# Chow groups of surfaces of lines in cubic fourfolds

D. Huybrechts  Mathematisches Institut, Universität Bonn, Endenicher Allee 60,
53115 Bonn, Germany<EMAIL_ADDRESS>
  *     * scAbstract. The surface of lines in a cubic fourfold intersecting a fixed line splits motivically into two parts, one of which resembles a K3 surface. We define the analogue of the Beauville–Voisin class and study the push-forward map to the Fano variety of all lines with respect to the natural splitting of the Bloch–Beilinson filtration introduced by Mingmin Shen and Charles Vial.

scKeywords. Cubic fourfolds, Fano variety of lines, Chow groups, Bloch
conjecture, Beauville–Voisin class

sc2020 Mathematics Subject Classification. 14C15, 14J70, 14J28, 14J29

  *
cApril 28, 2023Received by the Editors on December 5, 2022.
Accepted on May 21, 2023.

Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn,
Germany

sce-mail<EMAIL_ADDRESS>
This research was supported by the ERC Synergy Grant HyperK, Grant agreement
ID 854361, and the Hausdorff Center for Mathematics, Bonn (Germany’s
Excellence Strategy–EXC-2047/1–390685813).

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 Introduction
  2. 2 Zero-cycles on hyperkähler fourfolds
  3. 3 Special surfaces
  4. 4 Lines intersecting a fixed line

## 1\. Introduction

Fano varieties $F=F(X)$ of lines $L\subset X$ in smooth complex cubic
fourfolds $X\subset\mathbb P^{5}$ are hyperkähler manifolds of dimension four
and as such have many properties and features in common with K3 surfaces. In
particular, the Chow ring $\operatorname{CH}^{\ast}(F)$ of the Fano variety is
of great interest and has been investigated from various angles by Beauville
[Bea07], Voisin [Voi08, Voi16], M. Shen and Vial [SV16], J. Shen, Yin, and
Zhao [SYZ20, SY20], and many others.

### 1.1.

The first goal of this note is to study zero-dimensional cycles on $F$ from
the perspective of distinguished surfaces contained in $F$. More specifically,
we continue our investigation [Huy21] and consider the surface
$F_{L_{0}}\subset F$ of all lines $L\subset X$ intersecting a given fixed line
$L_{0}\subset X$. The surfaces $F_{L_{0}}$ and their Chow groups are of great
interest in their own right, but here, in order to gain a better understanding
of the Chow group of zero-dimensional cycles
$\operatorname{CH}^{4}(F)=\operatorname{CH}_{0}(F)$ on $F$, we first have a
closer look at the push-forward map
$\operatorname{CH}_{0}(F_{L_{0}})\to\operatorname{CH}_{0}(F)$.

The target and source of this map both come with a natural decomposition, and
our first result explains how they are related.

  * •

There exists a natural involution $\iota\colon F_{L_{0}}\xrightarrow{\vbox
to0.0pt{\vss\hbox{$\scriptstyle\sim$}\vskip-2.0pt}}F_{L_{0}}$, and its
quotient $F_{L_{0}}\to D_{L_{0}}\coloneqq F_{L_{0}}/_{\pm}$ describes a
quintic surface $D_{L_{0}}\subset\mathbb P^{3}$ with $16$ nodes. Homologically
trivial zero-cycles on $D_{L_{0}}$ can be identified with $\iota$-invariant
homologically trivial zero-cycles on $F_{L_{0}}$; i.e.
$\operatorname{CH}_{0}(D_{L_{0}})_{\hom}$ is the first summand of the
$\iota$-eigenspace decomposition

(1.1)
$\operatorname{CH}_{0}\left(F_{L_{0}}\right)_{\hom}=\operatorname{CH}_{0}\left(F_{L_{0}}\right)_{\hom}^{+}\oplus\operatorname{CH}_{0}\left(F_{L_{0}}\right)^{-}.$

The second summand gives back the interesting part of the Chow group of the
cubic itself, namely
$\operatorname{CH}_{0}(F_{L_{0}})^{-}\simeq\operatorname{CH}_{1}(X)_{\hom}$
via the Fano correspondence.

  * •

On the Fano side, the group $A\coloneqq\operatorname{CH}_{0}(F(X))$ splits as

(1.2) $A=A_{4}\oplus A_{2}\oplus A_{0}.$

Here, $A_{0}$ is generated by a distinguished class $c_{F}$, the analogue of
the Beauville–Voisin class on a K3 surface, and $A_{4}$ is the deepest part of
the Bloch–Beilinson filtration. This decomposition was studied in depth by
Shen and Vial [SV16]; see Section 2.2 for more details and references.

Our first result clarifies the link between the two decompositions (1.1) and
(1.2). The result confirms certain aspects of the Bloch–Beilinson philosophy
in the present situation.

######  Theorem 1.1.

Assume $L_{0}\subset X$ is a general line in a smooth cubic fourfold
$X\subset\mathbb P^{5}$.

  1. (i)

The push-forward map of invariant homologically trivial cycles

(1.3)
$\operatorname{CH}_{0}\left(D_{L_{0}}\right)_{\hom}\simeq\operatorname{CH}_{0}\left(F_{L_{0}}\right)^{+}_{\hom}\to
A$

takes image in $A_{4}\subset A$. For very general $X$ and very general
$L_{0}\subset X$, the image is non-trivial. Furthermore, the image of the
composition

$\operatorname{CH}_{0}\left(D_{L_{0}}\right)\longrightarrow
A\relbar\joinrel\twoheadrightarrow A/A_{4}=A_{2}\oplus A_{0}$

is spanned by the class $c_{F}(L_{0})\coloneqq 2c_{F}-[L_{0}]_{2}$, where
$[L_{0}]_{2}$ denotes the $A_{2}$-component of $[L_{0}]$.

  2. (ii)

The composition of the push-forward map and the projection
$A\twoheadrightarrow A_{2}\oplus A_{0}$ describes an isomorphism

$\operatorname{CH}_{0}\left(F_{L_{0}}\right)^{-}\xrightarrow{\;\vbox
to0.0pt{\vss\hbox{$\scriptstyle\sim$}\vskip-2.0pt}\;}A_{2}$

with its inverse induced by the map $[L]\mapsto(-1/2)[F_{L}]|_{F_{L_{0}}}$.
Furthermore, if $L_{0}$ is very general, the image of
$\operatorname{CH}_{0}(F_{L_{0}})^{-}\to A$ is not contained in $A_{2}$.

### 1.2.

The second goal of the paper is to investigate the K3 nature of the ‘negative
half ’ $F_{L_{0}}^{-}$ of the surface $F_{L_{0}}$. The surface $F_{L_{0}}$
‘decomposes’ into two parts, and one component behaves very much like a K3
surface. On the level of Chow groups, the two parts correspond to the
$\iota$-invariant and the $\iota$-anti-invariant cycles, and for rational Chow
motives we have the decomposition ${\mathfrak h}(F_{L_{0}})\simeq{\mathfrak
h}(F_{L_{0}})^{+}\oplus{\mathfrak h}(F_{L_{0}})^{-}$; see the discussion in
[Huy21, Section 4.6]. The K3 nature of the anti-invariant part is best seen by
the observation that $H^{2}(F_{L_{0}},\mathbb Z)^{-}_{\operatorname{pr}}$ is a
Hodge structure of K3 type of rank $22$, which, up to a factor and a Tate
twist, is in fact Hodge isometric to the primitive middle cohomology
$H^{4}(X,\mathbb Z)_{\operatorname{pr}}$ of the cubic fourfold; cf. [Huy21,
Theorem 0.2].

One distinguished feature, cf. [BV04], of the Chow ring of a complex K3
surface $S$ is that despite $\operatorname{CH}^{2}(S)$ being big, all
intersection products $\alpha_{1}\cdot\alpha_{2}$ of classes
$\alpha_{1},\alpha_{2}\in\operatorname{CH}^{1}(S)$ are proportional. They are
multiples of the Beauville–Voisin class $c_{S}\in\operatorname{CH}^{2}(S)$,
which is realised by any point contained in a rational curve. Transplanted in
our context, this triggers the question whether products
$\alpha_{1}\cdot\alpha_{2}$ of two anti-invariant classes
$\alpha_{1},\alpha_{2}\in\operatorname{CH}^{1}(F_{L_{0}})^{-}$ are all
multiples of a distinguished class
$c_{L_{0}}\in\operatorname{CH}^{2}(F_{L_{0}})$, which would necessarily be
invariant. Our second result shows that this is the case for primitive classes
and after push-forward to $F$.

######  Theorem 1.2.

For a general line $L_{0}\subset X$ in a smooth cubic fourfold, the
composition

$\operatorname{CH}^{1}\left(F_{L_{0}}\right)_{\operatorname{pr}}^{-}\times\operatorname{CH}^{1}\left(F_{L_{0}}\right)_{\operatorname{pr}}^{-}\longrightarrow\operatorname{CH}^{2}\left(F_{L_{0}}\right)^{+}\longrightarrow
A$

takes image in $\mathbb Z\cdot c_{F}(L_{0})$ with
$c_{F}(L_{0})=2c_{F}-[L_{0}]_{2}\in A_{2}\oplus A_{0}\subset A$ as above.

This suggests that we should view

$\operatorname{CH}^{\ast}\left(F_{L_{0}}^{-}\right)\coloneqq\mathbb
Z\oplus\operatorname{CH}^{1}\left(F_{L_{0}}\right)^{-}_{\operatorname{pr}}\oplus\left(\operatorname{CH}^{2}\left(F_{L_{0}}\right)^{-}\oplus\mathbb
Z\cdot c_{F}(L_{0})\right)$

as the Chow ring of $F_{L_{0}}^{-}$, i.e. of the ‘K3 half ’ of the surface
$F_{L_{0}}$. Coming back to Theorem 1.1, the natural isomorphism
$\operatorname{CH}_{0}(F_{L_{0}})^{-}\simeq A_{2}$ highlights the K3 character
of the middle part $A_{2}$ of the decomposition (1.2).

######  Remark 1.3.

One could ask whether
$\operatorname{CH}^{2}(F_{L_{0}}^{-})=\operatorname{CH}^{2}(F_{L_{0}})^{-}\oplus\mathbb
Z\cdot c_{F}(L_{0})$ can be realised simply as the image of
$\operatorname{CH}_{0}(F_{L_{0}})\to\operatorname{CH}_{0}(F)$, so that the
hyperkähler fourfold $F$ singles out the ‘K3 half ’ of $F_{L_{0}}$. However,
this fails due to the slightly counter-intuitive non-triviality of (1.3) in
Theorem 1.1(i).

In the opposite direction, one could wonder whether (1.3) is maybe injective.
This would have the effect that $c_{F}(L_{0})$ lifts uniquely to a class
$c_{L_{0}}\in\operatorname{CH}_{0}(F_{L_{0}})^{+}$ such that

$\mathbb
Z\oplus\operatorname{CH}^{1}\left(F_{L_{0}}\right)^{-}_{\operatorname{pr}}\oplus\left(\operatorname{CH}^{2}\left(F_{L_{0}}\right)^{-}\oplus\mathbb
Z\cdot c_{L_{0}}\right)\subset\operatorname{CH}^{*}\left(F_{L_{0}}\right)$

is a subring, i.e. such that $\alpha_{1}\cdot\alpha_{2}\in\mathbb Z\cdot
c_{L_{0}}$ for all
$\alpha_{1},\alpha_{2}\in\operatorname{CH}^{1}(F_{L_{0}})_{\operatorname{pr}}^{-}$.
The class $c_{L_{0}}$ on $F_{L_{0}}$ would then be the geometric realisation
of the Beauville–Voisin class for the K3 half $F_{L_{0}}^{-}$; see Remark 4.9
for further comments.

The main tool to prove these results is the map
$\psi\colon\operatorname{CH}_{0}(F)\to\operatorname{CH}_{2}(F)$,
$L\mapsto[F_{L}]$, which has been studied and used before; see [SV16, SYZ20,
SY20, Voi16]. Further restricting to surfaces contained in $F$ allows one to
transform zero-cycles on $F$ into zero-cycles on surfaces. However, since
$\psi$ annihilates the deepest part of the Bloch–Beilinson filtration
$A_{4}\subset\operatorname{CH}_{0}(F)$, this technique does not provide
control over all zero-cycles, on the Fano variety or on surfaces contained in
it.

Acknowledgement.

I wish to thank Claire Voisin for helpful comments and inspiration in general.
I am particularly indebted to Charles Vial for many helpful comments. In
particular, Remark 4.6(ii) follows his suggestions.

## 2\. Zero-cycles on hyperkähler fourfolds

It is expected that the Chow groups of an arbitrary smooth projective variety
$Z$ are endowed with natural (Bloch–Beilinson) filtrations

$\cdots\subset F^{2}\subset F^{1}\subset\operatorname{CH}^{k}(Z)$

that enjoy certain functoriality properties. Also, the filtrations should be
finite, more precisely $F^{i}=0$ for $i>k$, and the behaviour of algebraic
correspondences on the graded parts $F^{i}/F^{i+1}$ should be determined by
their action on the corresponding pieces of the Hodge structure of $Z$.

For example, any correspondence
${}_{\ast}\colon\operatorname{CH}_{0}(Z_{1})\to\operatorname{CH}_{0}(Z_{2})$
should respect the filtration, i.e. ${}_{*}(F^{i}(Z_{1}))\subset
F^{i}(Z_{2})$, and the induced map
${\operatorname{Gr}}^{i}({}_{\ast})\colon(F^{i}/F^{i+1})(Z_{1})\to(F^{i}/F^{i+1})(Z_{2})$
should be zero if and only if the cohomological action ${}^{\ast}\colon
H^{0}(Z_{2},{}_{Z_{2}}^{i})\to H^{0}(Z_{1},{}_{Z_{1}}^{i})$ is trivial. This
is often seen as a generalisation of Bloch’s conjecture; see [Voi02,
Conjecture 23.22].

There are various suggestions for such filtrations proposed by H. Saito
[Sai92], S. Saito [Sai96], Nori [Nor93], and Voisin [Voi04], but proving the
desired properties in general seems out of reach for now.

######  Example 2.1.

For zero-cycles on a smooth projective surface $S$, the various requirements
determine the Bloch–Beilinson filtration completely: concretely,
$F^{1}\operatorname{CH}_{0}(S)$ is the subgroup of all cycles of degree zero
or, equivalently, of those that are algebraically or, still equivalently,
homologically trivial, while $F^{2}\subset F^{1}$ is the kernel of the
Albanese morphism $F^{1}\to{\operatorname{Alb}}(S)$.

### 2.1.

For a hyperkähler manifold $Z$ of dimension four, the conjectural filtration
on zero-cycles would be of the form

$0\subset F^{4}\subset F^{3}\subset F^{2}\subset F^{1}\subset
F^{0}=A\coloneqq\operatorname{CH}_{0}(Z).$

Recall that by Roitman’s result [Roj80] the group $A$ is torsion-free, and, by
using the divisibility of Jacobians of curves contained in $Z$, the kernel of
the degree map $\deg\colon A\to\mathbb Z$ is known to be divisible.

Bloch’s conjecture applied to the identity correspondence
$[\Delta]_{\ast}={\operatorname{id}}$ leads to the following picture. The
vanishings $H^{0}(Z,{}_{Z}^{3})=0=H^{0}(Z,{}_{Z}^{1})$ should imply
$F^{4}=F^{3}$ and $F^{2}=F^{1}$, and the non-vanishings
$H^{0}(Z,{}_{Z}^{4})\neq 0\neq H^{0}(Z,{}_{Z}^{2})$ should give $F^{4}\neq 0$
and $F^{3}\neq F^{2}$. So the conjectural Bloch–Beilinson filtration in this
case would actually look like

(2.1) $0\subsetneqq F^{4}=F^{3}\subsetneqq F^{2}=F^{1}\subsetneqq
F^{0}=A=\operatorname{CH}_{0}(Z).$

In full generality, only $F^{2}=F^{1}$ is rigorously defined. Also note that a
generalisation of an argument of Mumford for surfaces shows that $F^{1}$ is
not trivial and in fact not representable by any finite-type scheme; see
[Voi02, Theorem 22.15].

Taking inspiration from the case of K3 surfaces [BV04], Beauville [Bea07]
asked whether the hyperkähler structure of $Z$ gives rise to a certain
splitting of the conjectural filtration (2.1). This would lead to a
decomposition

$A=A_{4}\oplus A_{2}\oplus A_{0},$

with $F^{4}=F^{3}=A_{4}$, $F^{2}=F^{1}=A_{4}\oplus A_{2}$, and $\deg\colon
A_{0}\xrightarrow{\vbox
to0.0pt{\vss\hbox{$\scriptstyle\sim$}\vskip-2.0pt}}\mathbb Z$. The generator
of degree one of $A_{4}$ is called the Beauville–Voisin class $c_{Z}\in A$.

Shen and Vial [SV16] suggested to use a certain lift
$\tilde{q}\in\operatorname{CH}^{2}(Z\times Z)$ of the class $q_{Z}\in
S^{2}H^{2}(Z,\mathbb Q)\subset H^{4}(Z\times Z,\mathbb Q)$ induced by the
Beauville–Bogomolov–Fujiki pairing as the analogue of the Poincaré bundle for
abelian varieties to define the decomposition as

$A_{k}\coloneqq\left\\{\alpha\in
A\mid\exp(\tilde{q})(\alpha)\in\operatorname{CH}^{k}(Z)\right\\}.$

An alternative and more geometric definition was proposed by Voisin [Voi16]:
One first introduces the orbit ${\rm O}_{x}\subset Z$ of any point $x\in Z$ as
the set of all points that are rationally equivalent to $x$. The orbit turns
out to be a countable union of closed subsets of dimensions at most two. Then,
$A_{0}\subset A$ and $A_{2}\oplus A_{0}$ should be generated by those points
with $\dim{\rm O}_{x}\geq 2$ and $\dim{\rm O}_{x}\geq 1$, respectively.

### 2.2.

Typically, conjectures concerning hyperkähler manifolds (of dimension four)
are first checked for the Hilbert scheme $S^{[2]}$ of a K3 surface $S$ and for
the Fano variety $F=F(X)$ of lines contained in a smooth cubic fourfold
$X\subset\mathbb P^{5}$. This has been undertaken for Beauville’s question by
Beauville [Bea07] himself, by Voisin [Voi08, Voi16], by M. Shen and Vial
[SV16], and more recently by J. Shen, Yin, and Zhao [SY20, SYZ20]. We briefly
review some of the results for the Fano variety $F=F(X)$ and its group of
zero-cycles $A=\operatorname{CH}_{0}(F)$.

Firstly, there is a natural candidate for the Beauville–Voisin class, i.e. the
generator of $A_{0}$, namely the unique class
$c_{F}\in\operatorname{CH}_{0}(F)$ satisfying

$6\cdot c_{F}=g\cdot[C_{x}].$

Here, $g\in\operatorname{CH}^{1}(F)$ is the Plücker polarisation and
$C_{x}\coloneqq\\{L\mid x\in L\\}\subset F$, which is a curve for the general
point $x\in X$. Observe that, since the cubic $X$ is unirational, the class of
the curve $[C_{x}]\in\operatorname{CH}^{3}(F)$ does not depend on the choice
of the general point $x$. The class $c_{F}$ indeed has the desired
multiplicative property: according to a result of Voisin [Voi08, Theorem 0.4],
the degree four component of any polynomial expression involving only the
Chern classes ${\rm c}_{2}(F)$ and ${\rm c}_{4}(F)$ and classes
$\alpha\in\operatorname{CH}^{1}(F)$ is a multiple of $c_{F}$.

Secondly, one exploits the Voisin map $f\colon F\dashrightarrow F$ and its
induced action on the Chow group $f_{\ast}\colon A\to A$. Recall that the
Voisin map is the endomorphism of degree 16 that maps a line of the first type
$L$ to the residual line $L^{\prime}$ of the intersection $L\subset P_{L}\cap
X$ with the unique plane $P_{L}\simeq\mathbb P^{2}$ tangent to $L\subset X$;
i.e. $P_{L}\cap X=2L\cup L^{\prime}$. Shen and Vial [SV16, Theorem 21.9]
proved that the induced action $f_{\ast}$ on $\operatorname{CH}_{0}$ has
eigenvalues $4$, $-2$, and $1$ and declare the corresponding eigenspaces to be
$A_{4}$, $A_{2}$, and $A_{0}=\mathbb Z\cdot c_{F}$.

Thirdly, mapping a line $L$ to the surface $F_{L}$ induces a map

(2.2) $\psi\colon
A\longrightarrow\operatorname{CH}_{2}(F),\quad[L]\longmapsto[F_{L}],$

and one defines $A_{4}$ as the kernel of $\psi$. Alternatively, $A_{4}$ can
also be described as the homologically trivial part of the subgroup generated
by all triangles, i.e. by sums $[L_{0}]+[L_{1}]+[L_{2}]$ of triples of lines
$L_{0},L_{1},L_{2}\subset X$ spanning a plane. Yet another possibility is to
describe $A_{4}$ as the kernel of the Fano correspondence $\varphi\colon
A=\operatorname{CH}_{0}(F)\to\operatorname{CH}_{1}(X)$ mapping the class of a
point in $F$ corresponding to a line in $X$ to the class of that line. In
fact, $\varphi$ factors through an isomorphism:

(2.3) $\varphi\colon A\relbar\joinrel\twoheadrightarrow A/A_{4}\simeq
A_{2}\oplus A_{0}\xrightarrow{\;\vbox
to0.0pt{\vss\hbox{$\scriptstyle\sim$}\vskip-2.0pt}\;}\operatorname{CH}_{1}(X).$

It is a priori not clear that the various definitions of the deepest part of
the Bloch–Beilinson filtration

(2.4) $A_{4}=\left\\{\alpha\mid
f_{\ast}\alpha=4\cdot\alpha\right\\}=\ker(\psi)=\ker(\varphi)=\left\langle[L_{0}]+[L_{1}]+[L_{2}]\right\rangle_{\hom}$

or of its complement

$A_{2}\oplus A_{0}=\left\\{\alpha\mid
f_{*}\alpha=-2\cdot\alpha\right\\}\oplus\left\\{\alpha\mid
f_{*}\alpha=\alpha\right\\}=\left\langle[L]\mid\dim{\rm O}_{L}\geq
1\right\rangle$

describe the same subgroups. The often geometrically quite intricate arguments
have been provided by Shen and Vial [SV16] and Voisin [Voi16]. Note that
$A_{4}$ and $A_{2}$ are both non-trivial, but this is not automatic. In fact,
both parts should be thought of as ‘big’, i.e. non-representable, although
$A_{2}$ is surface-like while $A_{4}$ ‘lives’ in dimension four; see below for
more on this.

Of the many results in the comprehensive monograph of Shen and Vial [SV16],
the following will be important for our discussion:

  1. (i)

The first concerns the multiplication
$\operatorname{CH}^{2}(F)\times\operatorname{CH}^{2}(F)\to\operatorname{CH}^{4}(F)=A$.
For a triangle $L_{0}\cup L_{1}\cup L_{2}\subset X$, the classes
$[F_{L_{i}}]\in\operatorname{CH}^{2}(F)$ satisfy, cf. [SV16, Proposition
20.7],

(2.5) $\left[F_{L_{1}}\right]\cdot\left[F_{L_{2}}\right]=6\cdot
c_{F}+[L_{0}]-[L_{1}]-[L_{2}].$

  2. (ii)

The second is about the image of the distinguished class $c_{F}$ under (2.2).
According to [SV16, Lemma A.5], one has

(2.6) $3\cdot\psi(c_{F})=g^{2}-{\rm c}_{2}(\mathcal{S}_{F})=(1/8)(5g^{2}+{\rm
c}_{2}(F)),$

where $\mathcal{S}_{F}$ is the universal subbundle on $F\subset\mathbb
G(1,\mathbb P^{5})$.

  3. (iii)

For every homologically trivial class $\gamma\in{\Im}(\psi)$, which according
to [SV16, Section 21] is equivalent to being an algebraically trivial class in
$\operatorname{CH}^{2}(F)$, and every class
$\alpha\in\operatorname{CH}^{1}(F)$ we have

(2.7) $\alpha^{2}\cdot\gamma\in A_{2}.$

This follows from combining $A_{2}=\operatorname{CH}^{1}(F)_{0}^{\cdot
2}\cdot\operatorname{CH}^{2}(F)_{2}$, see [SV16, Proposition 22.2],
$\operatorname{CH}^{1}(F)_{0}=\operatorname{CH}^{1}(F)$, see [SV16, Theorem
2], $\operatorname{CH}^{2}(F)_{2}=V^{2}_{-2}$, see [SV16, Theorem 21.9(iii)],
and $V^{2}_{-2}={\mathcal{A}}_{\hom}={\Im}(\psi)_{\hom}$, see [SV16,
Definition 20.1 and Proposition 21.10].

## 3\. Special surfaces

Apart from the surfaces $F_{L}\subset F=F(X)$, there are other interesting
types of surfaces in $F$. Most notably, there are the surface
$F^{\prime}\subset F$ of lines of the second type and the surfaces
$F(Y)\subset F$ of lines contained in general hyperplane sections
$Y=X\cap\mathbb P^{4}$. Before turning to cycles on these surfaces, let us
recall a few geometric facts.

All three surfaces $F_{L}$, $F^{\prime}$, and $F(Y)$ are smooth surfaces of
general type.(1)(1)(1)For $F^{\prime}$ to be smooth, one needs to assume $X$
general. Their basic numerical invariants are as follows:

  1. (1)

$p_{g}(F(Y))=10$ and $q(F(Y))=5$; see [Huy23, Chapter 5] for references.

  2. (2)

$p_{g}(F^{\prime})=449$ and $q(F^{\prime})=0$; see [GK23] and [Huy23, Section
6.4].

  3. (3)

$p_{g}(F_{L})=5$ and $q(F_{L})=0$. In fact, $\pi_{1}(F_{L})=\\{1\\}$, see
[Huy21, Lemma 1.2 and Appendix].

### 3.1.

Voisin proved [Voi92, Example 3.7] that the surfaces $F(Y)\subset F$ are
Lagrangian; cf. [Huy23, Lemma 6.4.5]. In other words, the pull-back map
$H^{0}(F,{}^{2}_{F})\to H^{0}(F(Y),{}_{F(Y)}^{2})$ is zero.

The Bloch–Beilinson filtration for zero-cycles on the surface $F(Y)$ is of the
form

$0\subset F^{2}\subset F^{1}\subset\operatorname{CH}_{0}(F(Y)),$

where the Albanese map $F^{1}/F^{2}\xrightarrow{\vbox
to0.0pt{\vss\hbox{$\scriptstyle\sim$}\vskip-2.0pt}}{\operatorname{Alb}}(F(Y))$
is an isomorphism and $F^{2}$ is big, i.e. not representable; see Example 2.1.

As always for the push-forward map from a surface,
$\operatorname{CH}_{0}(F(Y))\to A$ respects the filtration. However, since
$F(Y)\subset F$ is Lagrangian, the Bloch–Beilinson conjecture predicts that
the image of $F^{2}\operatorname{CH}_{0}(F(Y))\to A_{4}\oplus
A_{2}=F^{2}\subset A$ is actually contained in $F^{4}=F^{3}=A_{4}$. This
prediction can be confirmed as follows:(2)(2)(2)Thanks to C. Voisin for the
argument. As the Fano correspondences for $X$ and for the hyperplane section
$Y\subset X$ are compatible, the natural diagram

$\textstyle{\operatorname{CH}_{0}(F(Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{Y}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{X}}$$\textstyle{\operatorname{CH}_{1}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{CH}_{1}(X)}$

commutes. By the definition of the Bloch–Beilinson filtration for the surface
$F(Y)$, we know that $F^{2}\operatorname{CH}_{0}(F(Y))$ is the kernel of the
Albanese map. Since
${\operatorname{Alb}}(F(Y))\simeq\operatorname{CH}_{1}(Y)_{\hom}$, by a result
of Beauville and Murre, cf. [Huy23, Corollary 5.3.16], this shows that
$F^{2}\operatorname{CH}_{0}(F(Y))$ is contained in the kernel of
$\varphi_{Y}$. Therefore, its image in $A$ is contained in the kernel of
$\varphi_{X}$, which is $F^{4}=A_{4}\subset A$ by (2.4).

Note that for the general hyperplane section $Y\subset X$, the surface
$F(Y)\subset F$ is not a constant cycle surface; i.e. the image of
$\operatorname{CH}_{0}(F(Y))\to A$ is not of rank one. Indeed, otherwise the
generator $c_{Y}$ would stay constant, since $Y$ varies in the projective
space $|\mathcal{O}(1)|$, but as every line in $X$ is contained in some
hyperplane section, this would imply the triviality of $A_{4}\oplus A_{2}$.

A stronger statement is in fact true. Namely, for the very general hyperplane
section $Y$, the map $F^{2}\operatorname{CH}_{0}(F(Y))\to A_{4}$ is not
trivial. Here is a sketch of the argument:(2) The surfaces $F(Y)$ cover $F$
since every line in $X$ is contained in some hyperplane section. In fact, the
family of all $F(Y)$ can be seen as a $\mathbb P^{3}$-bundle $\mathcal{F}\to
F$ with a projection $\mathcal{F}\to|\mathcal{O}(1)|$. Restricting to a
general two-dimensional family $\pi\colon\mathcal{F}_{\mathbb P^{2}}\to\mathbb
P^{2}\subset|\mathcal{O}(1)|$ gives a fourfold with a dominant morphism
$\mathcal{F}_{\mathbb P^{2}}\to F$ inducing an injection
$H^{4,0}(F)\,\hookrightarrow H^{4,0}(\mathcal{F}_{\mathbb P^{2}})$. However,
over a dense open subset $U\subset\mathbb P^{2}$, we have
${}^{4}_{\mathcal{F}_{U}}\simeq\pi^{\ast}{}_{U}^{2}\otimes{}_{\pi}^{2}$, but
if $F^{2}\operatorname{CH}_{0}(F(Y))\to A$ were zero, then the component in
${}_{\pi}^{2}$ would be trivial.

######  Remark 3.1.

Although for the very general hyperplane section, the surface $F(Y)\subset F$
is not a constant cycle surface, special ones might be. Imitating the study of
constant cycle curves in ample linear systems on K3 surfaces, see [Huy14], it
should be interesting to study those in more detail.(3)(3)(3)Thanks to E.
Sertöz for a related question. In fact, Voisin [Voi08, Lemma 2.2] shows that
the surface of lines contained in a hyperplane section with five nodes is
rational (and singular) and hence a constant cycle surface.

### 3.2.

Now let $X$ be general such that the surface $F^{\prime}\subset F$ of lines of
the second type is smooth. Then
$F^{2}=F^{1}\subset\operatorname{CH}_{0}(F^{\prime})$, which the push-forward
map sends to $F^{2}=A_{4}\oplus A_{2}\subset A$. Since $F^{\prime}\subset F$
is not Lagrangian, see [Huy23, Section 6.4.4], the projection to the graded
part gives a non-trivial map $F^{2}\to A_{4}\oplus A_{2}\twoheadrightarrow
A_{2}$. In fact, according to a result of Shen and Vial [SV16, Proposition
19.5], the surface $F^{\prime}$ avoids $A_{4}$ and covers $A_{2}$. More
precisely,

(3.1) ${\Im}\left(\operatorname{CH}_{0}(F^{\prime})\longrightarrow
A\right)=A_{2}\oplus A_{0}\subset A.$

Again by Mumford’s argument, using $p_{g}(F^{\prime})>1$ and that
$F^{\prime}\subset F$ is not Lagrangian, the kernel and image of the push-
forward map are not representable. However, a geometric understanding of those
cycles on $F^{\prime}$ that become rationally trivial on $F$ is not available.
Also, unlike the case
$\operatorname{CH}_{0}(F_{L_{0}})^{-}\subset\operatorname{CH}_{0}(F_{L_{0}})$
studied in this paper, there does not seem to be any distinguished subgroup of
$\operatorname{CH}_{0}(F^{\prime})$ that maps isomorphically onto $A_{2}$.

As was observed by Shen and Vial [SV16, Theorem 3], (3.1) also implies
$A_{4}\neq 0$. Indeed, otherwise $\operatorname{CH}_{0}(F)$ would be
concentrated on the surface $F^{\prime}$, which by Bloch–Srinivas [BS83] would
contradict $H^{4,0}(F)\neq 0$.

######  Remark 3.2.

Observe that (3.1) fits Voisin’s description of $A_{2}\oplus A_{0}$ (and was
in fact used for its proof). Indeed, the rational endomorphism $f\colon
F\dashrightarrow F$ is resolved by a blow-up of $F$ in $F^{\prime}\subset F$.
Thus, for a line of the second type $L\in F^{\prime}$, the class $f_{\ast}[L]$
is realised by all points $L^{\prime}$ in the image of the exceptional line of
the blow-up $\tau\colon{\operatorname{Bl}}_{F^{\prime}}(F)\to F$ over $L$,
which clearly satisfy $\dim{\rm O}_{L^{\prime}}\geq 1$. Hence, by Voisin’s
description, we have $f_{*}[L]\in A_{2}\oplus A_{0}$ and, therefore, $[L]\in
A_{2}\oplus A_{0}$. For the other inclusion observe that the image of the
exceptional divisor of $\tau$ is an ample divisor in $F$ which therefore
intersects any curve. Hence, every line $L^{\prime}\in F$ with $\dim{\rm
O}_{L^{\prime}}\geq 1$, and so in particular $[L^{\prime}]\in A_{2}\oplus
A_{0}$ according to Voisin, gives rise to a class of the form $f_{\ast}[L]$
for some $L\in F^{\prime}$.

## 4\. Lines intersecting a fixed line

Let us now fix a general line $L_{0}\subset X$ and consider the surface
$F_{L_{0}}\subset F=F(X)$ of lines intersecting $L_{0}$. Then the quotient by
the standard involution defines a morphism $\pi\colon
F_{L_{0}}\twoheadrightarrow D_{L_{0}}$ onto a quintic surface
$D_{L_{0}}\subset\mathbb P^{3}$. The 16 fixed points, i.e. the 16 lines
$L\subset X$ with $\overline{LL_{0}}\cap X=2L\cup L_{0}$, give rise to 16
ordinary double points of $D_{L_{0}}$; see the original [Voi86] or the
discussion in [Huy23, Section 6.4.5] or in [Huy21].

The involution $\iota$ acts on $\operatorname{CH}_{0}(F_{L_{0}})$, and we
define

$\operatorname{CH}_{0}\left(F_{L_{0}}\right)^{\pm}\coloneqq\\{\alpha\mid\iota^{\ast}\alpha=\pm\alpha\\}\subset\operatorname{CH}_{0}\left(F_{L_{0}}\right).$

Then the pull-back map induces isomorphisms

$\operatorname{CH}_{0}\left(D_{L}\right)_{\hom}\xrightarrow{\;\vbox
to0.0pt{\vss\hbox{$\scriptstyle\sim$}\vskip-2.0pt}\;}\operatorname{CH}_{0}\left(F_{L}\right)_{\hom}^{+}\quad\text{
and
}\quad\operatorname{CH}_{0}\left(F_{L_{0}}\right)_{\hom}^{-}\simeq\operatorname{CH}_{0}\left(F_{L_{0}}\right)_{\hom}/\operatorname{CH}_{0}(D_{L})_{\hom}.$

Moreover,
$\operatorname{CH}_{0}(F_{L_{0}})^{-}=\operatorname{CH}_{0}(F_{L_{0}})_{\hom}^{-}=\ker(\pi_{\ast})={\Im}(1-\iota^{\ast})$;
see [Huy21, Section 4.1].

Of course,
$\operatorname{CH}_{0}(D_{L_{0}})/\operatorname{CH}_{0}(D_{L_{0}})_{\hom}\simeq\mathbb
Z$, but this isomorphism comes without a canonical split; i.e. no analogue of
the Beauville–Voisin class exists for the surface $D_{L_{0}}$. Theorem 1.1(i)
can be seen as a replacement: the image of $\operatorname{CH}_{0}(D_{L_{0}})$
in $A/A_{4}\simeq A_{2}\oplus A_{0}$ is generated by a distinguished class
depending on $L_{0}$.

### 4.1.

Before entering the proof of Theorem 1.1(i), we observe that the
correspondence

$D_{L_{0}}\joinrel\twoheadleftarrow\joinrel\relbar
F_{L_{0}}\lhook\joinrel\relbar\joinrel\rightarrow F$

induces the trivial map $H^{0}(F,{}_{F}^{2})\to
H^{0}(D_{L_{0}},{}_{D_{L_{0}}}^{2})$ since the restriction of the holomorphic
two-form on $F$ to $F_{L_{0}}$ is anti-invariant; see [Huy21, Section 2.3]. In
particular, according to the Bloch–Beilinson philosophy, one expects the map

$\operatorname{CH}_{0}\left(D_{L_{0}}\right)_{\hom}=\left(F^{2}/F^{3}\right)\left(D_{L_{0}}\right)\longrightarrow\left(F^{2}/F^{3}\right)(F)=A_{2}$

to be trivial. Equivalently, the composition of pull-back and push-forward
should map the group of zero-cycles
$\operatorname{CH}_{0}(D_{L_{0}})_{\hom}=\operatorname{CH}_{0}(F_{L_{0}})_{\hom}^{+}$
on $D_{L_{0}}$ to $F^{3}=A_{4}\subset A$. In this sense, the first assertion
Theorem 1.1(i) can be seen as a confirmation of the Bloch–Beilinson
philosophy.

###### Proof of Theorem 1.1(i).

A point $t\in D_{L_{0}}$ corresponds to a pair of lines $L_{1},L_{2}\subset X$
such that $L_{0}\cup L_{1}\cup L_{2}$ forms a triangle, i.e. all three lines
are contained in a single plane.

Under push-forward, the class $[t]\in\operatorname{CH}_{0}(D_{L_{0}})$ of the
point $t\in D_{L_{0}}$ is then mapped to $[L_{1}]+[L_{2}]\in A$ and, after
composing further with $\psi\colon A\to\operatorname{CH}_{2}(F)$,
$[L]\mapsto[F_{L}]$, to

(4.1)
$\left[F_{L_{1}}\right]+\left[F_{L_{2}}\right]=\varphi\left(h^{3}\right)-\left[F_{L_{0}}\right].$

Here, $h$ is the hyperplane class on $X$, $h^{3}$ can be represented by the
plane $\mathbb P^{2}=\overline{L_{0}L_{1}L_{2}}$, and
$\varphi\colon\operatorname{CH}_{1}(X)\to\operatorname{CH}_{2}(F)$ is the Fano
correspondence. As $\varphi(h^{3})-[F_{L_{0}}]$ is independent of the point
$t\in D_{L_{0}}$ and $\operatorname{CH}_{0}(D_{L_{0}})_{\hom}$ is spanned by
classes of the form $[t]-[t^{\prime}]$, $t,t^{\prime}\in D_{L_{0}}$, we find
that the image of $\operatorname{CH}_{0}(D_{L_{0}})_{\hom}\to A$ is contained
in the kernel of $\psi$, which is $A_{4}$.

Next let us show that the image of the composition
$\operatorname{CH}_{0}(D_{L_{0}})\to A\to A/A_{4}\simeq A_{2}\oplus A_{0}$ is
spanned by the class $c_{F}(L_{0})=2c_{F}-[L_{0}]_{2}$, where $[L_{0}]_{2}$ is
the degree two part of

$[L_{0}]=[L_{0}]_{4}+[L_{0}]_{2}+[L_{0}]_{0}\in A_{4}\oplus A_{2}\oplus
A_{0}.$

In order to prove this, observe that the image of
$3c_{F}-([L_{0}]_{2}+[L_{0}]_{0})=c_{F}(L_{0})$ under the injection
$\bar{\psi}\colon A/A_{4}\,\hookrightarrow\operatorname{CH}_{2}(F)$ can be
computed by means of (2.6) as

$\psi\colon A\longrightarrow\operatorname{CH}_{2}(F),\quad
c_{F}(L_{0})\longmapsto g^{2}-{\rm
c}_{2}(\mathcal{S}_{F})-\left[F_{L_{0}}\right].$

The latter equals $\varphi(h^{3})-[F_{L_{0}}]$ by [Huy23, Proposition 6.4.1],
and to conclude we apply (4.1), which proves
$\varphi(h^{3})-[F_{L_{0}}]=[F_{L}]+[F_{\iota(L)}]$ for any $L\in F_{L_{0}}$.
Hence, the classes $c_{F}(L_{0})$ and $[L]+[\iota(L)]$ modulo $A_{4}$ have the
same image under the injection $\bar{\psi}$. Therefore, the image of

$\mathbb
Z\simeq\frac{\operatorname{CH}_{0}\left(D_{L_{0}}\right)}{\operatorname{CH}_{0}\left(D_{L_{0}}\right)_{\hom}}\longrightarrow
A/A_{4}\simeq A_{2}\oplus A_{0}$

is indeed $\mathbb Z\cdot c_{F}(L_{0})$.

Before proving the rest of Theorem 1.1(i), we note the following.

######  Claim.

Assume $\rho(F)>1$. Then the class $c_{F}(L_{0})$ is contained in the image of
$\operatorname{CH}_{0}(D_{L_{0}})\to A$.

###### Proof.

It suffices to prove the existence of an invariant class
$\eta\in\operatorname{CH}_{0}(D_{L_{0}})^{+}$ with a non-trivial push-forward
contained in $A_{2}\oplus A_{0}$. By Lemma 4.7, the proof of which is
independent of the rest of the discussion here, this holds for the restriction
of the square $\eta=(\alpha|_{F_{L_{0}}})^{2}$ of any primitive class
$0\neq\alpha\in\operatorname{CH}^{1}(F)_{\operatorname{pr}}$. ∎

It remains to prove that $\operatorname{CH}_{0}(D_{L_{0}})_{\hom}\to A_{4}$ is
not zero for the very general $X$ and the very general line $L_{0}\subset X$.
Clearly, it suffices to show this for one cubic, but using the above claim, we
will in fact show it for all cubics with $\rho(F)>1$.

So we assume $\rho(F)>1$ and suppose that for the very general choice of
$L_{0}$ and hence for all $L_{0}$, the map
$\operatorname{CH}_{0}(D_{L_{0}})_{\hom}\to A_{4}$ is zero. Then, using that
$\operatorname{CH}_{0}(D_{L_{0}})\to A/A_{4}$ has rank one and the fact that
by the claim $c_{F}(L_{0})\in A_{2}\oplus A_{0}$ is contained in the image of
the push-forward map, all of $\operatorname{CH}_{0}(D_{L_{0}})$ would map to
$A_{2}\oplus A_{0}$ (with its image generated by $c_{F}(L_{0})$). Now consider
a fixed point $L\in F_{L_{0}}$ of the covering involution $F_{L_{0}}\to
D_{L_{0}}$, i.e. a line $L\subset X$ such that

(4.2) $2L\cup L_{0}=\mathbb P^{2}\cap X$

for some plane $\mathbb P^{2}\subset\mathbb P^{5}$. Then
$2[L]\in\operatorname{CH}_{0}(D_{L_{0}})\subset\operatorname{CH}_{0}(F_{L_{0}})^{+}$,
and for $L$ as a point in $F$, one then has $[L]\in A_{2}\oplus A_{0}$.
However, for every line $L\in F$, there exists a line $L_{0}$ satisfying
(4.2), and for $L$ general $L_{0}$ also is general (and so the surface
$F_{L_{0}}$ is smooth). Hence, the triviality of
$\operatorname{CH}_{0}(D_{L_{0}})_{\hom}\to A_{4}$ for the general $L_{0}$
would prove that all points in $F$ define classes in $A_{2}\oplus A_{0}$, i.e.
$A_{4}=0$, which is absurd. ∎

######  Remark 4.1.

The last part of the argument also shows that in general the class of a fixed
point $L\in F_{L_{0}}$ of the covering involution $\iota$ is not mapped to the
distinguished class $(1/2)c_{F}(L_{0})\in A_{2}\oplus A_{0}\subset A_{4}\oplus
A_{2}\oplus A_{0}$. Hence, at least for generic choices, none of the 16 fixed
points of $\iota$ is a candidate for the Beauville–Voisin class
$c_{L_{0}}\in\operatorname{CH}_{0}(F_{L_{0}})^{+}$ in Remark 1.3; see also
Remark 4.9.

######  Remark 4.2.

The Beauville–Voisin class on a K3 surface is by definition of degree one,
while the analogue $c_{F}(L_{0})\in A_{2}\oplus A_{0}$ is of degree two. There
are two reasons for this. First, the projection $F_{L_{0}}\to D_{L_{0}}$ is of
degree two, and thus the push-forward of any class on $F_{L_{0}}$ that is
pulled back from $D_{L_{0}}$ has even degree. Second, the intersection pairing
$(\alpha_{1}|_{F_{L_{0}}}.\alpha_{2}|_{F_{L_{0}}})$ of any two primitive
classes $\alpha_{1},\alpha_{2}\in\operatorname{CH}^{1}(F)_{\operatorname{pr}}$
is even; see [Huy21, Corollary 1.7].

### 4.2.

The proof of Theorem 1.1(ii) starts with the following key technical lemma,
which is based on an excess intersection computation.

######  Lemma 4.3.

We consider the composition

$\beta\colon\operatorname{CH}_{0}\left(F_{L_{0}}\right)\longrightarrow\operatorname{CH}_{0}(F)\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\operatorname{CH}_{2}(F)\stackrel{{\scriptstyle\operatorname{res}}}{{\longrightarrow}}\operatorname{CH}_{0}\left(F_{L_{0}}\right)$

of the push-forward map
$\operatorname{CH}_{0}(F_{L_{0}})^{-}\to\operatorname{CH}_{0}(F)$, the map
$\psi\colon\operatorname{CH}_{0}(F)\to\operatorname{CH}_{2}(F)$,
$[L]\mapsto[F_{L}]$, and the restriction map
${\operatorname{res}}\colon\operatorname{CH}_{2}(F)\to\operatorname{CH}_{0}(F_{L_{0}})$.
Then, $\beta=-2\cdot{\operatorname{id}}$.

###### Proof.

The group $\operatorname{CH}_{0}(F_{L_{0}})^{-}$ is generated by classes of
the form $[L_{1}]-[L_{2}]$, where $L_{1}\in F_{L_{0}}$ is any point and
$L_{2}=\iota(L_{1})$ is its image under the covering involution of
$F_{L_{0}}\to D_{L_{0}}$. The map $\beta$ sends such a class to the
restriction $([F_{L_{1}}]-[F_{L_{2}}])|_{F_{L_{0}}}$.

First observe that the push-forward to $A$ turns
$([F_{L_{1}}]-[F_{L_{2}}])|_{F_{L_{0}}}$ into
$([F_{L_{1}}]-[F_{L_{2}}])\cdot[F_{L_{0}}]$, which by (2.5) is indeed
$-2([L_{1}]-[L_{2}])\in A$. We need to show that this equality holds before
pushing forward, i.e.
$([F_{L_{1}}]-[F_{L_{2}}])|_{F_{L_{0}}}=-2([L_{1}]-[L_{2}])\in\operatorname{CH}_{0}(F_{L_{0}})^{-}$,
and for this we have to find a geometric interpretation of
$[F_{L_{i}}]|_{F_{L_{0}}}$.

The intersection $F_{L_{1}}\cap F_{L_{0}}$ is the set of all lines
simultaneously intersecting $L_{1}$ and $L_{0}$. Hence,

$F_{L_{1}}\cap F_{L_{0}}=\\{L_{2}\\}\sqcup C_{1}.$

Here, $C_{1}\coloneqq\\{L\mid x_{1}\in L\\}$, with $x_{1}$ the point of
intersection of $L_{1}$ and $L_{0}$. A similar statement holds for
$F_{L_{2}}\cap F_{{L_{0}}}$. Hence,

$\left(\left[F_{L_{1}}\right]-\left[F_{L_{2}}\right]\right)|_{F_{L_{0}}}=[L_{2}]-[L_{1}]+E_{1}-E_{2},$

where $E_{1}$ and $E_{2}$ are the contributions from the excess intersections
$C_{1}$ and $C_{2}$. We will show that
$E_{1}-E_{2}=[L_{2}]-[L_{1}]\in\operatorname{CH}_{0}(F_{L_{0}})$.

To compute $E_{1},E_{2}$ we recall that the excess intersection
$[S]|_{T}\in\operatorname{CH}_{0}(S_{2})$ of two smooth surfaces $S,T\subset
F$ intersecting in a smooth curve $C=S\cap T$ is the push-forward of ${\rm
c}_{1}(\mathcal{E})$, where $\mathcal{E}$ is the line bundle
$\mathcal{N}_{S/F}|_{C}/\mathcal{N}_{C/T}\simeq\mathcal{N}_{T/F}|_{C}/\mathcal{N}_{C/S}$;
cf. [Ful98, Theorem 9.2] or [EH16, Section 13.3]. In other words,
$\mathcal{E}$ is part of a commutative diagram of short exact sequences

$\textstyle{\mathcal{N}_{C/T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{N}_{S/F}|_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}}$$\textstyle{\mathcal{T}_{T}|_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}_{F}|_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{N}_{T/F}|_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}_{S}|_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{N}_{C/S}\hbox
to0.0pt{.\hss}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Hence, as a line bundle on $C$, one has

$\displaystyle\mathcal{E}$ $\displaystyle\simeq$
$\displaystyle\det(\mathcal{N}_{S/F})|_{C}\otimes\mathcal{N}_{C/T}^{\ast}$
$\displaystyle\simeq$
$\displaystyle\omega_{F}^{\ast}|_{C}\otimes\omega_{S}|_{C}\otimes\omega_{T}|_{C}\otimes\omega_{C}^{\ast}.$

In our situation, we let $S=F_{L_{i}}$ and $T=F_{L_{0}}$. Their canonical
bundles are $\pi_{i}^{\ast}\mathcal{O}(1)$ and $\pi_{0}^{\ast}\mathcal{O}(1)$,
where the $\pi_{i}\colon F_{L_{i}}\to D_{L_{i}}\subset\mathbb P^{3}$,
$i=0,1,2$, denote the projections. Hence, the excess contribution for the
intersection $[F_{L_{i}}]|_{F_{L_{0}}}$ comes from the line bundle

(4.3)
$\mathcal{E}_{i}\simeq\pi_{i}^{\ast}\mathcal{O}(1)|_{C_{i}}\otimes\pi_{0}^{\ast}\mathcal{O}(1)|_{C_{i}}\otimes\omega_{C_{i}}^{\ast}.$

######  Claim.

If $\mathcal{O}_{F}(1)$ denotes the Plücker polarisation on $F$ and the two
lines $L_{0},L_{i}$ are considered as points in $C_{i}$, then

$\mathcal{E}_{i}\simeq\mathcal{O}_{F}(2)|_{C_{i}}\otimes\mathcal{O}_{C_{i}}(-L_{0}-L_{i})\otimes\omega_{C}^{\ast}.$

###### Proof.

To prove the claim it suffices to show that
$\pi_{j}^{\ast}\mathcal{O}(1)|_{C_{i}}\simeq\mathcal{O}_{F}(1)|_{C_{i}}\otimes\mathcal{O}_{C_{i}}(-L_{j})$
for $j=0$ and $j=i$. It is known that
$\mathcal{O}_{F}(1)|_{F_{L_{j}}}\simeq\pi_{j}^{\ast}\mathcal{O}(1)\otimes\mathcal{O}_{F_{j}}(C_{i})$;
see the original [Voi86, Section 3, Lemma 2] or [Huy23, Equation (4.5) in
Remark 6.4.13]. Thus, we need to prove that
$\mathcal{O}_{F_{L_{j}}}(C_{i})|_{C_{i}}\simeq\mathcal{O}_{C_{i}}(L_{j})$. To
see this, observe that the blow-up of $F_{L_{j}}$ in the point $L_{j}$ comes
with the projection $q\colon{\operatorname{Bl}}_{L}(F_{L_{j}})\to L_{j}$, so
that $C_{i}$ is the fibre over $x_{i}\in L_{j}$. In particular, the linear
equivalence of $\mathcal{O}_{F_{L_{j}}}(C_{i})$ is independent of the point
$x_{i}\in L_{j}$. As for two distinct points $x\neq x^{\prime}\in L_{j}$, the
two curves $C_{x}$ and $C_{x^{\prime}}$ only intersect in the point $L_{j}\in
F_{L_{j}}$ and do so transversally, this concludes the proof of the claim.∎

The claim immediately shows
$E_{1}-E_{2}=[L_{2}]-[L_{1}]\in\operatorname{CH}_{0}(F_{L_{0}})$, which
concludes the proof of the lemma. ∎

######  Remark 4.4.

With the notation in the above proof, we have actually shown that

$\psi([L_{1}])|_{F_{L_{0}}}=[L_{2}]+E_{1}=[L_{2}]+2g\cdot
C_{1}-[L_{0}]-[L_{1}]-[\omega_{C_{1}}].$

Since
$\omega_{C_{1}}\simeq(\omega_{F_{L_{0}}}\otimes\mathcal{O}(C_{1}))|_{C_{1}}\simeq(\pi_{0}^{\ast}\mathcal{O}(1)\otimes\mathcal{O}(C_{1}))|_{C_{1}}\simeq\mathcal{O}_{F}(1)|_{C_{1}}$,
this gives

$\psi([L_{1}]+[L_{2}])|_{F_{L_{0}}}=2(g\cdot
C-[L_{0}])\in\operatorname{CH}_{0}\left(F_{L_{0}}\right),$

where $C\subset F_{L_{0}}$ is any curve linearly equivalent to $C_{1}\subset
F_{L_{0}}$ or, equivalently, to $C_{2}\subset F_{L_{0}}$.

######  Corollary 4.5.

The push-forward map

$\operatorname{CH}_{0}\left(F_{L_{0}}\right)^{-}\,\lhook\joinrel\relbar\joinrel\rightarrow
A$

and its composition with the projection $A\twoheadrightarrow A/A_{4}\simeq
A_{2}\oplus A_{0}$

(4.4)
$\operatorname{CH}_{0}\left(F_{L_{0}}\right)^{-}\,\lhook\joinrel\relbar\joinrel\rightarrow
A\relbar\joinrel\twoheadrightarrow A/A_{4}$

are both injective. ∎

###### Proof of Theorem 1.1(ii).

It remains to prove that for a very general line $L_{0}$, the image of the
push-forward $\operatorname{CH}_{0}(F_{L_{0}})^{-}\to A$ is not contained in
$A_{2}$ and that the composition (4.4) induces an isomorphism
$\operatorname{CH}_{0}(F_{L_{0}})^{-}\xrightarrow{\vbox
to0.0pt{\vss\hbox{$\scriptstyle\sim$}\vskip-2.0pt}}A_{2}\subset A/A_{4}$.

For the first assertion, we choose a line $L_{0}$ such that for the
decomposition

$[L_{0}]=[L_{0}]_{4}+[L_{0}]_{2}+[L_{0}]_{0}\in A_{4}\oplus A_{2}\oplus
A_{0},$

we have $[L_{0}]_{4}\neq 0$. Then we view $L_{0}$ as a point in $F_{L_{0}}$
and consider its image $\iota(L_{0})\in F_{L_{0}}$ under the involution. Their
difference defines a class
$[\iota(L_{0})]-[L_{0}]\in\operatorname{CH}_{0}(F_{L_{0}})^{-}$. By the
definition of the Voisin endomorphism $f\colon F\dashrightarrow F$, the push-
forward of this class is $f_{\ast}[L_{0}]-[L_{0}]\in A$, which by the
definitions of $A_{4}$, $A_{2}$, and $A_{0}$ equals
$3([L_{0}]_{4}-[L_{0}]_{2})$. This proves that the push-forward of
$[\iota(L_{0})]-[L_{0}]\in\operatorname{CH}_{0}(F_{L_{0}})^{-}$ is not
contained in $A_{2}\oplus A_{0}$.

The surjectivity of the composition (4.4) follows from (2.3), showing
$A_{2}\xrightarrow{\vbox
to0.0pt{\vss\hbox{$\scriptstyle\sim$}\vskip-2.0pt}}\operatorname{CH}_{1}(X)_{\hom}$
via the Fano correspondence, and the isomorphism
$\operatorname{CH}_{0}(F_{L_{0}})^{-}\xrightarrow{\vbox
to0.0pt{\vss\hbox{$\scriptstyle\sim$}\vskip-2.0pt}}\operatorname{CH}_{1}(X)_{\hom}$,
obtained as the composition of the push-forward and the Fano correspondence;
cf. [Huy21, Theorem 0.4]. ∎

######  Remark 4.6.

  1. (i)

Since the image of the other projection

(4.5)
$\operatorname{CH}_{0}\left(F_{L_{0}}\right)^{-}\,\lhook\joinrel\relbar\joinrel\rightarrow
A\relbar\joinrel\twoheadrightarrow A_{4}$

contains the component $[L_{0}]_{4}\in A_{4}$ and since $A_{4}$ is not
generated by points in a single surface, the image of (4.5) does indeed depend
on $L_{0}$. It is natural to wonder whether the composition (4.5), similarly
to $\operatorname{CH}_{0}(D_{L_{0}})_{\hom}\to A_{4}$ in Remark 1.3, is
actually injective for general $L_{0}$.

  2. (ii)

As pointed out by Charles Vial, for $[L_{0}]=c_{F}$ the map (4.5) is actually
trivial and in particular not injective. To prove this one first notices that
for $[L_{0}]=c_{F}$, the endomorphism
$\delta\colon\operatorname{CH}_{0}(F)\to\operatorname{CH}_{0}(F)$,
$\alpha\mapsto[F_{L_{0}}]\cdot\psi(\alpha)=(1/3)(g^{2}-{\rm
c}_{2}(\mathcal{S}_{F}))\cdot\psi(\alpha)$ preserves $A_{2}$. Using the
notation of [SV16], this follows from $g,{\rm
c}_{2}(\mathcal{S}_{F})\in\operatorname{CH}^{2}(F)_{0}$, see [SV16, Theorem
21.9], $\operatorname{CH}^{2}(F)_{2}=\psi(A_{4}\oplus A_{2})=\psi(A_{2})$, see
[SV16, Proposition 21.10], and
$\operatorname{CH}^{2}(F)_{0}\cdot\operatorname{CH}^{2}(F)_{2}=A_{2}$, see
[SV16, Proposition 22.3]. Then by Lemma 4.3, $\delta$ maps the image
$\alpha_{4}+\alpha_{2}\in A_{4}\oplus A_{2}$ of a class in
$\operatorname{CH}_{0}(F_{L_{0}})^{-}$ to $-2\cdot(\alpha_{4}+\alpha_{2})$.
Since $\delta(\alpha_{4}+\alpha_{2})=\delta(\alpha_{2})\in A_{2}$, this
implies $\alpha_{4}=0$.

In fact, by Lemma 4.3 and the above proof of Theorem 1.1(ii), the push-forward
then defines an isomorphism
$\operatorname{CH}_{0}(F_{L_{0}})^{-}\xrightarrow{\vbox
to0.0pt{\vss\hbox{$\scriptstyle\sim$}\vskip-2.0pt}}A_{2}$, and
$\delta|_{A_{2}}=-2\cdot{\operatorname{id}}$. Note that this shows in
particular that $A_{2}$ is spanned by classes $[L_{1}]-[L_{2}]$ forming a
triangle with a fixed line $L_{0}$ with $[L_{0}]=c_{F}$.

  3. (iii)

Following a suggestion of Charles Vial, see also [SV16, Theorem 20.2(ii)], we
observe that $A_{4}$ is spanned by the images of all the maps (4.5) for
varying $L_{0}$. Indeed, since $A_{4}\oplus A_{2}$ is spanned by classes of
the form $[L_{1}]-[L_{2}]$ and since for any two lines $L_{1},L_{2}$, there
exists a line $L$ intersecting both, by writing
$[L_{1}]-[L_{2}]=([L_{1}]-[L])-([L_{2}]-[L])$ one reduce to the case that
$L_{1},L_{2}$ are planar. In this case, if $L_{0}$ denotes the residual line
of $L_{1}\cup L_{2}\subset\overline{L_{1}L_{2}}\cap X$, then
$L_{1}=\iota(L_{2})$ and, therefore, $([L_{1}]-[L_{2}])_{4}$ is contained in
the image of (4.5).

### 4.3.

As the first step towards a proof of Theorem 1.2, we recall from [Huy21,
Theorem 0.2], see also [Iza99, Theorem 3] and [She12, Theorem 4.7 and
Corollary 4.8], that the restriction of line bundles induces an isomorphism

(4.6) $\operatorname{CH}^{1}(F)_{\operatorname{pr}}\xrightarrow{\;\vbox
to0.0pt{\vss\hbox{$\scriptstyle\sim$}\vskip-2.0pt}\;}\operatorname{CH}^{1}\left(F_{L_{0}}\right)^{-}_{\operatorname{pr}}.$

Here, the primitive parts on the two sides are defined with respect to the
Plücker polarisation $\mathcal{O}_{F}(1)$ on $F$ and its restriction to
$F_{L_{0}}$. Thus, to establish Theorem 1.2, it suffices to show that for any
$\alpha\in\operatorname{CH}^{1}(F)_{\operatorname{pr}}$, the class
$\alpha^{2}\cdot[F_{L_{0}}]$ is a multiple of
$c_{F}(L_{0})=2c_{F}-[L_{0}]_{2}$.

The next step can be seen as a warm-up. The result was used already in the
proof of Theorem 1.1(i).

######  Lemma 4.7.

For any class $\alpha\in\operatorname{CH}^{1}(F)$, the push-forward of
$(\alpha|_{F_{L_{0}}})^{2}\in\operatorname{CH}_{0}(F_{L_{0}})$ to $F$ is a
class in $A_{2}\oplus A_{0}$.

###### Proof.

The assertion is equivalent to $\alpha^{2}\cdot[F_{L_{0}}]\in A_{2}\oplus
A_{0}$. To prove this, recall that the two classes $\psi([L_{0}])=[F_{L_{0}}]$
and $\psi(c_{F})=(1/3)(g^{2}-{\rm c}_{2}(\mathcal{S}_{F}))$ in
$\operatorname{CH}^{2}(F)$ are homologically equivalent; cf. [Huy23,
Proposition 6.4.1].

Hence, $[F_{L_{0}}]-\psi(c_{F})$ is contained in the homologically trivial
part of ${\Im}(\psi)$. Therefore, by (2.7) we have
$\alpha^{2}\cdot([F_{L_{0}}]-\psi(c_{F}))\in A_{2}$. Now, we apply a result of
Voisin [Voi08, Theorem 0.4] and use the well-known fact that ${\rm
c}_{2}(\mathcal{S}_{F})$ is a linear combination of ${\rm
c}_{2}(\mathcal{T}_{F})$ and $g^{2}$, cf. [Huy23, Proposition 6.4.1], to
conclude that $3\alpha^{2}\cdot\psi(c_{F})=\alpha^{2}\cdot(g^{2}-{\rm
c}_{2}(\mathcal{S}_{F}))\in A_{0}$. Hence, $\alpha^{2}\cdot[F_{L_{0}}]\in
A_{2}\oplus A_{0}$. ∎

###### Proof of Theorem 1.2.

We apply Lemma 4.7 to an arbitrary anti-invariant primitive class
$\delta\in\operatorname{CH}^{1}(F_{L_{0}})^{-}_{\operatorname{pr}}$, which by
(4.6) can be written as $\alpha|_{F_{L_{0}}}$ for a unique primitive class
$\alpha$ on $F$. Then the push-forward of
$\delta^{2}\in\operatorname{CH}^{2}(F_{L_{0}})^{+}$ is contained in
$A_{2}\oplus A_{0}$. As by Theorem 1.1(i) the image of the composition
$\operatorname{CH}^{2}(D_{L_{0}})\to A\twoheadrightarrow A/A_{4}=A_{2}\oplus
A_{0}$ is spanned by $c_{F}(L_{0})$, this shows that the push-forwards of
$\delta^{2}$ and $c_{F}(L_{0})$ are proportional.

To conclude, note that all numerical intersection products
$(\alpha_{1}|{F_{L_{0}}}.\alpha_{2}|{F_{L_{0}}})$ for primitive classes
$\alpha_{1},\alpha_{2}\in\operatorname{CH}^{1}(F)_{\operatorname{pr}}$ are
even; see [Huy21, Corollary 1.7]. This implies that
$\alpha_{1}|_{F_{L_{0}}}\cdot\alpha_{1}|_{F_{L_{0}}}\in\operatorname{CH}^{2}(F_{L_{0}})^{+}$
is contained in the index two subgroup
$\operatorname{CH}^{2}(D_{L_{0}})\subset\operatorname{CH}^{2}(F_{L_{0}})^{+}$.
∎

######  Remark 4.8.

The result cannot be extended to the whole $\operatorname{CH}^{1}(F_{L_{0}})$.
Indeed, the push-forward of $g^{2}|_{F_{L_{0}}}$, i.e.
$g^{2}\cdot[F_{L_{0}}]$, is not a multiple of $c_{F}(L_{0})$, for by [SV16,
Lemma 18.2] we have

$\displaystyle g^{2}\cdot\left[F_{L_{0}}\right]$ $\displaystyle=$
$\displaystyle[f(L_{0})]-4[L_{0}]+24c_{F}$ $\displaystyle=$
$\displaystyle(4[L_{0}]_{4}-2[L_{0}]_{2}+[L_{0}]_{0})-4[L_{0}]_{4}-4[L_{0}]_{2}-4[L_{0}]_{0}+24c_{F}$
$\displaystyle=$ $\displaystyle-6[L_{0}]_{2}-3[L_{0}]_{0}+24c_{F},$

which can be a multiple of $c_{F}(L_{0})$ only if $[L_{0}]_{2}=0$. By virtue
of [SYZ20, Theorem 3.4], the latter is in fact equivalent to $[L_{0}]=c_{F}$,
so excluded for general $L_{0}$.

Also, extending to just the full anti-invariant part is not possible. Indeed,
as used before, the restriction of the Plücker polarisation satisfies
$\mathcal{O}_{F}(1)|_{F_{L_{0}}}\simeq\pi^{\ast}_{0}\mathcal{O}(1)\otimes\mathcal{O}(C_{x})$
for any point $x\in L_{0}$. Hence, we have
$(g_{F_{L_{0}}}-\iota^{\ast}(g_{F_{L_{0}}}))^{2}=C_{x}\cdot
C_{x}+\iota(C_{x}\cdot C_{x})-2(C_{x}\cdot\iota(C_{x}))$ on $F_{L_{0}}$, which
under push-forward to $F$ becomes
$[L_{0}]+f_{\ast}[L_{0}]-2(C_{x}\cdot\iota(C_{x})=[L_{0}]+f_{\ast}[L_{0}]-2(6c_{F}-2[L_{0}])$,
which for general $L_{0}$ is not contained in $A_{2}\oplus A_{0}$. The last
equality uses a formula in the proof of [SV16, Lemma 18.2].

######  Remark 4.9.

It also seems unlikely that the Beauville–Voisin class $c_{F}(L_{0})\in
A_{2}\oplus A_{0}\subset\operatorname{CH}_{0}(F)$ can be lifted to a
Beauville–Voisin class $c_{L_{0}}\in\operatorname{CH}_{0}(F_{L_{0}})$. Indeed,
as a consequence of Remark 4.4, we know that

$\psi\left(c_{F}(L_{0})\right)|_{F_{L_{0}}}=2(g\cdot
C-[L_{0}])\in\operatorname{CH}_{0}\left(F_{L_{0}}\right)$

with
$C\in|\mathcal{O}_{F}(1)|_{F_{L_{0}}}\otimes\pi_{0}^{\ast}\mathcal{O}(-1)|$.
So, if there exists a Beauville–Voisin class in
$\operatorname{CH}_{0}(F_{L_{0}})^{+}$ as in Remark 1.3, then a natural guess
would be that it must be proportional to $g\cdot C-[L_{0}]$. Note, however,
that the push-forward of $g\cdot C-[L_{0}]$ is typically not contained in
$A_{2}\oplus A_{0}$, so not proportional to $c_{F}(L_{0})$. Even worse, the
class $g\cdot C-[L_{0}]\in\operatorname{CH}_{0}(F_{L_{0}})$ is not even
invariant.

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