathcal{O}_{c}(1)]$ and
$\operatorname{pr}[\mathcal{O}_{p}]$ for a very general $X$, where $c$ is a
conic, $p$ is a point on $X$ and $\operatorname{pr}$ is the projection

$\operatorname{pr}:K_{0}(X)\rightarrow K_{0}(\mathcal{A}_{X}).$

Unlike the cubic fourfold case, the dual double EPW sextic determines the GM
fourfold only up to period partners and the Chern class
$c_{3}(\operatorname{pr}[\mathcal{O}_{p}])$ of a skyscraper sheaf might lie in
different pieces of the filtration with respect to different GM fourfolds. We
define a modification

$p:K_{0}(\mathcal{A}_{X})\rightarrow\operatorname{CH}_{1}(X)$

of $c_{3}$ by dropping the effect of $\operatorname{pr}[\mathcal{O}_{p}]$, see
Section 6 for details.

We propose the following conjecture relating the $K3$ category
$\mathcal{A}_{X}$ to the filtration ${\rm S}_{\bullet}(X)$ as a parallel to
the case of a cubic fourfold.

######  Conjecture 1.2.

For any object $\mathcal{E}\in\mathcal{A}_{X}$, we have

$p(\mathcal{E})\in{\rm S}_{d(\mathcal{E})}(X).$

Let $i^{*}$ be the left adjoint functor of the natural inclusion
$\mathcal{A}_{X}\hookrightarrow D^{b}(X)$. We show that the classes of lines
and conics on a very general GM fourfold $X$ are in ${\rm S}_{2}(X)$, which
implies our second main result concerning sheaves supported on low-degree
rational curves (Theorem 6.3):

######  Theorem 1.3.

We assume that $X$ is very general. If $\mathcal{F}$ is supported on lines or
conics on $X$, conjecture 1.2 holds for $\mathcal{E}=i^{*}\mathcal{F}$.

Our approach to deducing Theorem 1.1 has another application, concerning with
the relation between $\operatorname{CH}_{0}(\tilde{Y}^{\vee})$ and
$\operatorname{CH}_{1}(X)$. When $\tilde{Y}^{\vee}$ is birational to $S^{[2]}$
for some K3 surface $S$, it has been shown in [17] that
$\operatorname{CH}_{0}(\tilde{Y}^{\vee})$ with $\mathbb{Q}$-coefficients has a
natural decomposition for the Chow group. For 0-cycles, the decomposition is

$\operatorname{CH}_{0}(\tilde{Y}^{\vee})=\mathbb{Q}\cdot
o\oplus\operatorname{CH}_{0}(S)_{\rm
hom}\oplus\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{+}_{\rm hom},$

where $\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{+}_{\rm hom}$ is the
$\iota$-invariant part of $\operatorname{CH}_{0}(\tilde{Y}^{\vee})_{\rm hom}$.
We show that in the general case, the group $\operatorname{CH}_{1}(X)$ would
play the role of the group $\operatorname{CH}_{0}(S)$.

######  Theorem 1.4.

(Theorem 5.5) When the dual double EPW sextic $\tilde{Y}^{\vee}$ is general,
we have the following isomorphism of groups:

$\mathbb{Z}\cdot o\oplus\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{-}_{\rm
hom}\cong\operatorname{CH}_{1}(X),$

where $\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{-}_{\rm hom}$ is the
$\iota$-anti-invariant part of $\operatorname{CH}_{0}(\tilde{Y}^{\vee})_{\rm
hom}$.

In particular, the group $\operatorname{CH}_{1}(X)$ is torsion-free and we see
that if $X^{\prime}$ is another GM fourfold with the same dual double EPW
sextic as $X$, then their groups of 1-cycles are isomorphic.

Our results rely on Iliev and Manivel’s construction and hence all dual double
EPW sextics are assumed to be general in the rest of the paper.

##  2\. Notation

The following notations appear frequently in the paper, so we gather them
together for the sake of convenience.

  * •

$V_{m}$ is an $m$-dimensional vector space, $X$ is a smooth ordinary GM
fourfold and $\tilde{Y}^{\vee}$ is the dual double sextic associated with $X$.

  * •

$\iota$ is the involution of $\tilde{Y}^{\vee}$ and $Z\subset\tilde{Y}^{\vee}$
is the invariant locus of $\iota$.

  * •

$c$ is a conic on $X$, $F(X)$ is the Hilbert scheme of conics on $X$ and
$\alpha:F(X)\rightarrow\tilde{Y}^{\vee}$ is the birational
$\mathbb{P}^{1}$-fibration. $\alpha$ induces an isomorphism
$\alpha_{*}:\operatorname{CH}_{0}(F(X))\cong\operatorname{CH}_{0}(\tilde{Y}^{\vee}).$

  * •

$P$ is the universal conic, $p,q$ are the projections to $F(X)$ and $X$.
$\Phi=p_{*}q^{*}:\operatorname{CH}^{*}(X)\rightarrow\operatorname{CH}^{*}(F)$.
$\Psi$ is the composition of $q_{*}p^{*}$ and $\alpha_{*}^{-1}$:

$\Psi=q_{*}p^{*}\alpha_{*}^{-1}:\operatorname{CH}_{0}(\tilde{Y}^{\vee})\rightarrow\operatorname{CH}_{1}(X),$

sending $\alpha(c)$ to its class $[c]\in\operatorname{CH}_{1}(X)$.

  * •

$S_{V_{4}}=\operatorname{Gr}(2,V_{4})\cap X$ for some $V_{4}\subset V_{5}$,
which is a degree $4$ Del Pezzo surface.

  * •

$D_{c}$ is the subvariety of $F(X)$ parameterizing conics intersecting the
given conic $c$. $I\subset F(X)\times F(X)$ is the subvariety of $F(X)\times
F(X)$, parameterizing two intersecting conics.

##  3\. Preliminaries

Let $X=\operatorname{Gr}(2,V_{5})\cap H\cap Q$ be a very general GM fourfold
and $\tilde{Y}^{\vee}\rightarrow Y^{\vee}$ be its associated double cover of
the dual EPW sextic. Denote by
$\iota:\tilde{Y}^{\vee}\rightarrow\tilde{Y}^{\vee}$ the associated involution
and by $Z$ the invariant locus of $\iota$. We recall some facts about GM
varieties and EPW sextics in this section. Throughout the paper, $V_{m}$ is an
$m$-dimensional vector space.

###  3.1. EPW sextics

Debarre and Kuznetsov gave a way to associate a GM variety with an EPW sextic
in [3] via linear algebra. We will follow [4] to recall some basic facts about
EPW sextics. A GM variety
$X_{n}=\operatorname{Gr}(2,V_{5})\cap\mathbb{P}(W)\cap Q$ of dimension $n$ is
a smooth intersection of a linear subspace of dimension $n+4$, a quadratic
hypersurface and $\operatorname{Gr}(2,V_{5})$ inside
$\mathbb{P}(\bigwedge^{2}V_{5})$. It has an associated GM data
$(W_{n+5},V_{6},V_{5},q)$, where

  * •

$V_{6}$ is a $6$-dimensional vector space, consisting of quadrics containing
$X$.

  * •

$V_{5}$ is a hyperplane section of $V_{6}$, consisting of quadrics containing
$\operatorname{Gr}(2,V_{5})$.

  * •

$W_{n+5}\subset\bigwedge^{2}V_{5}$ is a subspace of dimension $n+5$,
corresponding to the linear subspace in the definition of GM variety.

  * •

$q:V_{6}\rightarrow\mathrm{Sym}^{2}W_{n+5}^{\vee}$, is a linear map such that
(here we choose an isomorphism $\bigwedge^{5}V_{5}\cong\mathbb{C}$)

$\forall v\in V_{5},q(v)(w,w)=v\wedge w\wedge w.$

Then $X=\bigcap_{v\in V_{6}}Q(v)\subset\mathbb{P}(W_{n+5})$, where $Q(v)$ is
defined by $q(v)$. For a general GM data, $X$ is a smooth GM variety.

On the other hand, $\bigwedge^{3}V_{6}$ can be endowed with a symplectic form
by wedge product. Debarre and Kuznetsov defined the Lagrangian data set
$(V_{6},V_{5},A)$ with $A\subset\bigwedge^{3}V_{6}$ a Lagrangian subspace and
$V_{5}\subset V_{6}$ a hyperplane. It gives an EPW data

$Y_{A}^{\geq
l}:=\\{[v]\in\mathbb{P}(V_{6})~{}|~{}\mathrm{dim}(A\cap(v\wedge\bigwedge^{2}V_{6}))\geq
l\\}$

and an extra $V_{5}$. Here, $Y_{A}^{\geq 1}$ is a sextic in
$\mathbb{P}(V_{6})$ and $Y_{A}^{\geq 2}$ is a surface. In [20], O’Grady showed
that

######  Theorem 3.2.

If $A$ contains no decomposable vectors and $Y_{A}^{\geq 3}=\emptyset$, there
is a double cover $\tilde{Y}_{A}\rightarrow Y_{A}^{\geq 1}$ branched along the
surface $Y_{A}^{\geq 2}$. $\tilde{Y}_{A}$ is a smooth IHS variety, called the
double EPW sextic.

Iliev and Manivel described in [14] another way to construct the EPW sextic
associated with a GM variety $X$. Define $\widetilde{\mathrm{Disc}}(X)$ to be
the subscheme of $\mathbb{P}(V_{6})$ of singular quadrics containing $X$. It
is a hypersurface of degree $n+5$ and the multiplicity of the hyperplane
$\mathbb{P}(V_{5})$ of Plücker quadrics is at least $n-1$. Then

$\mathrm{Disc}(X):=\widetilde{\mathrm{Disc}}(X)-(n-1)\mathbb{P}(V_{5})$

equals the EPW sextic $Y_{A}$.

One can use the same construction for the Lagrangian subspace
$A^{\vee}\subset\bigwedge^{3}V_{6}^{\vee}$, the corresponding
$\tilde{Y}_{A^{\vee}}$ is called the dual double EPW sextic. We mainly deal
with the dual double EPW sextic associated with a GM variety and abbreviate it
as $\tilde{Y}^{\vee}$ in the following sections.

There is a bijection between the GM data sets and the Lagrangian data sets by
[4, Theorem 2.2]. However, the GM fourfolds associated with a fixed dual EPW
sextic are not unique. Debarre and Kuznetsov proved the following [7][4,
Theorem 2.6]:

######  Theorem 3.3.

Let $M^{GM}$ and $M^{EPW}$ be the coarse moduli spaces of ordinary GM
$n$-folds and EPW sextics. Then there exists a surjective morphism:

$\pi:M^{GM}\longrightarrow M^{EPW}$

with fibre $\pi^{-1}(A^{\vee})=Y_{A}^{5-n}$.

If two GM fourfolds are in the same fiber of $\pi$, we say that they are
period partners.

###  3.4. Lines and Conics on $X$

In this subsection, we gather some facts about lines and conics on a general
GM fourfold $X$ following [14, Section 3] and [6].

Let $F(X)$ be the Hilbert scheme of conics and $F_{1}(X)$ the Hilbert scheme
of lines on $X$. $F(X)$ is a smooth fivefold. For a conic $c\subset X$, we
denote by $\langle c\rangle$ the plane spanned by $c$. For $V_{4}\subset
V_{5}$, we set

$S_{V_{4}}=\operatorname{Gr}(2,V_{4})\cap H\cap
Q\subset\mathbb{P}(\bigwedge^{2}V_{4})\cap H\cong\mathbb{P}^{4},$

which is a degree $4$ Del Pezzo surface.

The hyperplane $H$ can be viewed as a two-form $\omega$ on $V_{5}$ with a one-
dimensional kernel $W_{1}$. Let $V_{i}\subset V_{5}$ be a subspace of
dimension $i$. There are three types of conics on $X$:

  1. (1)

a $\tau$-conic: The plane $\langle c\rangle$ spanned by the conic $c$ is not
contained in $\operatorname{Gr}(2,V_{5})$. The normal bundle of a smooth
$\tau$-conic $c$ is
$N_{c/X}=\mathcal{O}_{c}\oplus\mathcal{O}_{c}(1)\oplus\mathcal{O}_{c}(1)$. A
general conic is a $\tau$-conic.

  2. (2)

a $\rho$-conic: The plane $\langle c\rangle$ is of the form
$\mathbb{P}(V_{3}^{\vee})$ and contained in $H$, or equivalently,
$W_{1}\subset V_{3}$ and $V_{3}$ is isotropic for $\omega$. For a general $X$,
the normal bundle of a smooth $\rho$-conic $c$ is
$N_{c/X}=\mathcal{O}_{c}\oplus\mathcal{O}_{c}(1)\oplus\mathcal{O}_{c}(1)$.
They form a $3$-dimensional subvariety in $F(X)$.

  3. (3)

a $\sigma$-conic: The plane $\langle c\rangle$ is of the form
$\mathbb{P}(V_{1}\bigwedge V_{4})$ and contained in $H$, or equivalently,
$V_{4}\subset V_{1}^{\perp_{\omega}}$. For $V_{1}\neq W_{1}$, dim
$V_{1}^{\perp_{\omega}}=4$. They are parameterized by
$Bl_{[W_{1}]}(\mathbb{P}(V_{5}))$ and form a $4$-dimensional subvariety in
$F(X)$.

We see that all the $\rho$-conics (resp. $\sigma$-conics) are represented by
the same class in $\operatorname{CH}_{1}(X)$ and denote it by $\rho$ (resp.
$\sigma$) for abuse of notations.

Lines on $X$ are related to the double EPW sextic $\tilde{Y}_{A}$. The general
$\sigma$-conics which are unions of two lines are parameterized by

$[V_{1}]\in Y_{A}^{\geq 1}\cap\mathbb{P}(V_{5})$

and a general line is one of the components of a degenerate $\sigma$-conic by
[6, Theorem 4.7]. More precisely, the Hilbert scheme of lines $F_{1}(X)$ is
the blow-up of $\tilde{Y}_{A}\times_{\mathbb{P}(V_{6})}\mathbb{P}(V_{5})$ at a
point. There is a rational involution

$\iota_{1}:F_{1}(X)\dashrightarrow F_{1}(X),$

with $l\cup\iota_{1}(l)$ a $\sigma$-conic.

The Fano variety of conics $F(X)$ admits birationally a
$\mathbb{P}^{1}$-fibration to $\tilde{Y}^{\vee}$ in the following way. A conic
$c\subset X$ spans a degree $2$ surface in $\mathbb{P}(V_{5})$, which is
degenerate and hence contained in some $\mathbb{P}(V_{4})$. Therefore,
$c\subset S_{V_{4}}$. One may check that if $c$ is not of $\rho$-type, the
corresponding $V_{4}$ is unique.

The linear system $|c|$ in $S_{V_{4}}$ is one-dimensional, which defines the
birational $\mathbb{P}^{1}$-fibration to the dual double EPW sextic, see [14,
Proposition 4.18]:

$\alpha:F(X)\rightarrow\tilde{Y}^{\vee}.$

The fibration maps all $\sigma$ and $\rho$ conics to two points $y_{1}$ and
$y_{2}=\iota(y_{1})$ on $\tilde{Y}^{\vee}$ and further to the Plücker
hyperplane $[V_{5}]\in Y\subset\mathbb{P}(V_{6}^{\vee})$, by [14, Lemma 4.12].
In conclusion, we have the following diagram.

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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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(3.1)

where $P$ is the universal conic. The fiber of $\alpha$ for $y\neq
y_{1},y_{2}$ is $\mathbb{P}^{1}$ and hence $\alpha$ induces an isomorphism

$\alpha_{*}:\operatorname{CH}_{0}(F(X))\cong\operatorname{CH}_{0}(\tilde{Y}^{\vee}).$

We will frequently use some maps between Chow groups: Let
$\Phi=p_{*}q^{*}:\operatorname{CH}^{*}(X)\rightarrow\operatorname{CH}^{*}(F)$
and $\Psi$ be the composition of $q_{*}p^{*}$ and $\alpha_{*}^{-1}$:

$\Psi=q_{*}p^{*}\alpha_{*}^{-1}:\operatorname{CH}_{0}(\tilde{Y}^{\vee})\rightarrow\operatorname{CH}_{1}(X),$

sending $\alpha(c)$ to its class $[c]\in\operatorname{CH}_{1}(X)$.

The involution $\iota$ can be described generically as follows. For
$y\in\tilde{Y}^{\vee}$, we take a conic $c$ in the fiber $\alpha^{-1}(y)$ and
it is contained in $S_{V_{4}}\subset\mathbb{P}^{4}$. Then taking a hyperplane
in $\mathbb{P}^{4}$ containing $c$, it intersects with $S_{V_{4}}$ along
another conic $c^{\prime}$. Then

$\iota(y)=\alpha(c^{\prime}).$

One can check that for a different choice of $c$ and a hyperplane in
$\mathbb{P}^{4}$, the corresponding $\alpha(c^{\prime})$ is the same.

Finally, we need to describe the Hilbert scheme of conics $F(X_{3})$ on a
linear section $X_{3}$ of $X$. Let $B\subset\bigwedge^{3}V_{6}$ be the
Lagrangian subspace corresponding to $X_{3}$. Then

$Y^{\geq 2}_{B^{\vee}}\subset Y^{\geq 1}_{A^{\vee}}$

and $F(X_{3})$ is birational to the pull-back of $Y^{\geq 2}_{B^{\vee}}$ in
$\tilde{Y}^{\vee}$, see [14, Section 5.1].

######  Remark 3.5.

In [33, Section 2.4], it was claimed that the locus of $\sigma$-conics is
contracted to a constant cycle surface on $\tilde{Y}^{\vee}$. However, it is
not true, because, by Iliev and Manivel’s construction, the locus of
$\sigma$-conics should be contracted to a point.

###  3.6. The Beauville-Voisin conjecture

The Beauville-Voisin conjecture implies that the cycle class map of an IHS
variety is injective when restricted to the subring generated by divisors in
the Chow ring.

For a double EPW sextic, the Beauville-Voisin conjecture for zero cycles was
proved in [10] and [17]. The result is slightly stronger, that is, the cycle
class map restricted to the subring

$R^{*}(\tilde{Y}^{\vee}):=\langle\operatorname{CH}^{1}(\tilde{Y}^{\vee}),\operatorname{CH}^{2}(\tilde{Y}^{\vee})^{+},c_{j}(\tilde{Y}^{\vee})\rangle$
(3.2)

generated by divisors, $\iota$-invariant $2$-cycles and Chern classes is
injective for $i=4$. In particular, there is a distinguished 0-cycle $o$ on
$\tilde{Y}^{\vee}$. For a very general double EPW sextic,
$\operatorname{CH}^{1}(\tilde{Y}^{\vee})$ is generated by $h$, which is the
polarization induced by the double cover of the sextic hypersurface
$Y\subset\mathbb{P}^{5}$. Let $Z$ be the invariant locus of the involution.
The following relations have been obtained in [10, Proposition 6.1][10, Lemma
9.3]:

$3Z=15h^{2}-c_{2},Z^{2}=192,h^{4}=12,h^{2}Z=40.$ (3.3)

Moreover, the classes of $Z^{2},h^{4}$ and $h^{2}Z$ are all proportional to
the class $o$.

##  4\. A constant cycle surface on $\tilde{Y}^{\vee}$

This section is devoted to proving Theorem 1.1. In the following parts of the
paper, we assume that $X$ is general.

###  4.1. Two facts about $\operatorname{CH}_{1}(X)$

We first show that $\operatorname{CH}_{1}(X)$ is rationally generated by
conics on $X$ and those conics represented by points on the invariant surface
$Z$ have the same class in $\operatorname{CH}_{1}(X)$.

######  Proposition 4.2.

The group $\operatorname{CH}_{1}(X)_{\mathbb{Q}}$ is generated by conics and
$\operatorname{CH}_{1}(X)$ is generated by lines on $X$.

###### Proof.

We first show that $X$ is connected by chains of conics. Every two general
points $x_{1},x_{2}$ on $X$ correspond to two $2$-dimensional subspaces in
$V_{5}$, hence the two subspaces are in some $4$-dimensional $V_{4}\subset
V_{5}$. Thus the two points are contained in $S_{V_{4}}$, which is a degree
$4$ Del Pezzo surface. Choosing two conic fibrations of $S_{V_{4}}$, we see
that the $x_{1}$ and $x_{2}$ are connected by a chain of $3$ conics. Hence by
[27, Proposition 3.1], there is a non-zero integer $N$, such that for any
1-cycle $D$ on $X$,

$N\cdot D\sim{\rm sum~{}of~{}conics}.$

Hence $\operatorname{CH}_{1}(X)_{\mathbb{Q}}$ is generated by conics.

For integral coefficients, it has been shown in [27, Theorem 1.3] that
$\operatorname{CH}_{1}(X)$ is generated by rational curves. By bend and break,
a rational curve $C$ of degree $\geq 3$ on $X$ is algebraically equivalent to
a sum of lower-degree rational curves. Moreover, a conic on a Del Pezzo
surface is linearly equivalent to a sum of lines, therefore $C$ is
algebraically equivalent to a sum of lines $\sum_{i}n_{i}L_{i}$. The argument
in [27, Lemma 3.4] shows that $\operatorname{CH}_{1}(X)_{\rm alg}$ is
divisible. Therefore there exists a 1-cycle $C^{\prime}$ such that

$N\cdot C^{\prime}=C-\sum_{i}n_{i}L_{i}\in\operatorname{CH}_{1}(X)_{\rm alg},$

since $N\cdot C^{\prime}$ is rationally equivalent to a linear combination of
lines, we find that $\operatorname{CH}_{1}(X)$ is generated by lines with
$\mathbb{Z}$-coefficients. ∎

######  Lemma 4.3.

For $y\in\tilde{Y}^{\vee}$, we have $\Psi(y+\iota(y))$ is constant in
$\operatorname{CH}_{1}(X)$. In particular, $\Psi(2y)$ is constant in
$\operatorname{CH}_{1}(X)$ if $y$ is on the invariant surface $Z$.

###### Proof.

Since any $0$-cycle is rationally equivalent to some $0$-cycle supported on an
open subscheme, we may assume that $y\in\tilde{Y}^{\vee}$ is a general point
such that $y\notin Z$. We recall the description of the map

$F(X)\xrightarrow{\alpha}\tilde{Y}^{\vee}\rightarrow Y^{\vee},$

in [14, Proposition 4.9]. Then the conics in $\alpha^{-1}(y)$ are identical to
conics in a linear system in $S_{V_{4}}$ for some $V_{4}$. The involution
$\iota$ sends conics in $S_{V_{4}}$ to their residual conics.

It suffices to show that for a general $V_{4}$ and any $c,c^{\prime}$ residual
in $S_{V_{4}}$, the sums $c+c^{\prime}$ are constant in
$\operatorname{CH}_{1}(X)$, which follows from the fact that the hyperplane
sections in such $S_{V_{4}}$’s are parameterized by a $\mathbb{P}^{3}$-bundle
over $\mathbb{P}(V_{5}^{\vee})$. ∎

###  4.4. Relations in $\operatorname{CH}^{*}(F)$ of incidences

We introduce an incidence $I\in\operatorname{CH}^{2}(F\times F)$ and compute
$I^{2}\in\operatorname{CH}^{4}(F\times F)$, where $F=F(X)$. Denote by $c_{s}$
the corresponding conic to $s\in F$. The morphisms are the same as in diagram
(3.1).

######  Definition 4.5.

  1. (1)

The incidence subscheme $I\subset F\times F$ is defined to be

$I:=\\{(s,t)\in F\times F~{}|~{}c_{s}\cap c_{t}\neq\emptyset\\},$

which is endowed with the reduced closed subscheme structure and of dimension
$8$.

  2. (2)

$D_{c}=p(q^{-1}(c))\subset F$ is the locus in $F$ consisting of conics
intersecting the given conic $c$ and $D_{c}^{\prime}=q^{-1}(c)$.

  3. (3)

$\Sigma_{1}\subset F\times F$ is defined to be

$\Sigma_{1}:=\\{(s,t)\in F\times
F~{}|~{}c_{s}~{}\text{and}~{}c_{t}~{}\text{are union of two lines and have a
common component}\\},$

which is of dimension $5$.

  4. (4)

$\Sigma_{2}\subset F\times F$ consists of $(s,t)\in F\times F$, where neither
$c_{s}$ nor $c_{t}$ is of $\tau$-type.

  5. (5)

$W\subset F\times F$ is defined to be

$W:=\\{(s,t)~{}|~{}\exists V_{4},c_{s}~{}\text{and}~{}c_{t}~{}\text{are
residual in}~{}S_{V_{4}}\\}.$

For general $s,t$, that means the two conics meet at two points. Since a conic
has a pencil of residual conics in $S_{V_{4}}$, we have $\text{dim}~{}W=6$.

######  Lemma 4.6.

For a general GM fourfold $X$, two different conics $c_{1}$ and $c_{2}$ span
different planes. If $c$ is of $\tau$-type, then $\langle
c\rangle\cap\operatorname{Gr}(2,V_{5})=c$.

###### Proof.

A general $X$ contains no planes. Since $X$ is the intersection of quadrics,
$\langle c\rangle$ is not contained in some quadric $Q_{0}\supset X$. Hence,
$\langle c\rangle\cap X\subset Q_{0}\cap\langle c\rangle=c$, which means that
$\langle c\rangle$ determines $c$.

The second statement follows from the fact that the plane spanned by a
$\tau$-conic is not contained in $\operatorname{Gr}(2,V_{5})$ and
$\operatorname{Gr}(2,V_{5})$ is also the intersection of quadrics. ∎

According to Lemma 4.6, we obtain an injective morphism

$\displaystyle F\rightarrow\operatorname{Gr}(3,10),$ (4.1)

sending $t\in F$ to $\langle
c_{t}\rangle\subset\mathbb{P}(\bigwedge^{2}V_{5})$. Let
$\mathbb{P}=\mathbb{P}(E)$ be the universal plane of $\operatorname{Gr}(3,10)$
restricted to $F$, we have the following diagram:

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38.43922pt\raise-68.30444pt\hbox{\hbox{\kern 0.0pt\raise
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59.4033pt\raise-33.32582pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.38196pt\hbox{$\scriptstyle{q^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
69.97394pt\raise-39.06888pt\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
69.97394pt\raise-39.06888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{P}(\bigwedge^{2}V_{5})}$}}}}}}}{\hbox{\kern-3.0pt\raise-78.13776pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
31.5295pt\raise-78.13776pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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(4.2)

######  Lemma 4.7.

The evaluation morphism $q$ is flat for a general $X$. In particular, the
class $I=(p\times p)_{*}(q\times
q)^{*}\Delta_{X}\subset\operatorname{CH}^{2}(F\times F)$. Equivalently, the
correspondence $I=$ ${}^{t}P\circ P$.

###### Proof.

The universal conic $P$ is Cohen-Macaulay since it is a divisor in a smooth
variety $\mathbb{P}$, hence it suffices to show that the dimensions of fibers
of $q$ are constant. Suppose on the contrary that there is a point $x\in X$
such that $q^{-1}(x)$ has a component $K$ of dimension $\geq 3$. We claim that

######  Claim.

There is a smooth conic $c$ passing through $x$ such that $(c,x)\in P$ belongs
to $K$ and $c$ is of $\tau$-type or $\rho$-type.

######  Proof of Claim.

Assume that $x=V_{2}$. It suffices to show that the dimension of the locus of
$\sigma$-conics and singular conics passing through $x$ is less than $3$.

In fact, the $\sigma$-conics are parameterized by the planes

$\mathbb{P}(V_{1}\bigwedge V_{4})\subset H,$

where $V_{1}\subset V_{4}$. When $V_{1}\neq W_{1}$, the locus of
$\sigma$-conics passing through $x$ are equal to the image of
$\mathbb{P}(V_{2})$ in $Bl_{[W_{1}]}(\mathbb{P}(V_{5}))$, which is
$1$-dimensional. When $W_{1}=V_{1}$, a $\sigma$-conic passing through $x$
corresponds to a plane $\langle c\rangle=\mathbb{P}(V_{1}\bigwedge V_{4})$
with $V_{2}\subset V_{4}\subset V_{5}$, and there is a $2$-dimensional choice
of $V_{4}$. For singular conics, there are at most $1$-dimensional lines
passing through a point on $X$ by [6, Theorem 4.7]. Therefore the claim
follows.

The fiber $p^{-1}(c)=c\subset P$ can be identified with $c\subset X$ via $q$
and

$N_{c/P}=\mathrm{H}^{0}(N_{c/X})\otimes\mathcal{O}_{c}.$

Considering the differential of $q$ restricted to $c\subset P$, we obtain the
following diagram, see also [16, section 2.3]:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
176.42519pt\raise-41.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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The normal bundle
$N_{c/X}=\mathcal{O}_{c}\oplus\mathcal{O}_{c}(1)\oplus\mathcal{O}_{c}(1)$ for
a smooth conic of $\tau$-type or $\rho$-type. Therefore,

$T_{(c,x)}q^{-1}(x)=\\{s\in\mathrm{H}^{0}(N_{c/X})~{}|~{}s_{x}=0\\},$

which is $2$-dimensional, contradicting to the fact that $\mathrm{dim}~{}K\geq
3$.

∎

######  Lemma 4.8.

Let $c_{1}$ and $c_{2}$ be two different conics on a general $X$ and assume
that at least one of them is of $\tau$-type. If $c_{1}$ and $c_{2}$ have no
common component and $\langle c_{1}\rangle\cap\langle c_{2}\rangle$ is a line
$l$, there exists a $4$-dimensional subspace $V_{4}\subset V_{5}$, such that
$\langle c_{1}\rangle\cup\langle
c_{2}\rangle\subset\mathbb{P}(\bigwedge^{2}V_{4})$. In particular,
$\iota(\alpha([c_{1}]))=\alpha([c_{2}])$.

###### Proof.

A general $X$ contains no planes and hence $l\not\subset X$. The conic $c_{i}$
is contained in $\mathbb{P}(\bigwedge^{2}V_{4i})$ for some $4$-dimensional
vector space $V_{4i}$. If $V_{41}\neq V_{42}$, then $V_{3}:=V_{41}\cap V_{42}$
is $3$-dimensional and

$l\subset\mathbb{P}(\bigwedge^{2}V_{3})\subset\operatorname{Gr}(2,V_{5}).$

In particular, $\langle c_{i}\rangle\cap\operatorname{Gr}(2,V_{5})\neq c_{i}$.
Therefore by Lemma 4.6, we have $\langle
c_{i}\rangle\subset\operatorname{Gr}(2,V_{5})$ and it means that the two
conics are of $\sigma$-type or $\rho$-type, contradicting to the fact that one
of the two conics is of $\tau$-type.

Hence $V_{41}=V_{42}=V_{4}$ and $c_{1},c_{2}$ are contained in $S_{V_{4}}$.
Then $\iota(\alpha(c_{1}))=\alpha(c_{2})$ follows from that $\iota$ sends a
conic to its residual conic in $S_{V_{4}}$. ∎

We define $\tilde{I}\subset P\times P$ and
$I^{\prime}\subset\mathbb{P}\times\mathbb{P}$ to be the incidence
correspondences, i.e.

$\tilde{I}=(q\times
q)^{-1}\Delta_{X}{\rm~{}and~{}}I^{\prime}=(q^{\prime}\times
q^{\prime})^{-1}\Delta_{\mathbb{P}(\bigwedge^{2}V_{5})}.$

Let $I_{0}=I\setminus(\Sigma_{1}\cup\Sigma_{2}\cup W)$,
$\tilde{I_{0}}=(p\times p)^{-1}I_{0}$ and $I_{0}^{\prime}=(p^{\prime}\times
p^{\prime})^{-1}I_{0}$. Then Lemma 4.8 says that there is a section from
$I_{0}$ to $I^{\prime}_{0}$, sending two intersecting conics to their common
point.

######  Lemma 4.9.

The inclusion $I_{0}\subset F\times F\setminus(\Sigma_{1}\cup\Sigma_{2}\cup
W)$ is a regular embedding.

###### Proof.

The inclusion $I^{\prime}\subset\mathbb{P}\times\mathbb{P}$ is a regular
embedding since it is obtained from
$\Delta_{\mathbb{P}(\bigwedge^{2}V_{5})}\subset\mathbb{P}(\bigwedge^{2}V_{5})\times\mathbb{P}(\bigwedge^{2}V_{5})$
via base change. By the discussion above, there exists a section from $I_{0}$
to $I^{\prime}_{0}$. We apply [13, B.7.5] and obtain that $I_{0}\subset
F\times F\setminus(\Sigma_{1}\cup\Sigma_{2}\cup W)$ is a regular embedding. ∎

Now we can compute the self-intersection $I^{2}$, the analogous case of cubic
fourfolds can be found in [30, Proposition 2.3] or [12, Appendix A.4].

######  Proposition 4.10.

The self-intersection

$I^{2}=aW+I\cdot A+B+C\in\operatorname{CH}^{4}(F\times F)$

for some constant $a$. Here, $A$ and $B$ are contained in
$pr_{1}^{*}\operatorname{CH}^{*}(F)\cdot
pr_{2}^{*}\operatorname{CH}^{*}(F)\subset\operatorname{CH}^{*}(F\times F)$,
where $pr_{i}$ is the projection and $C$ is supported on $\Sigma_{2}\subset
F\times F$.

###### Proof.

By Lemma 4.9, $I_{0}\subset F\times F\setminus(\Sigma_{1}\cup\Sigma_{2}\cup
W)$ is a regular embedding and the restriction $p_{0}$ of $p$ from
$\tilde{I_{0}}\subset P\times P$ to $I_{0}\subset F\times F$ is an
isomorphism. Then,

$I_{0}^{2}=c_{2}(N_{I_{0}/F\times F})$

and

$0\rightarrow pr_{1}^{*}T_{P/F}\oplus pr_{2}^{*}T_{P/F}\rightarrow
N_{\tilde{I}_{0}/P\times P}\rightarrow p_{0}^{*}(N_{I_{0}/F\times
F})\rightarrow 0.$

Let $q_{0}$ be the map $q_{0}:\tilde{I_{0}}\rightarrow\Delta_{X}=X$ which maps
two conics in $\tilde{I_{0}}$ to their common point. Since $\tilde{I}=(q\times
q)^{-1}\Delta_{X}$, we have

$N_{\tilde{I}_{0}/P\times
P}=q_{0}^{*}(T_{X})=pr_{1}^{*}(T_{X})\cdot\tilde{I}_{0}=pr_{2}^{*}(T_{X})\cdot\tilde{I}_{0}$

and

$c(p_{0}^{*}(N_{I_{0}/F\times
F}))=\frac{q_{0}^{*}c(T_{X})}{pr_{1}^{*}c(T_{P/F})\cdot
pr_{2}^{*}c(T_{P/F})}.$ (4.4)

Let $K=\sigma_{1,1}|_{X}$ be the class of $S_{V_{4}}$ and $H=\sigma_{1}|_{X}$
is the hyperplane class. We claim that $c_{2}(p_{0}^{*}(N_{I_{0}/F\times F}))$
is a degree 2 polynomial with variables in
$pr_{1}^{*}\operatorname{CH}^{*}(F)\cdot pr_{2}^{*}\operatorname{CH}^{*}(F)$
and $pr_{1}^{*}\langle H,K\rangle\cdot pr_{2}^{*}\langle H,K\rangle$
restricting to $\tilde{I}_{0}$. The following computations is standard:

  1. (1)

The Chern classes of $X$

$c(T_{X})=\frac{c(T_{\operatorname{Gr}(2,V_{5})})|_{X}}{(1+H)(1+2H)},$

where $c(T_{\operatorname{Gr}(2,V_{5})})$ is a polynomial in Schubert cycles.
Therefore, the degree $1$ and $2$ parts of $q_{0}^{*}c(T_{X})$ are polynomials
in $H$ and $\sigma_{1,1}$ via the two projections and restriction to
$\tilde{I}_{0}$.

  2. (2)

The Chern classes of the relative tangent bundle $T_{P|F}$

$c(T_{P|F})=\frac{c(T_{\mathbb{P}/F})|_{P}}{c(N/\mathbb{P})}=\frac{c(p^{*}(E)\otimes\mathcal{O}_{\mathbb{P}}(1))|_{P}}{c(\mathcal{O}_{P}(P))},$

where $c_{1}(\mathcal{O}_{\mathbb{P}}(1))|_{X}=H$ and
$P=2c_{1}(\mathcal{O}_{\mathbb{P}}(1))+p^{*}D\in\operatorname{CH}^{1}(\mathbb{P})$
for some $D\in\operatorname{CH}^{1}(F)$. So the degree $1$ and degree $2$
parts are polynomials in $H$ and $\operatorname{CH}^{*}(F)$.

The claim follows from the computations and (4.4). Next, the terms in
$p_{0}^{*}c_{2}(N_{I_{0}/F\times F})$ are of the following three types:

  1. (1)

$D\cdot\tilde{I}_{0}$, where $D\in pr_{1}^{*}\operatorname{CH}^{*}(F)\cdot
pr_{2}^{*}\operatorname{CH}^{*}(F)$.

  2. (2)

$(q\times q)^{*}pr_{i}^{*}H\cdot D\cdot\tilde{I}_{0}$ for $i=1$ or $2$, where
$D$ is of form $(q\times q)^{*}pr_{j}^{*}H$ or $(p\times
p)^{*}pr_{j}^{*}D_{1}$ for some $D_{1}\in\operatorname{CH}^{1}(F)$.

  3. (3)

$(q\times q)^{*}pr_{i}^{*}K\cdot\tilde{I}_{0}$ for $i=1$ or $2$.

Applying $p_{0*}$ to $p_{0}^{*}c_{2}(N_{I_{0}/F\times F})$, the terms of the
first type are of the form $I\cdot A$ by projection formula. For the other two
types, we use $\tilde{I}=(q\times q)^{*}\Delta_{X}$ and it’s enough to show
that $pr_{i}^{*}H\cdot pr_{j}^{*}H\cdot\Delta_{X}$ and
$pr_{i}^{*}K\cdot\Delta_{X}$ are in $pr_{1}^{*}\operatorname{CH}^{*}(X)\cdot
pr_{2}^{*}\operatorname{CH}^{*}(X)$. Denote by
$M=\operatorname{Gr}(2,V_{5})\cap H$ the Grassmanian hull of $X$ and consider

$j_{1}:X\times X\hookrightarrow M\times X.$

We know that $M$ is a linear variety (Schubert varieties are linear in the
sense of [28], since they are stratified by Schubert cells) and therefore by
[28, Proposition 1],

$\operatorname{CH}^{*}(M\times X)=pr_{1}^{*}\operatorname{CH}^{*}(M)\cdot
pr_{2}^{*}\operatorname{CH}^{*}(X).$

Hence we have

$j_{1}^{*}j_{1*}\Delta_{X}\in pr_{1}^{*}\operatorname{CH}^{*}(X)\cdot
pr_{2}^{*}\operatorname{CH}^{*}(X)$

Since $X$ is of class $2H$ in $M$, we have
$j_{1}^{*}j_{1*}\Delta_{X}=2pr_{1}^{*}H\cdot\Delta_{X}$. Hence
$pr_{1}^{*}H\cdot\Delta_{X}\in pr_{1}^{*}\operatorname{CH}^{*}(X)\cdot
pr_{2}^{*}\operatorname{CH}^{*}(X)$.

For the third type, we recall that the class $K\in\operatorname{CH}^{2}(X)$ is
represented by a Del Pezzo surface $S=S_{V_{4}}$ and
$pr_{i}^{*}K\cdot\Delta_{X}$ is the push-forward of the diagonal class
$\Delta_{S}$ in $\operatorname{CH}^{2}(S\times S)$. Then by the decomposition
of diagonal, we see that

$\Delta_{S}=\alpha\times S+\Gamma,$

where $\alpha\in\operatorname{CH}_{0}(S)$ and $\Gamma$ is supported on
$S\times V$ for some proper subvariety $V\subset S$. If $\dim V=0$, $\Gamma$
is of the form $S\times\beta$ for some $\beta\in\operatorname{CH}_{0}(X)$. If
$\dim V=1$, then $\Gamma\in\operatorname{CH}^{1}(V\times X)$. We see that
$\operatorname{CH}^{1}(V\times X)=pr_{1}^{*}\operatorname{CH}^{1}(V)\cdot
pr_{2}^{*}\operatorname{CH}^{1}(S)$ since ${\rm H}^{1}(S,\mathcal{O}_{S})=0$.
In both cases, we obtain:

$\Delta_{S}\in pr_{1}^{*}\operatorname{CH}^{*}(S)\cdot
pr_{2}^{*}\operatorname{CH}^{*}(S),$

which implies that $pr_{i}^{*}K\cdot\Delta_{X}$ are in
$pr_{1}^{*}\operatorname{CH}^{*}(X)\cdot pr_{2}^{*}\operatorname{CH}^{*}(X)$.

Finally, we get the conclusion by the localization exact sequence for Chow
groups.

∎

###  4.11. $Z$ is a constant cycle surface

Now we can prove that $Z$ is a constant cycle surface on $\tilde{Y}^{\vee}$,
and a point on $Z$ represents the distinguished 0-cycle $o$ that appeared in
(3.2). First, we notice that

$\Phi([c])=I_{*}([c])=D_{c}$

since $q$ is flat, where we recall that $\Phi=p_{*}q^{*}$. Hence the class of
$D_{c}$ depends only on the class $[c]\in\operatorname{CH}_{1}(X)$.

For a general $V_{4}$, $S_{V_{4}}$ is isomorphic to the blow-up of
$\mathbb{P}^{2}$ at five points. Let $H$ be the pullback of the hyperplane
class on $\mathbb{P}^{2}$ and $E_{i}$, $i=1,\dots 5$, be the five exceptional
curves. There are $10$ pencils of conics on $S_{V_{4}}$, which are $H-E_{i}$
and $2H-\sum_{j\neq i}E_{j}$. Denote by $L_{i},~{}i=1,\dots 10$ the
corresponding lines on $F$ and let $L_{i+10}=L_{i}$. Recall that $L_{i}$’s are
contracted to a point by $\alpha$ and $\alpha(L_{i})=\iota(\alpha(L_{i+5}))$.
By a straightforward computation in $S_{V_{4}}$, we have

$c_{i}\cdot c_{j}=1~{}{\rm for}~{}j\neq i,i+5;c_{i}\cdot c_{i+5}=2~{}{\rm
and}~{}c_{i}^{2}=0~{}{\rm for~{}any}~{}i.$ (4.5)

Here, $c_{i}$ is a conic in $L_{i}$. That is, conics in $L_{i+5}$ meet $c_{i}$
at two points and conics in $L_{j},j\neq i,i+5$ meet $c_{i}$ at one point. We
have the following computation:

######  Lemma 4.12.

Let $c_{i}$ be a conic on a general $S_{V_{4}}$ and $c_{i}\in L_{i}$, then

$D_{c_{i}}^{2}=\sum_{j\neq i,j\neq
i+5}L_{j}+4L_{i+5}+kL_{\sigma}\in\operatorname{CH}_{1}(F),$ (4.6)

where $L_{\sigma}\subset F$ is a curve contained in the locus of
$\sigma$-conics.

###### Proof.

We may assume that $S_{V_{4}}$ is general so that $W_{1}\not\subset V_{4}$ and
$S_{V_{4}}$ contains no $\sigma$-conics. Take another conic $c_{0}\in|c_{i}|$
and thus $D_{c_{0}}=D_{c_{i}}$ in $\operatorname{CH}^{2}(F)$. The intersection
$D_{c_{i}}\cap D_{c_{0}}$ consists of conics $c$ which meet both $c_{i}$ and
$c_{0}$. We distinguish between the two cases whether $c$ is contained in
$S_{V_{4}}$ or not.

If $c\in D_{c_{i}}\cap D_{c_{0}}$ and $c\subset S_{V_{4}}$, it is contained in
another linear system of conics, supported on $L_{j}$, for some $j\neq i$.
Conversely, $L_{j}\subset{\rm Supp}~{}D_{c_{i}}\cap D_{c_{0}}$ for $j\neq i$
by (4.5).

If $c\in D_{c_{i}}\cap D_{c_{0}}$ and $c\not\subset S_{V_{4}}$, we first show
that $c$ must be a $\sigma$-conic. Let $x\in c\cap c_{0}$ , $y\in c\cap c_{i}$
and $V_{x},V_{y}$ be the corresponding 2-dimensional subspaces in $V_{5}$. We
have $V_{x}\cup V_{y}$ is a three-dimensional space $V_{3}$, otherwise $c$
would be contained in $S_{V_{4}}$. It follows that $V_{x}\cap V_{y}$ is a one-
dimensional space $V_{1}$. Therefore

$\mathbb{P}(V_{1}\bigwedge V_{3})\subset\operatorname{Gr}(2,V_{5})\cap\langle
c\rangle.$

By Lemma 4.6, $c$ is a $\sigma$-conic or a $\rho$-conic. Since
$W_{1}\not\subset V_{4}$, we see that $W_{1}\not\subset V_{3}$ and $c$ is not
a $\rho$-conic. Therefore $c$ is a $\sigma$-conic.

Recall that $\sigma$-conics are parameterized by
$Bl_{[W_{1}]}(\mathbb{P}(V_{5}))$. Let $S_{c_{0}}$ and $S_{c_{i}}$ be the
surfaces swept out by lines parameterized by $c_{0}$ and $c_{i}$ in
$\mathbb{P}(V_{4})$. There is a natural morphism

$S_{c_{0}}\cap S_{c_{i}}\rightarrow Bl_{[W_{1}]}(\mathbb{P}(V_{5})),$

and the image is exactly the locus of $\sigma$-conics meeting $c_{i}$ and
$c_{0}$ by the previous discussion. The intersection $S_{c_{0}}\cap S_{c_{i}}$
is clearly one dimensional. Therefore the conics in $D_{c_{i}}\cap D_{c_{0}}$
not contained in $S_{V_{4}}$ are parameterized by a curve $L_{\sigma}\subset
F$.

Thus $D_{c_{i}}$ and $D_{c_{0}}$ intersect dimensionally transversely and it
remains to compute the multiplicity. We consider the differential $dq$ as in
(4.3). Let $c$ be a general smooth conic in $D_{c_{i}}\cap D_{c_{0}}$ and
$c\in L_{j},j\neq i,i+5$. Then $c$ meets $c_{i}$ (resp. $c_{0}$) at one point
$x$ (resp. $y$). We have $T_{x}c\subset T_{x}S_{V_{4}}$, hence the image of
$T_{x}c$ in $N_{c/X}$ is $N_{c/S_{V_{4}},x}$. Then by (4.3), we have

$T_{(c,x)}D_{c_{i}}^{\prime}=dq^{-1}(T_{x}c_{i})=\mathrm{H}^{0}(N_{c/X}\otimes
I_{x})\oplus\mathrm{H}^{0}(N_{c/S_{V_{4}},x}),$

which is $3$-dimensional. Thus $D_{c_{i}}^{\prime}$ is smooth at $(c,x)$.
Since $c_{i}$ and $c$ meet at a single point, $D_{c_{i}}^{\prime}$ maps
isomorphically to $D_{c_{i}}$ via $p$ around $c$. Hence,

$T_{c}D_{c_{i}}=\mathrm{H}^{0}(N_{c/X}\otimes
I_{x})\oplus\mathrm{H}^{0}(N_{c/S_{V_{4}}})\subset T_{c}F.$

Similarly for $c_{0}$, we obtain

$T_{c}D_{c_{0}}=\mathrm{H}^{0}(N_{c/X}\otimes
I_{y})\oplus\mathrm{H}^{0}(N_{c/S_{V_{4}}})\subset T_{c}F.$

Since $\mathrm{H}^{0}(N_{c/X}\otimes I_{x}\otimes I_{y})=0$, we have

$T_{c}D_{c_{i}}\cap
T_{c}D_{c_{0}}=\mathrm{H}^{0}(N_{c/S_{V_{4}}})=T_{c}L_{j}.$

That means $D_{c_{i}}$ and $D_{c_{0}}$ meet transversely at $c$, hence the
multiplicity of $L_{j}$ is $1$ for $j\neq i,i+5$.

For a general conic $c\in D_{c_{i}}\cap D_{c_{0}}\cap L_{i+5}$, $c$ meet
$c_{i}$ and $c_{0}$ at two points. Let $c\cap c_{i}=x_{1},x_{2}$ and $c\cap
c_{0}=y_{1},y_{2}$. Then $p:D_{c_{i}}^{\prime}\rightarrow D_{c_{i}}$ (resp.
$D_{c_{0}}^{\prime}\rightarrow D_{c_{0}}$) maps the two points $(c,x_{1})$ and
$(c,x_{2})$ (resp. $(c,y_{1})$ and $(c,y_{2})$ ) to $c$. We obtain that the
tangent cones

$\begin{split}C_{c}D_{c_{i}}&=\mathrm{H}^{0}(N_{c/X}\otimes
I_{x_{1}})\oplus\mathrm{H}^{0}(N_{c/S_{V_{4}}})\cup\mathrm{H}^{0}(N_{c/X}\otimes
I_{x_{2}})\oplus\mathrm{H}^{0}(N_{c/S_{V_{4}}})\\\
C_{c}D_{c_{0}}&=\mathrm{H}^{0}(N_{c/X}\otimes
I_{y_{1}})\oplus\mathrm{H}^{0}(N_{c/S_{V_{4}}})\cup\mathrm{H}^{0}(N_{c/X}\otimes
I_{y_{2}})\oplus\mathrm{H}^{0}(N_{c/S_{V_{4}}}).\end{split}$ (4.7)

Take a general hypersurface $G\subset F$ containing $c$ such that $G$ meets
$L_{i+5}$ transversely at $c$ and then the multiplicities

${\rm mult}_{c}D_{c_{i}}|_{G}={\rm mult}_{c}D_{c_{0}}|_{G}=2.$

Combining the fact that the direction of $\mathrm{H}^{0}(N_{c/S_{V_{4}}})$ is
in the fiber of $\alpha$, we obtain

$C_{c}D_{c_{i}}|_{G}\cap C_{c}D_{c_{0}}|_{G}=0$

by (4.7) and the description of the normal bundle of conics. By [8,
Proposition 1.29], the intersection multiplicity of $D_{c_{i}}|_{G}$ and
$D_{c_{i}}|_{G}$ at $c$ is $4$. It follows that the multiplicity of the
intersection of $D_{c_{0}}$ and $D_{c_{i}}$ at $L_{i+5}$ is $4$. ∎

Apply this Lemma to every $c_{i}\subset S_{V_{4}}$ and by a linear
combination, we have:

$6(D_{c_{i}}^{2}+D_{c_{i+5}}^{2})-\sum_{j=1}^{10}D_{c_{j}}^{2}=12(L_{i}+L_{i+5})+L^{\prime},$
(4.8)

where $L^{\prime}$ is a 1-cycle on $F$ consisting of $\sigma$-conics.

######  Lemma 4.13.

The constant $a$ in Proposition 4.10 is nonzero. If $n[c_{1}]\sim n[c_{2}]$ in
$\operatorname{CH}_{1}(X)$ for some nonzero integer $n$, then
$[c_{1}]\sim[c_{2}]$ in $\operatorname{CH}_{0}(\tilde{Y}^{\vee})$.

###### Proof.

We assume on the contrary that $a=0$. Then

$I^{2}=I\cdot A+B+C\in\operatorname{CH}^{4}(F\times F),$

where $A,B,C$ are as in Proposition 4.10. A correspondence $\Gamma$ of
codimension larger than $0$ in $pr_{1}^{*}\operatorname{CH}^{*}(F)\cdot
pr_{2}^{*}\operatorname{CH}^{*}(F)$ is of the form $[Z_{1}\times Z_{2}]$,
where either $Z_{1}$ or $Z_{2}$ is a proper subvariety of $F$. In both cases,
$\Gamma$ induces a constant map
$\operatorname{CH}_{0}(F)\rightarrow\operatorname{CH}^{*}(F)$. Further, by
Lemma 4.7, $I=$ ${}^{t}P\circ P$ and hence
$I_{*}:\operatorname{CH}_{0}(F)\rightarrow\operatorname{CH}_{3}(F)$ factor
through $\operatorname{CH}_{1}(X)$. Together with the torsion-freeness of
$\operatorname{CH}_{0}(F)$, it implies that if $a=0$, then for
$\xi,\zeta\in\operatorname{CH}_{0}(F)$,

$I^{2}_{*}(\xi)=I^{2}_{*}(\zeta)$ (4.9)

as long as $q_{*}p^{*}(n\xi)=q_{*}p^{*}(n\zeta)\in\operatorname{CH}_{1}(X)$,
where $p,q$ are the projections from the universal conic to $F$ and $X$.

According to [26, Lemma 17.3], we know that $I^{2}_{*}([c])=D_{c}^{2}$. Then
(4.9) together with Lemma 4.3 imply that $D_{c}^{2}+D_{\iota(c)}^{2}$ are
constant in $\operatorname{CH}_{1}(F)$ when varying $c\in F$. We want to
deduce a contradiction between the formula (4.8) and the constantness of
$D_{c}^{2}+D_{\iota(c)}^{2}$.

Take a very ample divisor $G\subset F$. Intersecting both sides of formula
(4.8) with $G$ and pushing forward to $\tilde{Y}^{\vee}$, then the
constantness of $D_{c}^{2}+D_{\iota(c)}^{2}$ implies that $([c]+\iota([c]))$
is constant in $\operatorname{CH}_{0}(\tilde{Y}^{\vee})$, i.e., points on

$\Delta^{\prime}=\\{(y,\iota(y))~{}|~{}y\in\tilde{Y}^{\vee}\\}\subset\tilde{Y}^{\vee}\times\tilde{Y}^{\vee}$

are constant in $\operatorname{CH}_{0}(\tilde{Y}^{\vee})$.

Then for any power $\sigma^{l}$ of the two-form $\sigma$ on
$\tilde{Y}^{\vee}$, we have
$(pr_{1}^{*}(\sigma^{l})+pr_{2}^{*}(\sigma^{l}))|_{\Delta^{\prime}}=0$ by
Mumford’s theorem, see [29, Proposition 10.24]. Then we obtain

$2pr_{1}^{*}(\sigma^{2})|_{\Delta^{\prime}}=0,$

but it is impossible since $pr_{1}$ is an isomorphism between
$\Delta^{\prime}$ and $\tilde{Y}^{\vee}$. Therefore, $a$ is nonzero.

For the second part, assume that $n[c_{1}]\sim n[c_{2}]$ and again take a very
ample divisor $G\subset F$. Let $m$ be the intersection number of $G$ and a
general fiber of $\alpha:F\rightarrow\tilde{Y}^{\vee}$. Then we have
$n^{2}I^{2}_{*}([c_{1}])=n^{2}D_{c_{1}}^{2}=n^{2}D_{c_{2}}^{2}=n^{2}I^{2}_{*}([c_{2}])$
again by [26, Lemma 17.3] and $nI_{*}([c_{1}])=nI_{*}([c_{2}])$. Hence due to
Proposition 4.10 ,

$\begin{split}0=&G\cdot n^{2}(I^{2}_{*}([c_{1}])-I^{2}_{*}([c_{2}]))\\\
=&G\cdot n^{2}(aW+I\cdot A+B+C)([c_{1}]-[c_{2}])\\\ =&G\cdot
an^{2}W([c_{1}]-[c_{2}])\\\
=&amn^{2}\cdot(\iota([c_{1}])-\iota([c_{2}])),\end{split}$

here we view the cycles in $\operatorname{CH}^{4}(F\times F)$ as morphisms
from $\operatorname{CH}_{0}(F)$ to $\operatorname{CH}_{1}(F)$. Then

$\iota([c_{1}])\sim\iota([c_{2}])$

again by the torsion-freeness of $\operatorname{CH}_{0}(\tilde{Y}^{\vee})$ and
the fact that $a\neq 0$. Since $\iota$ is an involution, the result follows.

∎

Combining Lemma 4.3 and Lemma 4.13, we can get the following result :

######  Theorem 4.14.

The invariant locus $Z$ of the involution is a constant cycle surface on
$\tilde{Y}^{\vee}$.

Thus by (3.2), $h^{2}\cdot Z=40o$ in
$\operatorname{CH}_{0}(\tilde{Y}^{\vee})$. That means $o$ is represented by a
point on the constant cycle surface $Z$. We show that $o$ is in fact the class
represented by a point on any constant cycle surface:

######  Proposition 4.15.

For a very general $\tilde{Y}^{\vee}$, if $Z^{\prime}\subset\tilde{Y}^{\vee}$
is another constant cycle surface, then for any point $z^{\prime}\in
Z^{\prime}$, $z^{\prime}$ is rationally equivalent to $o$.

###### Proof.

It suffices to show that for a very general $\tilde{Y}^{\vee}$, any two
surfaces on $\tilde{Y}^{\vee}$ intersect. We obtain this by showing that every
surface on $\tilde{Y}^{\vee}$ is strictly nef, i.e., having a positive
intersection number with any non-zero effective $2$-cycle.

Let $N_{2}(\tilde{Y}^{\vee})$ be the space of $2$-cycles modulo numerical
equivalence with $\mathbb{R}$-coefficients, which is a $2$-dimensional vector
space spanned by $h^{2}$ and $Z$. As in [21], the Lagrangian surface $Z$ is in
the boundary of the effective cone $\overline{Eff}_{2}(\tilde{Y}^{\vee})$, and
$c_{2}(\tilde{Y}^{\vee})$ is not contained in the interior of the effective
cone. Thus

$\overline{Eff}_{2}(\tilde{Y}^{\vee})\subset\langle\mathbb{R}_{\geq
0}(Z),\mathbb{R}_{\geq 0}(c_{2})\rangle.$

Then by (3.3), we have

$\begin{split}\langle\mathbb{R}_{\geq 0}(Z),\mathbb{R}_{\geq
0}(c_{2})\rangle&\subset\langle\mathbb{R}_{\geq
0}(24h^{2}-5Z),\mathbb{R}_{\geq 0}(-2h^{2}+5Z)\rangle\\\
&=(\langle\mathbb{R}_{\geq 0}(Z),\mathbb{R}_{\geq 0}(c_{2})\rangle)^{\vee}\\\
&\subset Nef_{2}(\tilde{Y}^{\vee}).\end{split}$

The first inclusion is such that the cones have no boundary in common by a
straightforward computation. Hence every surface on $\tilde{Y}^{\vee}$ is
strictly nef. ∎

##  5\. The filtration on $\text{CH}_{1}(X)$

###  5.1. Basics of the filtration

A uniruled divisor on a $2n$-dimensional IHS variety $M$ is a divisor $D$
which admits a rational map to a $(2n-2)$-dimensional variety $B$:

${D}$${M,}$${B}$$\scriptstyle{q}$

and the general fibers of $q$ are rational curves. By [2], there exists a
uniruled divisor on an IHS variety of $K3^{[n]}$-type with $n\leq 7$, in
particular, on the dual double EPW sextic $\tilde{Y}^{\vee}$.

The Beauville–Voisin filtration on $\tilde{Y}^{\vee}$ is determined by
uniruled divisors and constant cycle surfaces. Points on uniruled divisors
serve as the $1$st piece and points on constant cycle surfaces serve as the
$0$th piece. According to [24, Lemma 1.1] and Proposition 4.15, the filtration
does not depend on the choice of a uniruled divisor and a constant cycle
surface on a very general dual double EPW sextic.

We define a filtration on $\operatorname{CH}_{1}(X)$ for a general GM fourfold
induced by the Beauville–Voisin filtration on $\tilde{Y}^{\vee}$. In the rest
of the paper, we study some properties of the filtration.

######  Definition 5.2.

The $i$th piece of the increasing filtration ${\rm
S}_{\bullet}X\subset\operatorname{CH}_{1}(X)$ consists of
$z\in\operatorname{CH}_{1}(X)$, such that (i.e., the class of $az$ can be
represented by a sum of multiples of $i$ conics in a uniruled divisor and
$\theta$)

$az=a_{1}c_{1}+a_{2}c_{2}\ldots+a_{i}c_{i}+a_{0}\theta$

for some integer $a,a_{j}$, where $c_{j}$’s are conics with $\alpha(c_{j})$ in
a uniruled divisor $D\subset\tilde{Y}^{\vee}$ and $\theta:=\Psi(o)$ for $o$
the class of a point on the constant cycle surface $Z$.

The EPW sextics associated with period partners $X$ and $X^{\prime}$ are the
same. We have an immediate corollary according to the definition:

######  Corollary 5.3.

For any $t\in\operatorname{CH}_{0}(\tilde{Y}^{\vee})$, $\Psi(t)\in{\rm
S}_{i}(X)$ iff $\Psi^{\prime}_{*}(t)\in{\rm S}_{i}(X^{\prime})$, where
$\Psi^{\prime}:\operatorname{CH}_{0}(\tilde{Y}^{\vee})\rightarrow\operatorname{CH}_{1}(X^{\prime})$
is similarly defined as $\Psi$.

By the proof of Proposition 4.2, there exists an integer $N$ such that $N\cdot
z$ is a linear sum of conics for every $z\in\operatorname{CH}_{1}(X)$. At the
end of the section, we show in Proposition 5.10 that every conic is rationally
equivalent to a sum of conics in $D$. Hence,

$\bigcup_{i=0}^{\infty}{\rm S}_{i}(X)=\operatorname{CH}_{1}(X).$

Let $\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{-}_{\rm hom}$ be the
$\iota$-anti-invariant part of $\operatorname{CH}_{0}(\tilde{Y}^{\vee})_{\rm
hom}$ and $\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{+}_{\rm hom}$ be the
$\iota$-invariant part. There is a decomposition of
$\operatorname{CH}_{0}(\tilde{Y}^{\vee})$:

$\operatorname{CH}_{0}(\tilde{Y}^{\vee})=\mathbb{Z}\cdot
o\oplus\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{-}_{\rm
hom}\oplus\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{+}_{\rm hom}.$

We show that the filtration on $\operatorname{CH}_{1}(X)$ can actually be
defined on the $\iota$-anti-invariant part
$\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{-}$.

######  Proposition 5.4.

The morphism
$\Psi:\operatorname{CH}_{0}(\tilde{Y}^{\vee})\rightarrow\operatorname{CH}_{1}(X)$
is zero on $\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{+}_{\rm hom}$ and an
isomorphism onto $\operatorname{CH}_{1}(X)_{\rm hom}$ when restricting to
$\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{-}_{\rm hom}$.

###### Proof.

Due to the divisibility of $\operatorname{CH}_{0}(\tilde{Y}^{\vee})_{\rm
hom}$, the elements in $\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{+}_{\rm hom}$
are of the form $\sum(t_{i}+\iota(t_{i})-2o)$ and elements in
$\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{-}$ are of the form $\sum
t_{i}-\iota(t_{i})$, where $t_{i}\in\tilde{Y}^{\vee}$. This yields $\Psi$ is
zero on $\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{+}_{\rm hom}$ by Lemma 4.3.

For the second part, there exists an integer $N$, such that for any
$z\in\operatorname{CH}_{1}(X)$, $N\cdot z$ is a linear combination of conics.
Then $\operatorname{CH}_{1}(X)_{\rm hom}$ is generated by conics by the
divisibility of $\operatorname{CH}_{1}(X)_{\rm hom}$. Therefore, it suffices
to show that $\Psi$ is injective on
$\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{-}$.

Let $G\subset F$ be an ample divisor and $G.L=m$, where $L$ is a general fibre
of $\alpha:F\rightarrow\tilde{Y}^{\vee}$. Denote by $c_{i}$ and
$c_{i}^{\prime}$ the conics representing the class $\Psi(t_{i})$ and
$\Psi(t_{i}^{\prime})$. If we have the relation

$\Psi(\sum t_{i}-\iota(t_{i}))=\Psi(\sum
t_{i}^{\prime}-\iota(t_{i}^{\prime}))\in\operatorname{CH}_{1}(X),$

it implies that:

$\sum(D_{c_{i}}-D_{\iota(c_{i})})=\sum(D_{c_{i}^{\prime}}-D_{\iota(c_{i}^{\prime})}).$
(5.1)

By Lemma 4.12, we have:

$G\cdot(D_{c_{i}}-D_{\iota(c_{i})})\cdot(D_{c_{i}}+D_{\iota(c_{i})})=G\cdot(D_{c_{i}}^{2}-D_{\iota(c_{i})}^{2})=4m(\iota(t_{i})-t_{i}).$

Here, we identify $\operatorname{CH}_{0}(F)$ and
$\operatorname{CH}_{0}(\tilde{Y}^{\vee})$ via the isomorphism $\alpha_{*}$. By
Lemma 4.3, we know that $D_{c}+D_{\iota(c)}$ is constant in
$\operatorname{CH}^{2}(F)$, we denote it by $D_{0}$. Therefore we obtain

$G\cdot D_{0}\cdot\sum(D_{c_{i}}-D_{\iota(c_{i})})=4m\sum(\iota(t_{i})-t_{i})$

and

$G\cdot
D_{0}\cdot\sum(D_{c_{i}^{\prime}}-D_{\iota(c_{i}^{\prime})})=4m\sum(\iota(t_{i}^{\prime})-t_{i}^{\prime}).$

The left-hand sides are equal by (5.1). Therefore by torsion-freeness we
obtain:

$\sum t_{i}-\iota(t_{i})=\sum t_{i}^{\prime}-\iota(t_{i}^{\prime}).$

Hence $\Psi$ is injective on $\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{-}$. ∎

Consequently, we immediately deduce that

######  Theorem 5.5.

We have an isomorphism between groups

$\mathbb{Z}\cdot o\oplus\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{-}_{\rm
hom}\cong\mathbb{Z}\cdot\theta\oplus\operatorname{CH}_{1}(X)_{\rm
hom}\cong\operatorname{CH}_{1}(X).$ (5.2)

In particular, the group $\operatorname{CH}_{1}(X)$ is torsion-free.

When $\tilde{Y}^{\vee}$ is birational to $S^{[2]}$ for some K3 surface $S$, it
has been shown in [17] that
$\operatorname{CH}_{0}(\tilde{Y}^{\vee})_{\mathbb{Q}}$ with
$\mathbb{Q}$-coefficient has a natural decomposition for the Chow group. For
0-cycles, the decomposition is

$\operatorname{CH}_{0}(\tilde{Y}^{\vee})_{\mathbb{Q}}=\mathbb{Q}\cdot
o\oplus\operatorname{CH}_{0}(S)_{\rm
hom}\oplus\operatorname{CH}_{0}(\tilde{Y}^{\vee})^{+}_{\rm hom}.$

Theorem 5.5 can be viewed as a generalization of this decomposition, where the
group $\operatorname{CH}_{1}(X)$ takes the place of
$\operatorname{CH}_{0}(S)$.

###  5.6. A result for conics

In this subsection, we prove that the class of a conic on $X$ is in ${\rm
S}_{2}(X)$. The proof is similar to the case of cubic fourfolds by showing any
conic is contained in a certain singular cubic threefold. Here, we need the
flexibility to change the GM fourfold to its period partners, guaranteed by
Corollary 5.3.

We consider the following incidence relation

$\Omega:=\\{(A,B)~{}|~{}\mathrm{dim}(A\cap B)\geq
9\\}\subset\mathbb{L}\mathbb{G}(\bigwedge^{3}V_{6})\times\mathbb{L}\mathbb{G}(\bigwedge^{3}V_{6}).$

Denote the fibre of the projection over $A$ by $\Omega_{A}$ and
$F_{y}=y\wedge\bigwedge^{2}V_{6}\subset\bigwedge^{3}V_{6}$. Clearly, we have

$Y_{B}^{\geq 2}\subset Y_{A}^{\geq 1}$

for $(A,B)\in\Omega$. We let

$\Sigma=\\{A\in\mathbb{L}\mathbb{G}(\bigwedge^{3}V_{6})~{}|~{}A{\rm~{}contains~{}a~{}decomposable~{}vector}\\},$

and $\Sigma_{k}$ be the closure of the locus of $A$ that contains exactly $k$
decomposable vectors.

We can deduce the following lemmas.

######  Lemma 5.7.

For any $A,B\subset\bigwedge^{3}V_{6}$ with $(A,B)\in\Omega$ and $A\neq B$, we
can find $V_{5}\subset V_{6}$, such that the corresponding GM varieties
satisfy $X_{B}\subset X_{A}$ and are of dimension $3$ and $4$.

###### Proof.

It’s enough to take $V_{5}\in Y_{A^{\vee}}^{=1}\cap Y_{B^{\vee}}^{=2}$ by
Theorem 3.3. ∎

######  Lemma 5.8.

For a general $B\in\Sigma_{8}$ and $V_{5}\in Y_{B^{\vee}}^{=2}$, let
$F(X_{B})$ be the surface of conics on the corresponding GM threefold $X_{B}$,
then dim Alb$(F(X_{B}))=2$.

###### Proof.

For a general choice of $B\in\Sigma_{8}$ and $V_{5}$, we may assume that the
$8$ decomposable vectors are

$\bigwedge^{3}V_{31},\ldots\bigwedge^{3}V_{38}\in B\cap\bigwedge^{3}V_{6}$

and $V_{31},\ldots V_{38}\not\subset\bigwedge^{3}V_{5}$. Then, by [3,
Proposition 2.24], the Grassmaninan hull $\operatorname{Gr}(2,V_{5})\cap
W_{B}$ is smooth. $\bigwedge^{3}V_{31}$ can be written as $v\wedge v_{1}\wedge
v_{2}$ with $v\in V_{6}\setminus V_{5}$, and $O=v_{1}\wedge v_{2}$ is the
kernel of $q(v)$, by [3, Theorem 3.16]. Then by [5, Lemma 4.1], $X_{B}$ has a
node at $O$. Then we can project $X_{B}\subset\mathbb{P}^{7}$ to
$X_{O}\subset\mathbb{P}^{6}$ from $O$ as in [5].

The quadrics containing $X_{O}$ form a net $P$. Let $D$ be the discriminant
curve parameterizing singular quadrics. Then there is a line $L\subset D$
corresponding to quadrics containing the projection of
$\operatorname{Gr}(2,V_{5})$. Therefore,

$D=C\cup L$

for some degree $6$ curve $C\subset P$.

We see that the curve $\mathbb{P}(V_{31})\cap Y_{B}^{\geq 2}$ parameterizing
quadrics in $Y_{B}^{\geq 2}$ of corank $\geq 2$ whose vertices contain $O$,
thus it equals $C$. Then by [9, Proposition 2.20], $C$ has $7$ nodes
corresponding to the extra decomposable vectors. For a general $B$,

$Y_{B}^{=3}=\emptyset,$

hence the quadrics in $C$ are all of rank $6$. There is a étale double cover
$\tilde{C}\rightarrow C$, corresponding to the choice of a family of
$3$-planes contained in a quadric in $C$, see [5, section 4.2]. Let
$p^{\prime}:\tilde{N}\rightarrow N$ be the normalization of
$p:\tilde{C}\rightarrow C$.

There is a morphism $P^{\vee}\rightarrow C^{(6)}$, sending a line in $P$ to
its intersection with $C$. Denote $S^{\prime}$ the pull back of $P^{\vee}$ in
$\tilde{C}^{(6)}$ and $S^{\prime\prime}$ the further pull back in
$\tilde{N}^{(6)}$. By [18, Proposition 5.8], $S^{\prime}$ has two irreducible
components and $F(X_{B})$ is birational to one of the components. It follows
that $F(X_{B})$ is birational to a component of $S^{\prime\prime}$. Then by
[34, Theorem 8.19],

${\rm Alb}(F(X_{B}))\cong Pr(\tilde{N}/N),$

which is of dimension $2$ since $g(N)=3$. ∎

######  Lemma 5.9.

For a general point $y\in Y_{A}$, we can find $B\in\Omega_{A}\cap\Sigma_{8}$,
such that $y\in Y_{B}^{\geq 2}$.

###### Proof.

The proof is similar to [11, Proposition 5.1]. Let

$\Gamma=\Omega\cap\mathbb{L}\mathbb{G}(\bigwedge^{3}V_{6})\times\Sigma_{8}\cap\mathbb{L}\mathbb{G}(\bigwedge^{3}V_{6})\times\Omega_{y},$

where $\Omega_{y}=\\{B~{}|~{}\mathrm{dim}(B\cap F_{y})\geq 2\\}$ and
$F_{y}=y\wedge\bigwedge^{2}V_{6}$. We have the two projections:

${\Gamma}$${\mathbb{L}\mathbb{G}(\bigwedge^{3}V_{6})}$${\Sigma_{8}}$$\scriptstyle{\pi}$$\scriptstyle{\rho}$

The fiber of $\rho$ is a codimension-$2$ subvariety of $\Omega_{B}$, which is
$8$-dimensional, see [11, Lemma 5.3]. It yields that dim $\Gamma=$ dim
$\mathbb{L}\mathbb{G}(\bigwedge^{3}V_{6})$. Therefore it suffices to show that
$d\pi$ at a general point $(A,B)\in\Gamma$ is an isomorphism.

Let $(A,B)\in\Gamma$ be general such that the $8$ decomposable forms
$\alpha_{1},\ldots\alpha_{8}$ in $B$ are linearly independent and that
$\alpha_{1},\ldots\alpha_{8}$ and $F_{y}\cap B$ are not contained in $A$. Let
$U=A\cap B$. According to [11, Lemma 5.4], we have the description of tangent
spaces:

$\begin{split}T_{(A,B)}\Omega&=\\{(q_{A},q_{B})\in\mathrm{Sym}^{2}(A^{\vee})\times\mathrm{Sym}^{2}(B^{\vee})~{}|~{}q_{A}|_{U}=q_{B}|_{U}\\}\\\
T_{B}\Sigma_{8}&=\\{q_{B}\in\mathrm{Sym}^{2}(B^{\vee})~{}|~{}q_{B}(\alpha_{1})=\ldots=q_{B}(\alpha_{8})=0\\}\\\
T_{B}\Omega_{y}&=\\{q_{B}:B\rightarrow B^{\vee}~{}|~{}q_{B}(F_{y})\subset
B+F_{y}/B\\}.\end{split}$

We assume on the contrary that there is a nonzero
$t=(q_{A},q_{B})\in\mathrm{Ker}~{}d\pi$. It implies that $q_{A}=0$ and thus
Ker $q_{B}=U\cup U^{\prime}$ for some hyperplane $U^{\prime}\subset B$.
According to the assumptions and the description of tangent spaces, the $8$
decomposable forms and $F_{y}\cap B$ are contained in Ker $q_{B}$ hence
contained in $U^{\prime}$. Then for a general $B$, dim $U^{\prime}=10$, which
is a contradiction. ∎

Now we can deduce the main result of this subsection:

######  Proposition 5.10.

For any conic $c\subset X$, we have $c\in{\rm S}_{2}(X)$.

###### Proof.

We may assume that $c$ is a general conic. Combining the above Lemmas, there
exist a GM $3$-fold $X_{B}$ with $8$ nodes containing $c$ and a period partner
$X^{\prime}$ of $X$ such that $X_{B}\subset X^{\prime}$. Let $S^{\prime}$ be
the image of

$F(X_{B})\subset F(X)\xrightarrow{\alpha}\tilde{Y}^{\vee}$

and $\tilde{S}$ be the resolution of $S^{\prime}$. Then a conic lying in
$R^{\prime}$ is in ${\rm S}_{1}(X)$, where $R^{\prime}$ is the normalization
of $S^{\prime}\cap D$.

The argument of [24, Section 2.4] implies that there is a surjection

$R^{\prime(2)}\twoheadrightarrow Alb(\tilde{S}).$

For a resolution $\tilde{X}_{B}\rightarrow X_{B}$, we have
$\operatorname{CH}_{1}(\tilde{X}_{B})_{\rm hom}\cong J(\tilde{X}_{B})$ by [31,
Theorem 0.3]. Hence, $\operatorname{CH}_{0}(\tilde{S})_{\rm
hom}\rightarrow\operatorname{CH}_{1}(\tilde{X}_{B})_{\rm hom}$ factors through
$Alb(\tilde{S})$. By the surjectivity of $R^{\prime(2)}\rightarrow
Alb(\tilde{S})$, there exists two conics $c_{1}$ and $c_{2}\in{\rm
S}_{1}(X^{\prime})$ such that
$c=c_{1}+c_{2}-\theta\in\operatorname{CH}_{1}(\tilde{X}_{B})$. Then the result
follows from Corollary 5.3. ∎

##  6\. Sheaves supported on conics and lines

In this section, we introduce a link between the filtration on
$\operatorname{CH}_{1}(X)$ and the Kuznetsov component $\mathcal{A}_{X}$ of
the derived category of $X$ and prove Theorem 1.3. We assume that $X$ is very
general.

Let $i^{*}:D^{b}(X)\rightarrow\mathcal{A}_{X}$ be the left adjoint of the
inclusion $i_{*}:\mathcal{A}_{X}\rightarrow D^{b}(X)$ and
$\operatorname{pr}:K_{0}(X)\rightarrow K_{0}(\mathcal{A}_{X})$ be the
projection of Grothendieck groups. Let $\overline{\operatorname{pr}}$ be the
further projection to $K_{\rm num}(X)$. For a very general $X$, the numerical
Grothendieck group $K_{\rm num}(X)\cong\mathbb{Z}^{\oplus 2}$ and under the
basis $\lambda_{1}$, $\lambda_{2}$ the Euler form is given by (cf. [23, Lemma
2.4]):

$\begin{pmatrix}-2&0\\\ 0&-2\end{pmatrix},$

where $\lambda_{1}=\overline{\operatorname{pr}}[\mathcal{O}_{c}(1)]$ and
$\overline{\operatorname{pr}}[\mathcal{O}_{p}]=-\lambda_{1}-2\lambda_{2}$ for
a conic $c\subset X$ and a point $p\in X$.

######  Lemma 6.1.

$K_{0}(\mathcal{A}_{X})\otimes\mathbb{Q}$ is generated by
$\operatorname{pr}[\mathcal{O}_{c}(1)]$ and
$\operatorname{pr}[\mathcal{O}_{p}]$.

###### Proof.

Since $\operatorname{CH}_{0}(X)\cong\mathbb{Z}$, the class $[\mathcal{O}_{p}]$
in $K_{0}(X)$ is independent of the choice of $p\in X$ and the cycle class map
$cl:\operatorname{CH}^{i}(X)\rightarrow{\rm H}^{i}(X,\mathbb{Z})$ is injective
for $i\neq 3$, by [1, Theorem 1.1].

We have the following commutative diagram:

${K_{0}(\mathcal{A}_{X})\otimes\mathbb{Q}}$${\operatorname{CH}^{*}(X)\otimes\mathbb{Q}}$${K_{\rm
num}(\mathcal{A}_{X})\otimes\mathbb{Q}}$${{{\rm
H}^{*}(X,\mathbb{Q})},}$$\scriptstyle{\overline{\operatorname{pr}}}$$\scriptstyle{\operatorname{ch}}$$\scriptstyle{cl}$$\scriptstyle{\overline{\operatorname{ch}}}$

It’s enough to show that $cl^{-1}(\overline{\operatorname{ch}}(K_{\rm
num}(\mathcal{A}_{X})))$ is generated by
$\operatorname{ch}(\mathcal{O}_{c}(1))$ and
$\operatorname{ch}(\mathcal{O}_{p})$ and this follows from the injectivity of
the cycle class map for $i\neq 3$ and the fact that
$\operatorname{CH}_{1}(X)\otimes\mathbb{Q}$ is generated by conics. ∎

For a cubic fourfold, the link between the Kuznetsov component and 1-cycles is
by taking the Chern class $c_{3}$. However, the situation is slightly
different for GM fourfolds:

If $X$ and $X^{\prime}$ are period partners, their associated dual double EPW
sextics are the same by Theorem 3.3. Therefore, their Fano varieties of conics
both admit a birational $\mathbb{P}^{1}$-fibration to $\tilde{Y}^{\vee}$. Then
by Lemma 6.1 there is a group isomorphism

$K_{0}(\mathcal{A}_{X})_{\mathbb{Q}}\cong
K_{0}(\mathcal{A}_{X}^{\prime})_{\mathbb{Q}},$

which maps $\operatorname{pr}[\mathcal{O}_{p}]$ to
$\operatorname{pr}[\mathcal{O}_{p^{\prime}}]$ and
$\operatorname{pr}[\mathcal{O}_{c}(1)]$ to
$\operatorname{pr}[\mathcal{O}_{c^{\prime}}(1)]$. Here $c^{\prime}$ is a conic
mapping to the same point with $c$ in $\tilde{Y}^{\vee}$ via the fibration.

However, the classes $c_{3}(\operatorname{pr}[\mathcal{O}_{p}])$ and
$c_{3}(\operatorname{pr}[\mathcal{O}_{p^{\prime}}])$ do not coincide in
$\operatorname{CH}_{0}(\tilde{Y}^{\vee})$ via the isomorphism (5.5) even
modulo the class $o$. We want the link between the Kuznetsov components and
1-cycles intrinsic for the EPW sextic, so we modify the $c_{3}$ to be
compatible with (5.5) by dropping the term of
$\operatorname{pr}[\mathcal{O}_{p}]$:

If
$[\mathcal{E}]=\sum_{i}a_{i}\operatorname{pr}[\mathcal{O}_{c_{i}}(1)]+b\operatorname{pr}[\mathcal{O}_{p}]\in
K_{0}(\mathcal{A}_{X})_{\mathbb{Q}}$, we set

$p(\mathcal{E})=\sum a_{i}c_{i}\in\operatorname{CH}_{1}(X)\otimes\mathbb{Q}.$

It is independent of the choice of representations, since $b$ relies only on
the numerical class of $[\mathcal{E}]$. Let $\mathcal{F}$ be a sheaf supported
on a nonsingular connected rational curve $C\subset X$ of degree $e>0$. Then
$[\mathcal{F}]$ is numerically equivalent to

$re[\mathcal{O}_{l}(1)]+m[\mathcal{O}_{p}]$

in $K_{\rm num}(X)$ for some integers $r>0$ and $m$, where $l$ is a line on
$X$. Let

$d(i^{*}\mathcal{F}):=\frac{1}{2}\dim\operatorname{Ext}^{1}_{\mathcal{A}_{X}}(i^{*}\mathcal{F},i^{*}\mathcal{F})$

and we have:

######  Lemma 6.2.

If $\mathcal{F}$ is supported on a line or a conic, then
$d(i^{*}\mathcal{F})\geq 2$.

###### Proof.

By straightforward computation, we have
$\overline{\operatorname{pr}}[\mathcal{O}_{p}]=-\lambda_{1}-2\lambda_{2}$ and
$\overline{\operatorname{pr}}[\mathcal{O}_{l}(1)]=-\lambda_{1}$. Hence
$\overline{\operatorname{pr}}[\mathcal{E}]=-m\lambda_{1}-(2m+re)\lambda_{2}$.
Then

$d(i^{*}\mathcal{F})\geq\frac{1}{2}\chi(i^{*}\mathcal{F},i^{*}\mathcal{F})+1=5m^{2}+4mre+r^{2}e^{2}+1.$

For $e=1$ or $2$, we can easily deduce that $d(i^{*}\mathcal{F})\geq 2$. In
fact, there is a lower bound by a quadric polynomial in $e$, but we only deal
with $e=1$ or $2$. ∎

Now we can prove the main theorem:

######  Theorem 6.3.

If $\mathcal{F}$ is supported on a line or a conic and
$\mathcal{E}=i^{*}\mathcal{F}$, then

$p(\mathcal{E})\in{\rm S}_{d(\mathcal{E})}(X).$

###### Proof.

By the definition of $p$, it is enough to show that every line or conic
belongs to ${\rm S}_{2}(X)$. For conics, it is proved in Proposition 5.10. For
lines, the strategy is to degenerate the residual curve of a line on a certain
surface to a sum of conics and lines.

The locus in $F(X)$ and its image in $\tilde{Y}^{\vee}$ parameterizing double
lines is a surface by [14, Lemma 3.8]. Since $X$ is very general, this surface
meets $Z$ by Proposition 4.15. Hence, there exists a line $l_{0}\subset X$
such that $2l_{0}=\theta$. Take $\mathbb{P}(V_{1}^{\prime}\bigwedge
V_{3}^{\prime})=l^{\prime}=\iota_{1}(l_{0})$, we obtain

$2l^{\prime}=2\sigma-\theta,$

where $\iota_{1}$ is the rational involution of the Hilbert scheme of lines
$F_{1}(X)$.

For $l=\mathbb{P}(V_{1}\bigwedge V_{3})\subset X$ a general line, let
$V_{2}=V_{1}\oplus V_{1}^{\prime}$ and $V_{4}=V_{1}^{\prime}\oplus V_{3}$ and
$S_{V_{2}}$ be the surface parameterizing points $x\in X$ with $V_{x}\cap
V_{2}\neq\emptyset$, where $V_{x}$ is the two dimensional vector space
corresponding to $x$. We see that $S_{V_{2}}$ is a degree $6$ surface in
$\mathbb{P}(V_{2}\bigwedge V_{5})\cap H=\mathbb{P}^{5}$. By construction, $l$
and $l^{\prime}$ are contained in $S_{V_{2}}$ and $S_{V_{2}}\cap S_{V_{4}}$ is
a hyperplane section of $S_{V_{4}}\subset\mathbb{P}(\bigwedge^{2}V_{4})\cap
H$, which is the union of $l$ and a degree $3$ curve $C_{3}$. Hence

$l+C_{3}=2\theta.$

Since $V_{3}^{\prime}\cap V_{4}$ is 2-dimensional, $l^{\prime}$ meets
$S_{V_{4}}$ at the point corresponding to $V_{3}^{\prime}\cap V_{4}$. For a
general choice of $l$, it does not meet $l^{\prime}$. It follows that
$l^{\prime}$ and $C_{3}$ meet at a point, and therefore they span a
$\mathbb{P}^{4}\subset\mathbb{P}(V_{2}\bigwedge V_{5})\cap H$, which cuts
$S_{V_{2}}$ into a degree $6$ curve $l^{\prime}\cup C_{3}\cup c$ for some
conic $c$.

We claim that

$l^{\prime}+C_{3}+c=2\theta+\sigma.$

In fact, we can choose the hyperplane section
$\mathbb{P}^{4}\subset\mathbb{P}(V_{2}\bigwedge V_{5})\cap H$ to be the span
of $\mathbb{P}(V_{2}\bigwedge V_{4})\cap H$ and $\mathbb{P}(V_{1}\bigwedge
V_{1}^{\perp})$ for some $V_{1}\subset V_{2}$ with $V_{1}^{\perp}\neq V_{4}$.
Then the class of a section is $l+C_{3}+c^{\prime}$, which equals
$2\theta+\sigma$. Here $c^{\prime}$ is the $\sigma$-type conic associated with
$V_{1}$.

Finally, we obtain that $l=l^{\prime}+c-\sigma=c-l_{0}$, which is in
$S_{2}(X)$ by Proposition 5.10. ∎

## References

  * [1] Bloch S, Srinivas V. Remarks on correspondences and algebraic cycles. Amer. J. Math., 1983, 105: 1235–1253
  * [2] Charles F, Mongardi G, Pacienza G. Families of rational curves on holomorphic symplectic varieties and applications to zero-cycles. arXiv:1907.10970, 2019
  * [3] Debarre O, Kuznetsov A. Gushel-Mukai varieties: Classification and birationalities. Algebr. Geom., 2018, 5: 15–76
  * [4] Debarre O. Gushel-Mukai varieties. arXiv:2001.03485, 2020
  * [5] Debarre O, Iliev A, Manivel L. On nodal prime Fano threefolds of degree 10. Sci. China Math., 2011, 54: 1591–1609
  * [6] Debarre O, Iliev A, Manivel L. On the period map for prime Fano threefolds of degree 10. J. Algebraic Geom., 2012, 21: 21–59
  * [7] Debarre O, Kuznetsov A. Gushel–Mukai varieties: moduli. Internat. J. Math., 2020, 31: 2050013, 59
  * [8] Eisenbud D, Harris J. 3264 and all that: A second course in algebraic geometry. Cambridge University Press, 2016
  * [9] Ferretti A. The Chow ring of double EPW sextics. Rend. Mat. Appl. (7), 2011, 31: 69–217.
  * [10] Ferretti A. The Chow ring of double EPW sextics. Algebra Number Theory, 2012, 6: 539–560.
  * [11] Ferretti A. Special subvarieties of EPW sextics. Math. Z., 2012, 272: 1137–1164.
  * [12] Fu L, Laterveer R, Vial C, Shen M. The generalized Franchetta conjecture for some hyper-Kähler varieties. J. Math. Pures Appl. (9), 2019, 130: 1–35
  * [13] Fulton W. Intersection theory. Berlin: Springer-Verlag, 1984.
  * [14] Iliev A, Manivel L. Fano manifolds of degree ten and EPW sextics. Ann. Sci. Éc. Norm. Supér. (4), 2011, 44: 393–426.
  * [15] Kuznetsov A, Perry A. Derived categories of Gushel-Mukai varieties. Compos. Math., 2018, 154: 1362–1406
  * [16] Kuznetsov A, Prokhorov Y, Shramov C. Hilbert schemes of lines and conics and automorphism groups of Fano threefolds. Jpn. J. Math., 2018, 13: 109–185.
  * [17] Laterveer R, Vial C. Zero-cycles on double EPW sextics. Commun. Contemp. Math., 2021, 23: Paper No. 2050040, 22.
  * [18] Logachev D. Fano threefolds of genus 6. Asian J. Math., 2012, 16: 515–559.
  * [19] O’Grady K. Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics. Duke Math. J., 2016, 134: 99–137.
  * [20] O’Grady K. Double covers of EPW-sextics. Michigan Math. J., 2013, 62: 143–184.
  * [21] Ottem J. Nef cycles on some hyperkähler fourfolds. In: London Math. Soc. Lecture Note Ser., 2022, 473: 228–237.
  * [22] Perry A, Pertusi L, Zhao X L. Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties. arXiv:1912.06935, 2019
  * [23] Pertusi L. On the double EPW sextic associated to a Gushel–Mukai fourfold. J. Lond. Math. Soc. (2), 2019, 100: 83–106.
  * [24] Shen J L, Yin Q Z. $K3$ categories, one-cycles on cubic fourfolds, and the Beauville-Voisin filtration. J. Inst. Math. Jussieu, 2020, 19: 1601–1627.
  * [25] Shen J L, Yin Q Z, Zhao X L. Derived categories of K3 surfaces, O’Grady’s filtration, and zero-cycles on holomorphic symplectic varieties. Compos. Math., 2020, 156: 179–197
  * [26] Shen M M, Vial C. The Fourier transform for certain hyperKähler fourfolds. Mem. Amer. Math. Soc., 2016, 240: vii+163.
  * [27] Tian Z Y, Zong H R. One-cycles on rationally connected varieties. Compos. Math., 2014, 150: 396–408.
  * [28] Totaro B. Chow groups, Chow cohomology, and linear varieties. Forum Math. Sigma, 2014, 2: Paper No. e17, 25.
  * [29] Voisin C. Hodge theory and complex algebraic geometry. II, Cambridge: Cambridge University Press, 2007.
  * [30] Voisin C. On the Chow Ring of Certain Algebraic Hyper-Kähler Manifolds. Pure Appl. Math. Q., 2008, 4: 613–649.
  * [31] Voisin C. Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal. J. Algebraic Geom., 2013, 22: 141–174.
  * [32] Voisin C. Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-Kähler varieties, In: K3 surfaces and their moduli, 2016, 315: 365–399.
  * [33] Voisin C. Triangle varieties and surface decomposition of hyper-Kähler manifolds. arXiv:1810.11848, 2018.
  * [34] Welters G. Abel-Jacobi isogenies for certain types of Fano threefolds. Mathematical Centre Tracts, Amsterdam. 1981, 141: i+139.