\begin{document}

\title[One-cycles on Gushel-Mukai]{One-cycles on Gushel-Mukai fourfolds and the Beauville-Voisin filtration}
\date{\today}

\author{Ruxuan Zhang}
\address{Ruxuan Zhang, Shanghai Center for Mathematical Sciences, Fudan University, Jiangwan Campus, Shanghai, 200438, China}
\email{rxzhang18@fudan.edu.cn}

\begin{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 $\CH_1$ 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.
\end{abstract}

\subjclass[2020]{14J42 , 14C15}
\keywords{Beauville-Voisin filtration, irreducible holomorphic symplectic variety, Gushel-Mukai fourfold}

\maketitle

\section{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 
$$ X=\Gr(2,5)\cap \mathbb{P}^8\cap Q\subset \mathbb{P}^9,$$ where $\Gr(2,5)\subset \mathbb{P}^9$ is the Pl\"ucker 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 \cite{kuznetsov2018derived} 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 $\Gr(2,5)$ 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 \cite{debarre2018gushel}\cite{o2006irreducible}\cite{iliev2011fano} 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 \cite[Proposition 5.17]{perry2019stability}, 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 \cite{iliev2011fano}, 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 \cite{voisin2016remarks} that there exists a filtration of the Chow ring $\CH^*(M)$ (called Beauville–Voisin filtration), which can be viewed as an opposite to the conjectural Bloch-Beilinson filtration.
For 0-cycles, the filtration ${\rm S}_\bullet\CH_0(M)$ is defined by
$${\rm S}_{i}\CH_0(M):=\left<x\in M|~\dim O_y\geq n-i\right>,$$ where $O_y$ is the orbit of $y$ under the rational equivalence (here we follow the definition in \cite{shen2020categories}, 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 $\CH_0(M)$, 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 ${\rm S}_iM\subset M$ be the set of points with dim $O_y\geq n-i$. Voisin conjectured in \cite[Conjecture 0.4]{voisin2016remarks}  that
\begin{equation}\label{vconj}
    \mathrm{dim}~{\rm S}_i M= n+i.
\end{equation}
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  \cite[Speculation 0.1]{shen2020categories}. 

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 $\CH_0(S)$
$${\rm S}_i(S) :=\bigcup_{\substack{\deg([z])=i\\z ~\text{effective}}}\{ [z] + \mathbb{Z} \cdot [o_S] \}$$
serves as the universal filtration for all moduli space of stable objects. It has been shown in \cite{shen2020derived} 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  $\mathcal{E}\in {\rm S}_i\CH_0(M)$.

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 \cite{shen2020derived} a filtration ${\rm S}_{\bullet}(X)$ on $\CH_1(X)$ 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 $d=\frac{1}{2}\dim \Ext^1(\mathcal{E},\mathcal{E})$. This implies  \eqref{vconj}  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 $\CH_0(\tilde{Y}^\vee)$ 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 $\CH_0$ might be huge. However, it was asked whether $Z$ is a constant cycle surface on $\tilde{Y}^\vee$, see \cite[Remark 2.7]{laterveer2021zero}. 
Our first result is (Theorem \ref{constant}):

\begin{theorem}\label{mainthm1}
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 $Z'$ meets $Z$.
\end{theorem}
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 \ref{mainthm1} allows us to construct a filtration  ${\rm S}_{\bullet}(X)$ on $\CH_1(X)$ 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 $\pr [\mathcal{O}_c(1)]$ and $\pr [\mathcal{O}_p]$ for a very general $X$, where $c$ is a conic, $p$ is a point on $X$ and $\pr$ is the projection $$\pr:K_0(X)\rightarrow K_0(\mathcal{A}_X).$$
Unlike the cubic fourfold case, the dual double EPW sextic determines the GM fourfold only up to period partners and the Chern class $c_3(\pr [\mathcal{O}_p])$ of a skyscraper sheaf might lie in different pieces of the filtration with respect to different GM fourfolds. We define a modification $$p:K_0(\mathcal{A}_X)\rightarrow \CH_1(X)$$ of $c_3$ by dropping the effect of $\pr [\mathcal{O}_p]$, see Section \ref{S5} 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.

\begin{conjecture} \label{mainconj}
For any object $\mathcal{E} \in \mathcal{A}_X$, we have
\[
p(\mathcal{E}) \in {\rm S}_{d(\mathcal{E})}(X).
\]
\end{conjecture}
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 \ref{maint}):
\begin{theorem}\label{mainthm}
We assume that $X$ is very general.
If $\mathcal{F}$ is supported on lines or conics on $X$, conjecture \ref{mainconj} holds for $\mathcal{E}=i^*\mathcal{F}$. 
\end{theorem}

Our approach to deducing Theorem \ref{mainthm1} has another application, concerning with the relation between $\CH_0(\tilde{Y}^\vee)$ and $\CH_1(X)$. When $\tilde{Y}^\vee$ is birational to $S^{[2]}$ for some K3 surface $S$,  it has been shown in \cite{laterveer2021zero} that $\CH_0(\tilde{Y}^\vee)$ with $\mathbb{Q}$-coefficients has a natural decomposition for the Chow group. For 0-cycles, the decomposition is
$$\CH_0(\tilde{Y}^\vee)=\mathbb{Q}\cdot o\oplus\CH_0(S)_{\rm hom}\oplus\CH_0(\tilde{Y}^\vee)^+_{\rm hom},$$
where $\CH_0(\tilde{Y}^\vee)^+_{\rm hom}$ is the $\iota$-invariant part of $\CH_0(\tilde{Y}^\vee)_{\rm hom}$. We show that in the general case, the group $\CH_1(X)$ would play the role of the group $\CH_0(S)$.
\begin{theorem}(Theorem \ref{chowiso})
When the dual double EPW sextic $\tilde{Y}^\vee$ is general, we have the following isomorphism of groups:
\begin{equation*}
    \mathbb{Z}\cdot o\oplus \CH_0(\tilde{Y}^\vee)^{-}_{\rm hom} \cong \CH_1(X),
\end{equation*}
where $\CH_0(\tilde{Y}^\vee)^{-}_{\rm hom}$ is the $\iota$-anti-invariant part of $\CH_0(\tilde{Y}^\vee)_{\rm hom}$.
\end{theorem}
In particular, the group $\CH_1(X)$ is torsion-free and we see that if $X'$ 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.

\section{Notation}
The following notations appear frequently in the paper, so we gather them together for the sake of convenience.
\begin{itemize}
    \item $V_m$ is an $m$-dimensional vector space, $X$ is a smooth ordinary GM fourfold and $\tilde{Y}^\vee$ is the dual double sextic associated with $X$.
    \item $\iota$ is the involution of $\tilde{Y}^\vee$ and $Z\subset \tilde{Y}^\vee$ is the invariant locus of $\iota$. 
    \item $c$ 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  $\alpha_* : \CH_0(F(X))\cong \CH_0(\tilde{Y}^\vee).$
    \item $P$ is the universal conic, $p,q$ are the projections to $F(X)$ and $X$. $\Phi=p_*q^*: \CH^*(X)\rightarrow \CH^*(F)$. $\Psi$ is the composition of $q_*p^*$ and $\alpha_*^{-1}$: $$\Psi=q_*p^*\alpha_*^{-1}: \CH_0(\tilde{Y}^\vee)\rightarrow \CH_1(X),$$ sending $\alpha(c)$ to its class $[c]\in \CH_1(X)$.
    \item $S_{V_4}=\Gr(2,V_4)\cap X$ for some $V_4\subset V_5$, which is a degree $4$ Del Pezzo surface.
    \item $D_c$ 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.
\end{itemize}
   
\section{Preliminaries}
Let $X=\Gr(2, V_5)\cap H\cap Q$ 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. 

\subsection{EPW sextics}
Debarre and Kuznetsov gave a way to associate a GM variety with an EPW sextic in \cite{debarre2018gushel} via linear algebra. We will follow \cite{debarre2020survey} to recall some basic facts about EPW sextics. A GM variety $X_n=\Gr(2,V_5)\cap \mathbb{P}(W)\cap Q$ of dimension $n$ is a smooth intersection of a linear subspace of dimension $n+4$, a quadratic hypersurface and $\Gr(2,V_5)$ inside $\mathbb{P}(\bigwedge^2V_5)$.
It has an associated GM data $(W_{n+5}, V_6, V_5, q)$, where 
\begin{itemize}
    \item $V_6$ is a $6$-dimensional vector space, consisting of quadrics containing $X$.
    \item $V_5$ is a hyperplane section of $V_6$, consisting of quadrics containing $\Gr(2,V_5)$.
    \item $W_{n+5}\subset \bigwedge^2V_5$ is a subspace of dimension $n+5$, corresponding to the linear subspace in the definition of GM variety.
    \item $q:V_6\rightarrow \mathrm{Sym}^2W_{n+5}^{\vee}$, is a linear map such that (here we choose an isomorphism $\bigwedge^5V_5\cong \mathbb{C}$)
    $$\forall v\in V_5, q(v)(w,w)=v\wedge w\wedge w.$$
    
\end{itemize}
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, $\bigwedge^3V_6$ can be endowed with a symplectic form by wedge product.
Debarre and Kuznetsov defined the Lagrangian data set $(V_6,V_5,A)$ with $A\subset \bigwedge^3V_6$ a Lagrangian subspace and $V_5\subset V_6$ a hyperplane. It gives an EPW data $$Y_A^{\geq l}:=\{[v]\in \mathbb{P}(V_6)~|~\mathrm{dim}(A\cap (v\wedge\bigwedge ^2V_6))\geq l\}$$ 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 \cite{o2013double}, O'Grady showed that 

\begin{theorem}
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.
\end{theorem}
Iliev and Manivel described in \cite{iliev2011fano} 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 Pl\"ucker 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  $A^{\vee}\subset \bigwedge^3V_6^\vee$, 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  \cite[Theorem 2.2]{debarre2020survey}. 
However, the GM  fourfolds associated with a fixed dual EPW sextic are not unique. Debarre and Kuznetsov proved the following \cite{debarre2020gushel}\cite[Theorem 2.6]{debarre2020survey}:
\begin{theorem}\label{moduli}
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:
\begin{equation*}
    \pi : M^{GM} \longrightarrow M^{EPW}
\end{equation*}
with fibre $\pi^{-1}(A^{\vee})=Y_{A}^{5-n}$. 
\end{theorem}
If two GM fourfolds are in the same fiber of $\pi$, we say that they are {\it period partners}.

\subsection{Lines and Conics on \texorpdfstring{$X$}{X}}
In this subsection, we gather some facts about lines and conics on a general GM fourfold $X$ following \cite[Section 3]{iliev2011fano} and \cite{debarre2012period}.

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 $$S_{V_4}= \Gr(2,V_4)\cap H\cap Q \subset \mathbb{P}(\bigwedge^2V_4)\cap H\cong \mathbb{P}^4,$$ 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 $X$:
\begin{enumerate}
    \item a {\bf $\tau$-conic}: The plane $\langle c\rangle$ spanned by the conic $c$ is not contained in $\Gr(2,V_5)$. 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.
    \item a {\bf $\rho$-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 $F(X)$. 
    \item a {\bf $\sigma$-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 $F(X)$.
\end{enumerate}
We see that all the $\rho$-conics (resp. $\sigma$-conics) are represented by the same class in $\CH_1(X)$ 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 \cite[Theorem 4.7]{debarre2012period}. 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 \cite[Proposition 4.18]{iliev2011fano}: 
 $$\alpha : F(X)\rightarrow \tilde{Y}^\vee.$$ 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 Pl{\"u}cker hyperplane $[V_5]\in Y\subset \mathbb{P}(V_6^{\vee})$, by \cite[Lemma 4.12]{iliev2011fano}.
In conclusion, we have the following diagram.
\begin{equation}\label{notation}
    \xymatrix{
  & P \ar[d]^q \ar[r]^p & F(X) \ar[r]^\alpha &\tilde{Y}^\vee            \\
  & X                       },
\end{equation}
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  $$\alpha_* : \CH_0(F(X))\cong \CH_0(\tilde{Y}^\vee).$$ 
We will frequently use some maps between Chow groups:  Let $\Phi=p_*q^*: \CH^*(X)\rightarrow \CH^*(F)$ and $\Psi$ be the composition of $q_*p^*$ and $\alpha_*^{-1}$: $$\Psi=q_*p^*\alpha_*^{-1}: \CH_0(\tilde{Y}^\vee)\rightarrow \CH_1(X),$$ sending $\alpha(c)$ to its class $[c]\in \CH_1(X)$.

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 $c'$. Then $$\iota(y)=\alpha(c').$$ One can check that for a different choice of $c$ and a hyperplane in $\mathbb{P}^4$, the corresponding $\alpha(c')$ 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  $B\subset \bigwedge^3V_6$ 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 \cite[Section 5.1]{iliev2011fano}.

\begin{remark}
In \cite[Section 2.4]{voisin2018triangle}, 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. 
\end{remark}

\subsection{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 \cite{ferretti2012chow} and \cite{laterveer2021zero}. The result is slightly stronger, that is,  the cycle class map restricted to the subring \begin{equation}\label{weak splitting}
    R^*(\tilde{Y}^\vee):= \langle \CH^1(\tilde{Y}^\vee), \CH^2(\tilde{Y}^\vee)^+, c_j(\tilde{Y}^\vee)\rangle 
\end{equation}
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,  $\CH^1(\tilde{Y}^\vee)$ 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 \cite[Proposition 6.1]{ferretti2012chow}\cite[Lemma 9.3]{ferretti2012chow}:  
\begin{equation}\label{intersection number}
    3Z=15h^2-c_2, Z^2=192, h^4=12, h^2Z=40.
\end{equation}
Moreover, the classes of $Z^2,h^4$ and $h^2Z$ are all proportional to the class $o$.

\section{A constant cycle surface on \texorpdfstring{$\tilde{Y}^\vee$}{Y}}
This section is devoted to proving Theorem \ref{mainthm1}. In the following parts of the paper, we assume that $X$ is general.

\subsection{Two facts about \texorpdfstring{$\CH_1(X)$}{X}}
We first show that $\CH_1(X)$ is rationally generated by conics on $X$ and those conics represented by points on the invariant surface $Z$ have the same class in $\CH_1(X)$.  
\begin{proposition}\label{ch}
The group $\CH_1(X)_{\mathbb{Q}}$ is generated by conics and   $\CH_1(X)$  is generated by lines on $X$.
\end{proposition}

\begin{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 \cite[Proposition 3.1]{tian2014one}, 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 $\CH_1(X)_{\mathbb{Q}}$ is generated by conics.

For integral coefficients, it has been shown in \cite[Theorem 1.3]{tian2014one} that $\CH_1(X)$ 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 $\sum_i n_iL_i$. The argument in \cite[Lemma 3.4]{tian2014one} shows that $\CH_1(X)_{\rm alg}$ is divisible. Therefore there exists a 1-cycle $C'$ such that $$N\cdot C'= C-\sum_i n_iL_i \in \CH_1(X)_{\rm alg},$$ since $N\cdot C'$ is rationally equivalent to a linear combination of lines, we find that $\CH_1(X)$ is generated by lines with $\mathbb{Z}$-coefficients. 
\end{proof}

\begin{lemma}\label{2.1}
For $y\in \tilde{Y}^\vee$, we have $\Psi(y+\iota(y))$ is constant in $\CH_1(X)$. In particular, $\Psi(2y)$ is constant in $\CH_1(X)$ if $y$ is on the invariant surface $Z$.
\end{lemma}
\begin{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 \cite[Proposition 4.9]{iliev2011fano}.  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 $c, c'$ residual in  $S_{V_4}$, the sums $c+c'$ are constant in $\CH_1(X)$, which follows from the fact that the hyperplane sections in such $S_{V_4}$'s are parameterized by a $\mathbb{P}^3$-bundle over $\mathbb{P}(V_5^{\vee})$.
\end{proof}

\subsection{Relations in \texorpdfstring{$\CH^*(F)$}{F} of incidences}
We introduce an incidence $I\in \CH^2(F\times F)$ and compute $I^2\in \CH^4(F\times F)$, where $F=F(X)$. Denote by $c_s$ the corresponding conic to $s\in F$.
The morphisms are the same as in diagram \eqref{notation}.
\begin{definition}
\begin{enumerate}
    \item The incidence subscheme $I\subset F\times F$ is defined to be 
     \begin{equation*}
     I:=\{(s,t)\in F\times F~|~c_s\cap c_t\neq \emptyset\},
     \end{equation*}
    which is endowed with the reduced closed subscheme structure and of dimension $8$.
    \item $D_c= p(q^{-1}(c))\subset F$ is the locus in $F$ consisting of conics intersecting the given conic $c$ and $D_c'=q^{-1}(c)$.
    
    \item $\Sigma_1\subset F\times F$ is defined to be
    \begin{equation*}
        \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}\},
    \end{equation*}
     which is of dimension $5$.
    \item $\Sigma_2\subset F\times F$ consists of $(s,t)\in F\times F$, where neither $c_s$ nor $c_t$ is of $\tau$-type.
    \item $W\subset F\times F$ is defined to be 
    \begin{equation*}
        W:=\{(s,t)~|~\exists  V_4,c_s~\text{and}~c_t ~\text{are residual in}~S_{V_4}\}.
    \end{equation*}
    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 $\text{dim}~ W=6$.
\end{enumerate}
\end{definition}

\begin{lemma}\label{plane}
For a general GM fourfold $X$, two different conics $c_1$ and $c_2$ span different planes. If $c$ is of $\tau$-type, then $\langle c\rangle\cap \Gr(2,V_5)=c$.
\end{lemma}
\begin{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 $\Gr(2,V_5)$ and $\Gr(2,V_5)$ is also the intersection of quadrics.
\end{proof}
According to Lemma \ref{plane}, we obtain  an injective morphism  
\begin{align}
    F\rightarrow \Gr(3,10), 
\end{align} sending $t\in F$ to $\langle c_t\rangle\subset \mathbb{P}(\bigwedge^2V_5)$. Let $\mathbb{P}=\mathbb{P}(E)$ be the universal plane of $\Gr(3,10)$ restricted to $F$, we have the following diagram:
\begin{equation}
    \xymatrix{
   P \ar[r]^q \ar[rd]^i \ar[rdd]^p & X \ar[rd]             \\
  &\mathbb{P}\ar[d]^{p'}  \ar[r]^{q'} & \mathbb{P}(\bigwedge^2V_5)         \\
  & F                      }.
\end{equation}

\begin{lemma}\label{L2.3}
The evaluation morphism $q$ is flat for a general $X$. In particular, the class $I=(p\times p)_*(q\times q)^*\Delta_X\subset \CH^2(F\times F)$. Equivalently, the correspondence $I=$ $^{t}P \circ P$.
\end{lemma}
\begin{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
\begin{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.
\end{claim}
\begin{claimproof}
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 \cite[Theorem 4.7]{debarre2012period}. Therefore the claim follows. 
\end{claimproof}

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 \cite[section 2.3]{kuznetsov2018hilbert}:
\begin{equation}\label{differential}
    \xymatrix{
    &0\ar[r] &Tc\ar[r]\ar[d]^*[@180]{\cong} &TP|_c \ar[r]\ar[d]^{dq|_c} &\mathrm{H}^0(N_{c/X})\otimes \mathcal{O}_c\ar[r]\ar[d]^{ev} &0 \\
    &0\ar[r] &Tc\ar[r] &TX|_c \ar[r] &N_{c/X}\ar[r] &0
    }.
\end{equation}
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, 
\begin{equation*}
    T_{(c,x)}q^{-1}(x)=\{s\in \mathrm{H}^0(N_{c/X})~|~s_x=0\},
\end{equation*}
 which is $2$-dimensional, contradicting to the fact that $\mathrm{dim}~K\geq 3$.

\end{proof}

\begin{lemma} \label{sec}
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 $\langle c_1\rangle\cup \langle c_2\rangle\subset \mathbb{P}(\bigwedge^2V_4)$. In particular, $\iota(\alpha([c_1]))=\alpha([c_2])$.
\end{lemma}
\begin{proof}
A general $X$ contains no planes and hence $l\not\subset X$.
The conic $c_i$ is contained in $\mathbb{P}(\bigwedge^2V_{4i})$ 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 $$l\subset \mathbb{P}(\bigwedge^2V_3)\subset \Gr(2,V_5).$$ In particular, $\langle c_i\rangle\cap \Gr(2,V_5)\neq c_i$. Therefore
by Lemma \ref{plane}, we have $\langle c_i\rangle\subset \Gr(2,V_5)$ 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 $S_{V_4}$. 
\end{proof}

We define $\tilde{I}\subset P\times P$ and $I'\subset \mathbb{P}\times \mathbb{P}$ to be the incidence correspondences, i.e. 
    $$\tilde{I}=(q\times q)^{-1}\Delta_X {\rm ~and~} I'=(q'\times q')^{-1}\Delta_{\mathbb{P}(\bigwedge^2V_5)}.$$ 
 Let $I_0=I\setminus (\Sigma_1\cup\Sigma_2\cup W)$, $\tilde{I_0}=(p\times p)^{-1}I_0$ and $I_0'=(p'\times p')^{-1}I_0$. 
Then Lemma \ref{sec} says that there is a section from $I_0$ to  $I'_0$, sending two intersecting conics to their common point. 
\begin{lemma}\label{L2.5}
The inclusion $I_0\subset F\times F\setminus (\Sigma_1\cup\Sigma_2\cup W)$ is a regular embedding.
\end{lemma}
\begin{proof}
The inclusion $I'\subset \mathbb{P}\times\mathbb{P}$ is a regular embedding since it is obtained from $\Delta_{\mathbb{P}(\bigwedge^2V_5)}\subset \mathbb{P}(\bigwedge^2V_5)\times \mathbb{P}(\bigwedge^2V_5)$ via base change. By the discussion above, there exists a section from $I_0$ to $I'_0$. We apply \cite[B.7.5]{fulton2016intersection} and obtain that $I_0\subset F\times F \setminus (\Sigma_1\cup\Sigma_2\cup W)$ is a regular embedding. 
\end{proof}

Now we can compute the self-intersection $I^2$, the analogous case of cubic fourfolds can be found in \cite[Proposition 2.3]{voisin4chow} or \cite[Appendix A.4]{fu2019generalized}.
\begin{proposition}\label{formula}
The self-intersection $$I^2=aW+I\cdot A+B+C\in \CH^4(F\times F)$$ for some constant $a$.  Here, $A$ and $B$ are contained in $pr_1^*\CH^*(F)\cdot pr_2^*\CH^*(F)\subset \CH^*(F\times F)$, where $pr_i$ is the  projection and $C$ is supported on $\Sigma_2\subset F\times F$.
\end{proposition}
\begin{proof}
By Lemma \ref{L2.5}, $I_0\subset F\times F\setminus (\Sigma_1\cup\Sigma_2\cup W)$ 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 
\begin{equation*}
    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.
\end{equation*}
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 
\begin{equation*}
    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
\end{equation*}
and
\begin{equation}\label{normalbundle}
    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})}.
\end{equation}
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 $pr_1^*\CH^*(F)\cdot pr_2^*\CH^*(F)$ and $pr_1^*\langle H,K\rangle\cdot pr_2^*\langle H,K\rangle$ restricting to $\tilde{I}_0$.  The following  computations is standard:
\begin{enumerate}
    \item  The Chern classes of $X$  
           \begin{equation*} 
              c(T_X)=\frac{c(T_{\Gr(2,V_5)})|_X}{(1+H)(1+2H)}
          ,\end{equation*}
          where $c(T_{\Gr(2,V_5)})$ 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 $\tilde{I}_0$.
    \item The Chern classes of the relative tangent bundle $T_{P|F}$
           \begin{equation*}
              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))}   
          ,\end{equation*}    
          where $c_1(\mathcal{O}_{\mathbb{P}}(1))|_X=H$ and $P=2c_1(\mathcal{O}_{\mathbb{P}}(1))+p^*D\in \CH^1(\mathbb{P})$ for some $D\in \CH^1(F)$. So the degree $1$ and degree $2$ parts are polynomials in $H$ and $\CH^*(F)$.    
\end{enumerate}
The claim follows from the computations and \eqref{normalbundle}. 
Next, the terms in $p_0^*c_2(N_{I_0/F\times F})$ are of the following three types:
\begin{enumerate}
    \item $D\cdot \tilde{I}_0$, where $D\in pr_1^*\CH^*(F)\cdot pr_2^*\CH^*(F)$.
    \item $(q\times q)^*pr_i^*H\cdot D\cdot \tilde{I}_0$ 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 $D_1\in \CH^1(F)$.
    \item $(q\times q)^*pr_i^*K\cdot \tilde{I}_0$ for $i=1$ or $2$.
\end{enumerate}
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 $pr_1^*\CH^*(X)\cdot pr_2^*\CH^*(X)$. Denote by $M=\Gr(2,V_5)\cap H$  the Grassmanian hull of $X$ and consider
\begin{equation*}
    j_1:X\times X\hookrightarrow M\times X.
\end{equation*}
We know that $M$ is a linear variety (Schubert varieties are linear in the sense of \cite{totaro2014chow}, since they are stratified by Schubert cells) and therefore
by \cite[Proposition 1]{totaro2014chow}, $$\CH^*(M\times X)=pr_1^*\CH^*(M)\cdot pr_2^*\CH^*(X).$$ Hence we have 
\begin{equation*}
    j_1^*j_{1*}\Delta_X\in pr_1^*\CH^*(X)\cdot pr_2^*\CH^*(X)
\end{equation*}
Since $X$ is of class $2H$ in $M$, we have $ j_1^*j_{1*}\Delta_X=2pr_1^*H\cdot \Delta_X$. Hence $pr_1^*H\cdot\Delta_X\in pr_1^*\CH^*(X)\cdot pr_2^*\CH^*(X)$.

For the third type, we recall that the class $K\in \CH^2(X)$ 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 $\CH^2(S\times S)$.  Then by the decomposition of diagonal, we see that $$\Delta_S=\alpha\times S+\Gamma,$$ where $\alpha\in \CH_0(S)$ 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 $\beta\in \CH_0(X)$. If $\dim V=1$, then $\Gamma\in \CH^1(V\times X)$. We see that $\CH^1(V\times X)=pr_1^*\CH^1(V)\cdot pr_2^*\CH^1(S)$ since ${\rm H}^1(S,\mathcal{O}_S)=0$. In both cases, we obtain: 
$$\Delta_S\in pr_1^*\CH^*(S)\cdot pr_2^*\CH^*(S),$$  which implies that $pr_i^*K\cdot\Delta_X$ are in $pr_1^*\CH^*(X)\cdot pr_2^*\CH^*(X)$.

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

\end{proof}
\subsection{\texorpdfstring{$Z$}{Y} 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 \eqref{weak splitting}. 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 $[c]\in \CH_1(X)$. 

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 \begin{equation}\label{conicindelpezzo}
     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.
\end{equation} 
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:
\begin{lemma}\label{KeyLemma}
Let $c_i$ be a conic on a general $S_{V_4}$ and $c_i\in L_i$, then 
\begin{equation}
    D_{c_i}^2=\sum_{j\neq i,j\neq i+5} L_j+4L_{i+5}+kL_{\sigma} \in \CH_1(F),
\end{equation}
where $L_{\sigma}\subset F$ is a curve contained in the locus of $\sigma$-conics.
\end{lemma}
\begin{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 $\CH^2(F)$. 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 \eqref{conicindelpezzo}. 

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 $$\mathbb{P}(V_1\bigwedge V_3)\subset \Gr(2,V_5)\cap \langle c\rangle.$$ By Lemma \ref{plane}, $c$ 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 \eqref{differential}.
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 $T_x c\subset T_xS_{V_4}$, hence the image of $T_x c$ in $N_{c/X}$ is $N_{c/S_{V_4},x}$.
Then by \eqref{differential},  we have $$T_{(c,x)}D_{c_i}'=dq^{-1}(T_xc_i)=\mathrm{H}^0(N_{c/X}\otimes I_x)\oplus \mathrm{H}^0(N_{c/S_{V_4},x}),$$ which is $3$-dimensional. Thus  $D_{c_i}'$ is smooth at $(c,x)$. Since $c_i$ and $c$ meet at a single point, $D_{c_i}'$ maps isomorphically to $D_{c_i}$ via $p$ around $c$. Hence,
\begin{equation*}
    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.
\end{equation*}
Similarly  for $c_0$, we obtain 
\begin{equation*}
    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.
\end{equation*}
Since $\mathrm{H}^0(N_{c/X}\otimes I_x\otimes I_y)=0$, we have 
\begin{equation*}
    T_{c}D_{c_i}\cap T_{c}D_{c_0}=\mathrm{H}^0(N_{c/S_{V_4}})=T_{c}L_j.
\end{equation*}
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 $p: D_{c_i}'\rightarrow D_{c_i}$ (resp. $D_{c_0}'\rightarrow D_{c_0}$) 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{equation}\label{cone}
\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}
\end{equation}
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 \eqref{cone} and the description of the normal bundle of conics.  By \cite[Proposition 1.29]{eisenbud20163264}, 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 $4$.
\end{proof}
Apply this Lemma to every $c_i\subset S_{V_4}$ and by a linear combination, we have:
\begin{equation}\label{inverseformula}
    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'  ,
\end{equation}
where $L'$ is a 1-cycle on $F$ consisting of $\sigma$-conics.

\begin{lemma}\label{nonzero}
The constant $a$ in Proposition \ref{formula} is nonzero.  If $n[c_1]\sim n[c_2]$ in $\CH_1(X)$ for some nonzero integer $n$, then $[c_1]\sim[c_2]$ in $\CH_0(\tilde{Y}^\vee)$.
\end{lemma}
\begin{proof}
We assume on the contrary that $a=0$. Then $$I^2=I\cdot A+B+C\in \CH^4(F\times F),$$ 
where $A,B,C$ are as in Proposition \ref{formula}. 
A correspondence $\Gamma$ of codimension larger than $0$ in $pr_1^*\CH^*(F)\cdot pr_2^*\CH^*(F)$  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  $\CH_0(F)\rightarrow \CH^*(F)$. Further, by Lemma \ref{L2.3}, $I=$ $^{t}P \circ P$ and hence $I_*: \CH_0(F)\rightarrow \CH_3(F)$ factor through $\CH_1(X)$. Together with the torsion-freeness of $\CH_0(F)$, it implies that if $a=0$, then for $\xi,\zeta \in \CH_0(F)$, 
\begin{equation}\label{contrary}
    I^2_*(\xi)=I^2_*(\zeta)
\end{equation}
as long as $q_*p^*(n\xi)=q_*p^*(n\zeta)\in \CH_1(X)$, where $p,q$ are the projections from the universal conic to $F$ and $X$. 

According to \cite[Lemma 17.3]{shen2016fourier}, we know that $I^2_*([c])=D_c^2$. Then \eqref{contrary} together with Lemma \ref{2.1} imply that $D_c^2+D_{\iota(c)}^2$ are constant in $\CH_1(F)$ when varying $c\in F$. 
We want to deduce a contradiction between the formula \eqref{inverseformula} and the constantness of $D_c^2+D_{\iota(c)}^2$.

Take a very ample divisor $G\subset F$.  Intersecting both sides of formula \eqref{inverseformula} 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 $\CH_0(\tilde{Y}^\vee)$, i.e., points on $$\Delta'=\{(y,\iota(y))~|~y\in \tilde{Y}^\vee\}\subset \tilde{Y}^\vee\times \tilde{Y}^\vee$$ are constant in $\CH_0(\tilde{Y}^\vee)$. 

Then for any power $\sigma^l$ of the two-form $\sigma$ on $\tilde{Y}^\vee$, we have $(pr_1^*(\sigma^l)+pr_2^*(\sigma^l))|_{\Delta'}=0$ by Mumford's theorem, see  \cite[Proposition 10.24]{voisin2003hodge}. Then we obtain $$2pr_1^*(\sigma^2)|_{\Delta'}=0,$$ but it is impossible since $pr_1$ is an isomorphism between $\Delta'$ 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 $n^2I^2_*([c_1])=n^2D_{c_1}^2=n^2D_{c_2}^2=n^2I^2_*([c_2])$ again by \cite[Lemma 17.3]{shen2016fourier} and $nI_*([c_1])=nI_*([c_2])$. Hence due to Proposition \ref{formula} , 
\begin{equation*}
    \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^2W([c_1]-[c_2])\\=&amn^2\cdot (\iota([c_1])-\iota([c_2])),
    \end{split}
\end{equation*}
here we view the cycles in $\CH^4(F\times F)$ as morphisms from $\CH_0(F)$ to $\CH_1(F)$. Then $$\iota([c_1])\sim \iota([c_2])$$ again by the torsion-freeness of $\CH_0(\tilde{Y}^\vee)$ and the fact that $a\neq 0$. Since $\iota$ is an involution, the result follows.

\end{proof}

Combining Lemma \ref{2.1} and Lemma \ref{nonzero}, we can get the following result :
\begin{theorem}\label{constant}
The invariant locus $Z$ of the involution is a constant cycle surface on $\tilde{Y}^\vee$.
\end{theorem}
\iffalse
\begin{proof}
For a general GM fourfold, the assertion follows from Lemma \ref{2.1} and Lemma \ref{nonzero}. It remains to run a specialization argument.

If $\tilde{Y}^\vee_0$ is any dual double EPW sextic, we choose a family of dual double EPW sextics and denote by $\cZ$ the invariant locus of the involution
$$\cZ\hookrightarrow\tilde{\cY}\xrightarrow{p} T,$$ 
such that for a general point $t\in T$, $Z_t$ is a constant cycle surface in $\tilde{Y}^\vee_t$ and the fiber over $0\in T$ is the original $\tilde{Y}^\vee_0$.

\end{proof}
\fi

Thus by \eqref{weak splitting}, $h^2\cdot Z=40o$ in $\CH_0(\tilde{Y}^\vee)$. 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:
\begin{proposition}\label{o}
For a very general $\tilde{Y}^\vee$, if $Z'\subset \tilde{Y}^\vee$ is another constant cycle surface, then for any point $z'\in Z'$, $z'$ is rationally equivalent to $o$.
\end{proposition}

\begin{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 \cite{ottem2015nef}, 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 \eqref{intersection number}, we have 
\begin{equation*}
\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}
 \end{equation*}
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.
 \end{proof}
 
\section{The filtration on \texorpdfstring{$\text{CH}_1(X)$}{CH\_1(X)}}

\subsection{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 $B$:
\begin{equation*}
    \begin{tikzcd}
D \arrow[r, hook] \arrow[d, "q"', dashed] & M, \\
B                                         &  
\end{tikzcd}
\end{equation*}
and the general fibers of $q$ are rational curves. By \cite{charles2019families}, 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 \cite[Lemma 1.1]{shen2020categories} and Proposition \ref{o}, 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 $\CH_1(X)$ 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.
\begin{definition}
The $i$th piece of the increasing filtration ${\rm S}_\bullet X\subset \CH_1(X)$  consists of $z\in \CH_1(X)$, such that (i.e., the class of $az$ can be represented by a sum of multiples of $i$ conics in a uniruled divisor and $\theta$) 
\begin{equation*}
    az=a_1c_1+a_2c_2\ldots +a_ic_i+a_0\theta
\end{equation*}
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 $Z$.
\end{definition}

The EPW sextics associated with period partners $X$ and $X'$ are the same. We have an immediate corollary according to the definition:
\begin{corollary}\label{partner}
For any $t\in \CH_0(\tilde{Y}^\vee)$, $\Psi(t)\in {\rm S}_i(X)$ iff $\Psi'_*(t)\in {\rm S}_i(X')$, where $\Psi':\CH_0(\tilde{Y}^\vee)\rightarrow \CH_1(X')$ is similarly defined as $\Psi$.
\end{corollary}

By the proof of Proposition \ref{ch}, there exists an integer $N$  such that $N\cdot z$ is a linear sum of conics for every $z\in \CH_1(X)$. At the end of the section, we show in Proposition \ref{conic} that every conic is rationally equivalent to a sum of conics in $D$. Hence, 
\begin{equation*}
    \bigcup_{i=0}^{\infty} {\rm S}_i(X)=\CH_1(X).
\end{equation*}
 

Let $\CH_0(\tilde{Y}^\vee)^{-}_{\rm hom}$ be the $\iota$-anti-invariant part of $\CH_0(\tilde{Y}^\vee)_{\rm hom}$ and $\CH_0(\tilde{Y}^\vee)^{+}_{\rm hom}$ be the $\iota$-invariant part.
There is a decomposition of $\CH_0(\tilde{Y}^\vee)$:
\begin{equation*}
    \CH_0(\tilde{Y}^\vee)=\mathbb{Z}\cdot o\oplus \CH_0(\tilde{Y}^\vee)^{-}_{\rm hom}\oplus \CH_0(\tilde{Y}^\vee)^{+}_{\rm hom}.
\end{equation*}

We show that the filtration on $\CH_1(X)$ can actually be defined on the $\iota$-anti-invariant part $\CH_0(\tilde{Y}^\vee)^{-}$. 

\begin{proposition}\label{decomposition}
The morphism $\Psi:\CH_0(\tilde{Y}^\vee)\rightarrow \CH_1(X)$  is zero on $\CH_0(\tilde{Y}^\vee)^{+}_{\rm hom}$ and an isomorphism onto $\CH_1(X)_{\rm hom}$ when restricting to $\CH_0(\tilde{Y}^\vee)^{-}_{\rm hom}$.
\end{proposition}
\begin{proof}
Due to the divisibility of $\CH_0(\tilde{Y}^\vee)_{\rm hom}$, the elements in $\CH_0(\tilde{Y}^\vee)^{+}_{\rm hom}$ are of the form $\sum (t_i+\iota(t_i)-2o)$ and elements in $\CH_0(\tilde{Y}^\vee)^{-}$ are of the form $\sum t_i-\iota(t_i)$, where $t_i\in\tilde{Y}^\vee$.
This yields  $\Psi$ is zero on $\CH_0(\tilde{Y}^\vee)^{+}_{\rm hom}$ by Lemma \ref{2.1}. 

For the second part, there exists an integer $N$, such that for any $z\in \CH_1(X)$, $N\cdot z$ is a linear combination of conics. Then $\CH_1(X)_{\rm hom}$ is generated by conics by the divisibility of $\CH_1(X)_{\rm hom}$.
Therefore, it suffices to show that $\Psi$ is injective on $\CH_0(\tilde{Y}^\vee)^{-}$. 

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 $c_i'$ the conics representing the class $\Psi(t_i)$ and $\Psi(t_i')$.
If we have the relation $$\Psi(\sum t_i-\iota(t_i))=\Psi(\sum t_i'-\iota(t_i'))\in \CH_1(X),$$  it implies that:
\begin{equation}\label{ker}
    \sum (D_{c_i}-D_{\iota(c_i)})=\sum (D_{c_i'}-D_{\iota(c_i')}).
\end{equation}
By Lemma \ref{KeyLemma}, we have:
\begin{equation*}
     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).
\end{equation*}
Here, we identify $\CH_0(F)$ and $\CH_0(\tilde{Y}^\vee)$ via the isomorphism $\alpha_*$.
By Lemma \ref{2.1}, we know that $D_c+D_{\iota(c)}$ is constant in $\CH^2(F)$,
we denote it by $D_0$. Therefore we obtain
\begin{equation*}
      G\cdot D_0 \cdot \sum (D_{c_i}-D_{\iota(c_i)})=4m\sum (\iota(t_i)-t_i) 
\end{equation*}
and
\begin{equation*}
      G\cdot D_0 \cdot \sum (D_{c_i'}-D_{\iota(c_i')})=4m\sum(\iota(t_i')-t_i').
\end{equation*}
The left-hand sides are equal by \eqref{ker}. Therefore by torsion-freeness we obtain: $$\sum t_i-\iota(t_i)=\sum t_i'-\iota(t_i').$$ Hence $\Psi$ is injective on $\CH_0(\tilde{Y}^\vee)^{-}$.
\end{proof}

Consequently, we immediately deduce that
\begin{theorem}\label{chowiso}
We have an isomorphism between groups
\begin{equation}
    \mathbb{Z}\cdot o\oplus \CH_0(\tilde{Y}^\vee)^{-}_{\rm hom}\cong \mathbb{Z}\cdot \theta \oplus \CH_1(X)_{\rm hom} \cong \CH_1(X).
\end{equation}
In particular, the group $\CH_1(X)$ is torsion-free. 
\end{theorem}
When $\tilde{Y}^\vee$ is birational to $S^{[2]}$ for some K3 surface $S$,  it has been shown in \cite{laterveer2021zero} that $\CH_0(\tilde{Y}^\vee)_\mathbb{Q}$ with $\mathbb{Q}$-coefficient has a natural decomposition for the Chow group. For 0-cycles, the decomposition is
$$\CH_0(\tilde{Y}^\vee)_\mathbb{Q}=\mathbb{Q}\cdot o\oplus\CH_0(S)_{\rm hom}\oplus\CH_0(\tilde{Y}^\vee)^+_{\rm hom}.$$
Theorem \ref{chowiso} can be viewed as a generalization of this decomposition, where the group $\CH_1(X)$ takes the place of $\CH_0(S)$.

\iffalse
The $EPW$ sextics associated with period partners $X$ and $X'$ are the same. We show that the filtration is intrinsic for $\tilde{Y}^\vee$. The Lemma \ref{decomposition} implies the following corollary immediately.
\begin{corollary}\label{partner}
The group $\CH_1(X)$ is torsion-free and for any $t\in \CH_0(\tilde{Y}^\vee)$ and $\mathbb{Z}\cdot o\oplus \CH_0(\tilde{Y}^\vee)^{-}_{\rm hom}\cong \CH_1(X)$. $\Psi(t)\in {\rm S}_i(X)$ iff $R'_*(t)\in {\rm S}_i(X')$.
\end{corollary}
\begin{proof}
For the first part, there exists an integer $N$, such that for any $z\in \CH_1(X)$, $N\cdot z$ is a linear combination of conics.  Hence by the divisibility of $\CH_1(X)_{\rm hom}$, we see that $\CH_1(X)_{\rm hom}\cong B(X)_{\rm hom}\cong \CH_0(\tilde{Y}^\vee)_{\rm hom}^-$, which is torsion-free.

The second part is obvious since the filtration is defined by $Z$ and a uniruled divisor on $\tilde{Y}^\vee$ and $\CH_1(X)_{\rm hom}\cong \CH_0(\tilde{Y}^\vee)_{\rm hom}^-$.

\end{proof}
\fi
\subsection{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 \ref{partner}. 

We consider the following incidence relation $$\Omega:=\{(A,B)~|~\mathrm{dim} (A\cap B)\geq 9\}\subset \mathbb{L}\mathbb{G}(\bigwedge^3V_6)\times \mathbb{L}\mathbb{G}(\bigwedge^3V_6).$$ Denote the fibre of the projection over $A$ by $\Omega_A$ and $F_y=y\wedge \bigwedge^2V_6\subset \bigwedge^3V_6$. Clearly, we have $$Y_B^{\geq 2}\subset Y_A^{\geq 1}$$ for $(A,B)\in \Omega$. We let 
$$\Sigma=\{A\in \mathbb{L}\mathbb{G}(\bigwedge^3V_6)~|~ A {\rm ~contains~ a~ decomposable~ vector}\},$$
and $\Sigma_k$ be the closure of the locus of $A$ that contains exactly $k$ decomposable vectors. 

We can deduce the following lemmas.  
\begin{lemma}
For any $A,B\subset \bigwedge^3V_6$ 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 $4$.
\end{lemma}
\begin{proof}
It's enough to take $V_5\in Y_{A^{\vee}}^{=1}\cap Y_{B^{\vee}}^{=2}$ by Theorem \ref{moduli}.
\end{proof}

\begin{lemma}
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  \rm{dim} Alb$(F(X_B))=2$.
\end{lemma}
\begin{proof}
For a general choice of $B\in \Sigma_8$ and $V_5$, we may assume that the $8$ decomposable vectors are $$\bigwedge^3V_{31},\ldots \bigwedge^3V_{38}\in B\cap \bigwedge^3V_6$$ and $V_{31}, \ldots V_{38}\not\subset  \bigwedge^3V_5$. Then, by \cite[Proposition 2.24]{debarre2018gushel}, the Grassmaninan hull $\Gr(2,V_5)\cap W_B$ is smooth. 
$\bigwedge^3V_{31}$ 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 \cite[Theorem 3.16]{debarre2018gushel}. Then by \cite[Lemma 4.1]{debarre2011nodal}, $X_B$ 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 \cite{debarre2011nodal}.  

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 $\Gr(2,V_5)$.
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 \cite[Proposition 2.20]{ferretti2009chow}, $C$  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 \'etale double cover $\tilde{C}\rightarrow C$, corresponding to the choice of a family of $3$-planes contained in a quadric in $C$, see \cite[section 4.2]{debarre2011nodal}. Let $p':\tilde{N} \rightarrow N$ 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 $S'$ the pull back of $P^{\vee}$ in $\tilde{C}^{(6)}$ and $S''$ the further pull back in $\tilde{N}^{(6)}$. 
By \cite[Proposition 5.8]{logachev2012fano}, $S'$ 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 $S''$. Then by \cite[Theorem 8.19]{welters1981abel}, $${\rm Alb} (F(X_B))\cong Pr(\tilde{N}/N),$$ which is of dimension $2$ since $g(N)=3$.
\end{proof}
\begin{lemma}
For a general point $y\in Y_A$, we can find  $B\in \Omega_A\cap \Sigma_8$, such that $y\in Y_B^{\geq 2}$.
\end{lemma}
\begin{proof}
The proof is similar to \cite[Proposition 5.1]{ferretti2012special}.
Let $$\Gamma=\Omega\cap \mathbb{L}\mathbb{G}(\bigwedge^3V_6)\times \Sigma_8\cap \mathbb{L}\mathbb{G}(\bigwedge^3V_6)\times \Omega_y,$$ where $\Omega_y=\{B~|~\mathrm{dim}(B\cap F_y)\geq 2\}$ and $F_y=y\wedge \bigwedge^2V_6$.  We have the two projections:

\begin{equation*}
\begin{tikzcd}
                       & \Gamma \arrow[ld, "\pi"'] \arrow[rd, "\rho"] &          \\
\mathbb{L}\mathbb{G}(\bigwedge^3V_6) &                                              & \Sigma_8
\end{tikzcd}    
\end{equation*}
The fiber of $\rho$ is a codimension-$2$ subvariety of $\Omega_B$, which is $8$-dimensional, see \cite[Lemma 5.3]{ferretti2012special}. It yields that dim $\Gamma=$ dim $\mathbb{L}\mathbb{G}(\bigwedge^3V_6)$. 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 \cite[Lemma 5.4]{ferretti2012special}, we have the description of tangent spaces:
\begin{equation*}
\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}
\end{equation*}

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 $q_B=U\cup U'$ for some hyperplane $U'\subset B$. 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 $U'$. Then for a general $B$,  dim $U'=10$, which is a contradiction. 
\end{proof}
Now we can deduce the main result of this subsection:
\begin{proposition}\label{conic}
For any conic $c\subset X$, we have $c\in {\rm S}_2(X)$.
\end{proposition}
\begin{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 $X'$ of $X$ such that $X_B\subset X'$. Let $S'$ be the image of $$F(X_B)\subset F(X)\xrightarrow{\alpha} \tilde{Y}^\vee$$ and $\tilde{S}$ be the resolution of $S'$. Then a conic lying in $R'$ is in ${\rm S}_1(X)$, where $R'$ is the normalization of $S'\cap D$.

The argument of \cite[Section 2.4]{shen2020categories} implies that there is a surjection $$R'^{(2)}\twoheadrightarrow Alb(\tilde{S}).$$  
For a resolution $\tilde{X}_B\rightarrow X_B$, we have $\CH_1(\tilde{X}_B)_{\rm hom}\cong J(\tilde{X}_B)$ by \cite[Theorem 0.3]{voisin2013abel}. Hence, $\CH_0(\tilde{S})_{\rm hom}\rightarrow \CH_1(\tilde{X}_B)_{\rm hom}$ factors through $Alb(\tilde{S})$.  By the surjectivity of $R'^{(2)}\rightarrow Alb(\tilde{S})$, 
there exists two conics $c_1$ and $c_2\in {\rm S}_1(X')$ such that 
$c=c_1+c_2-\theta\in \CH_1(\tilde{X}_B)$. Then the result follows from Corollary \ref{partner}.
\end{proof}

\section{Sheaves supported on conics and lines}\label{S5}
In this section, we introduce a link between the filtration on $\CH_1(X)$ and the Kuznetsov component $\mathcal{A}_X$ of the derived category of $X$ and prove Theorem \ref{mainthm}. 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 $\pr: K_0(X)\rightarrow K_0(\mathcal{A}_X)$ be the projection of Grothendieck groups.  Let $\overline{\pr}$ 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. \cite[Lemma 2.4]{pertusi2019double}): 
$$\begin{pmatrix}
-2 & 0 \\ 0 & -2
\end{pmatrix},$$
where $\lambda_1=\overline{\pr} [\mathcal{O}_c(1)]$ and $\overline{\pr} [\mathcal{O}_p]=-\lambda_1-2\lambda_2$ for a conic $c\subset X$ and a point $p\in X$.  

\begin{lemma}\label{K0}
$K_0(\mathcal{A}_X)\otimes \mathbb{Q}$  is generated by $\pr [\mathcal{O}_c(1)]$ and $\pr [\mathcal{O}_p]$.
\end{lemma}
\begin{proof}
Since $\CH_0(X)\cong\mathbb{Z}$, the class $[\mathcal{O}_p]$ in $K_0(X)$ is independent of the choice of $p\in X$ and the cycle class map $cl: \CH^i(X)\rightarrow {\rm H}^i(X,\mathbb{Z})$ is injective for $i\neq 3$, by \cite[Theorem 1.1]{bloch1983remarks}. 

We have the following commutative diagram:
$$\begin{tikzcd}
K_0(\mathcal{A}_X)\otimes \mathbb{Q} \arrow[d, "\overline{\pr}"] \arrow[rr, "\ch"] &  & \CH^*(X)\otimes \mathbb{Q} \arrow[d, "cl"] \\
K_{\rm num}(\mathcal{A}_X)\otimes \mathbb{Q} \arrow[rr, "\overline{\ch}"]                         &  & {{\rm H}^*(X,\mathbb{Q})}    ,       
\end{tikzcd}$$
It's enough to show that $cl^{-1}(\overline{\ch}(K_{\rm num}(\mathcal{A}_X)))$ is generated by $\ch(\mathcal{O}_c(1))$ and $\ch(\mathcal{O}_p)$ and this follows from the injectivity of the cycle class map for $i\neq 3$ and the fact that  $\CH_1(X)\otimes \mathbb{Q}$ is generated by conics.
 \end{proof}
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 $X'$ are period partners, their associated dual double EPW sextics are the same by Theorem \ref{moduli}.
Therefore, their Fano varieties of conics both admit a birational $\mathbb{P}^1$-fibration to $\tilde{Y}^\vee$. Then by Lemma \ref{K0} there is a group isomorphism $$K_0(\mathcal{A}_X)_\mathbb{Q}\cong K_0(\mathcal{A}_X')_\mathbb{Q},$$ which maps $\pr [\mathcal{O}_p]$ to $\pr [\mathcal{O}_{p'}]$ and $\pr [\mathcal{O}_c(1)]$ to $\pr [\mathcal{O}_{c'}(1)]$. Here $c'$ is a conic mapping to the same point with $c$ in $\tilde{Y}^\vee$ via the fibration. 

However, the classes $c_3(\pr [\mathcal{O}_p])$ and $c_3(\pr [\mathcal{O}_{p'}])$ do not coincide in $\CH_0(\tilde{Y}^\vee)$ via the isomorphism \eqref{chowiso} 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 \eqref{chowiso} by dropping the term of $\pr [\mathcal{O}_p]$: 

If $[\mathcal{E}]=\sum_i a_i\pr[\mathcal{O}_{c_i}(1)]+b\pr[\mathcal{O}_p]\in K_0(\mathcal{A}_X)_\mathbb{Q}$, we set $$p(\mathcal{E})=\sum a_ic_i\in \CH_1(X)\otimes \mathbb{Q}.$$ 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 
$$d(i^*\mathcal{F}):=\frac{1}{2}\dim \Ext^1_{\mathcal{A}_X}(i^*\mathcal{F},i^*\mathcal{F})$$ and we have:

\begin{lemma}
If $\mathcal{F}$ is supported on a line or a conic, then $d(i^*\mathcal{F})\geq 2$.
\end{lemma}
\begin{proof}
By straightforward computation, we have $\overline{\pr} [\mathcal{O}_p]=-\lambda_1-2\lambda_2$ and $\overline{\pr} [\mathcal{O}_l(1)]=-\lambda_1$. Hence $\overline{\pr} [\mathcal{E}]=-m\lambda_1-(2m+re)\lambda_2$.
Then 
$$d(i^*\mathcal{F})\geq \frac{1}{2}\chi (i^*\mathcal{F},i^*\mathcal{F})+1=5m^2+4mre+r^2e^2+1.$$
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 $2$.
\end{proof}

Now we can prove the main theorem:
\begin{theorem}\label{maint}
If $\mathcal{F}$ is supported on a line or a conic and $\mathcal{E}=i^*\mathcal{F}$, then
\[
p(\mathcal{E}) \in {\rm S}_{d(\mathcal{E})}(X).
\]
\end{theorem}
\begin{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 \ref{conic}. 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 \cite[Lemma 3.8]{iliev2011fano}. Since $X$ is very general,  this surface meets $Z$  by Proposition \ref{o}. Hence, there exists a line $l_0\subset X$ such that $2l_0= \theta$. Take $\mathbb{P}(V_1'\bigwedge V_3')=l'=\iota_1(l_0)$,  we obtain
$$2l'=2\sigma-\theta,$$
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 $V_2=V_1\oplus V_1'$ and $V_4=V_1'\oplus V_3$ 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 $l'$ are contained in $S_{V_2}$ and $S_{V_2}\cap S_{V_4}$ is a hyperplane section of $S_{V_4}\subset \mathbb{P}(\bigwedge^2V_4)\cap H$, which is the union of $l$ and a degree $3$ curve $C_3$. Hence
$$l+C_3=2\theta .$$ 
Since $V_3'\cap V_4$ is 2-dimensional, $l'$ meets $S_{V_4}$ at the point corresponding to $V_3'\cap V_4$. For a general choice of $l$, it does not meet $l'$. It follows that $l'$ 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 $l'\cup C_3\cup c$ for some conic $c$. 

We claim that 
$$l'+C_3+c=2\theta+\sigma.$$
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 $l+C_3+c'$, which equals $2\theta+\sigma$. Here $c'$ is the $\sigma$-type conic associated with $V_1$.

Finally, we obtain that $l=l'+c-\sigma=c-l_0$, which is in $S_2(X)$ by Proposition \ref{conic}.
\end{proof}

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