# A remark on the generalized Franchetta conjecture for K3 surfaces

Arnaud Beauville  Université Côte d’Azur
CNRS – Laboratoire J.-A. Dieudonné
Parc Valrose
F-06108 Nice cedex 2, France<EMAIL_ADDRESS>
###### Abstract.

A family of K3 surfaces $\mathscr{X}\rightarrow B$ has the _Franchetta
property_ if the Chow group of 0-cycles on the generic fiber is cyclic. The
generalized Franchetta conjecture proposed by O’Grady asserts that the
universal family $\mathscr{X}_{g}\rightarrow\mathscr{F}_{g}$ of polarized K3
of degree $2g-2$ has the Franchetta property. While this is known only for
small $g$ thanks to [P-S-Y], we prove that for all $g$ there is a hypersurface
in $\mathscr{F}_{g}$ such that the corresponding family has the Franchetta
property.

##  1\. Introduction

In 1954, Franchetta stated that the only line bundles defined on the generic
curve of genus $g\geq 2$ are the powers of the canonical bundle [F]. Since the
proof was insufficient, the result became known as the _Franchetta conjecture_
; it was proved by Harer in [H], see also [A-C].

In [O’G], O’Grady proposed an analogue of this result for 0-cycles on K3
surfaces. Recall that the Chow group $\operatorname{CH}^{2}(X)$ of 0-cycles on
a K3 surface $X$ contains a canonical class $\mathfrak{o}_{X}$, the class of
any point lying on some rational curve in $X$; for any divisors $D$ and
$D^{\prime}$ on $X$, the product $D\cdot D^{\prime}$ in
$\operatorname{CH}^{2}(X)$ is a multiple of $\mathfrak{o}_{X}$ [B-V]. Let
$p:\mathscr{X}\rightarrow B$ be a map of smooth varieties whose general fiber
is a K3 surface. We say that the family $\mathscr{X}\rightarrow B$ has the
Franchetta property if for every smooth fiber $X$ of $p$ the image of the
restriction map
$\operatorname{CH}^{2}(\mathscr{X})\rightarrow\operatorname{CH}^{2}(X)$ is
contained in $\mathbb{Z}\cdot\mathfrak{o}_{X}$. Equivalently, the Chow group
$\operatorname{CH}^{2}(\mathscr{X}_{\eta})$ of the generic fiber is cyclic.

For $g\geq 2$, let $\mathscr{X}_{g}\rightarrow\mathscr{F}_{g}$ be the
universal family of polarized K3 surfaces of degree $2g-2$. The generalized
Franchetta conjecture of O’Grady is the assertion that this family has the
Franchetta property111Here one can view $\mathscr{F}_{g}$ as a stack, or
restrict to the open subset parametrizing K3 with trivial automorphism group..
It is proved for $g\leq 10$ and some higher values of $g$ in [P-S-Y]; the
general case seems far out of reach. We prove in this note a much weaker (and
much easier) statement:

######  Theorem.

There exists for every $g$ a hypersurface in $\mathscr{F}_{g}$ such that the
corresponding family satisfies the Franchetta property.

The key point of the proof is the construction, for each $g$, of a
$18$-dimensional family of polarized K3 surfaces of degree $2g-2$, which can
be realized as complete intersections in $\mathbb{P}^{1}\times\mathbb{P}^{n}$
for $n=2,3$ or $4$ (§3). Then a simple argument, already used in [P-S-Y],
shows that these families have the Franchetta property (§2). Here the crucial
property of our families is that they are parameterized by a linear space;
thus there is no chance that the method extends to the whole moduli space
$\mathscr{F}_{g}$, which is of general type for $g$ large enough [G-H-S].

##  2\. The method

We use the method of [P-S-Y], based on the following result. Let $P$ be a
smooth complex projective variety, $E$ a vector bundle on $P$, globally
generated by a subspace $V$ of $H^{0}(E)$. Consider the subvariety
$\mathscr{X}\subset\mathbb{P}(V)\times P$ of pairs $(\mathbb{C}s,x)$ with
$s(x)=0$ 222Here $\mathbb{P}(V)$ is the space of lines in $V$.; let $p,q$ be
the projections onto $\mathbb{P}(V)$ and $P$. For $s\in
V\smallsetminus\\{0\\}$, the fiber $p^{-1}(\mathbb{C}s)$ is the zero locus of
$s$ in $P$; for $x\in P$, the fiber $q^{-1}(x)$ is the space of lines
$\mathbb{C}s\subset V$ such that $s(x)=0$. Since $V$ generates $E$, the
projection $q:\mathscr{X}\rightarrow P$ is a projective bundle (in particular,
$\mathscr{X}$ is smooth).

######  Proposition.

For any smooth fiber $X$ of $p$, the image of the restriction map
$\operatorname{CH}(\mathscr{X})\rightarrow\operatorname{CH}(X)$ is equal to
the image of $\operatorname{CH}(P)$.

Proof : Let $h\in\operatorname{CH}^{1}(\mathbb{P}(V))$ be the class of a
hyperplane section. The class $p^{*}h\in CH^{1}(\mathscr{X})$ induces the
hyperplane class on a general fiber of $q$; since $q$ is a projective bundle,
it follows that $\operatorname{CH}(\mathscr{X})$ is generated by
$q^{*}\operatorname{CH}(P)$ and the powers of $p^{*}h$. But $p^{*}h$ vanishes
on the fibers, hence the result.∎

######  Corollary.

Assume that the smooth fibers of $p$ are K3 surfaces, and that the
multiplication map

$m_{P}:\operatorname{Sym}^{2}\operatorname{CH}^{1}(P)\rightarrow\operatorname{CH}^{2}(P)$
is surjective. Then the family $\mathscr{X}\rightarrow\mathbb{P}(V)$ has the
Franchetta property.

Proof : Let $X$ be a smooth fiber of $p$. The commutative diagram

$\textstyle{\operatorname{Sym}^{2}\operatorname{CH}^{1}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{P}}$$\textstyle{\operatorname{Sym}^{2}\operatorname{CH}^{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{X}}$$\textstyle{\operatorname{CH}^{2}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{CH}^{2}(X)}$

shows that the image of
$\operatorname{CH}^{2}(P)\rightarrow\operatorname{CH}^{2}(X)$ is contained in
the image of $m_{X}$, hence in $\mathbb{Z}\cdot\mathfrak{o}_{X}$.∎

##  3\. Proof of the theorem

Since $\dim\mathscr{F}_{g}=19$, we must construct for every $g$ a
$18$-dimensional family of polarized K3 surfaces $(S,L)$ with $(L)^{2}=2g-2$
satisfying the Franchetta property. We will need three different constructions
in order to cover every $g\geq 8$ (the small genus case follow from [P-S-Y]).
We will apply the Corollary with $P=\mathbb{P}^{1}\times\mathbb{P}^{n}$ for
$n=2,3$ or $4$ — note that the surjectivity of $m_{P}$ is trivially satisfied.
For $i,j\in\mathbb{N}$, we put
$\mathcal{O}_{P}(i,j):=\mathcal{O}_{\mathbb{P}^{1}}(i)\boxtimes\mathcal{O}_{\mathbb{P}^{n}}(j)$;
the vector bundle $E$ will be a direct sum of $n-1$ line bundles of this type,
so $S$ is a complete intersection of $n-1$ hypersurfaces in $P$. In order for
$S$ to be a K3 surface we must have
$\det(E)=K_{P}^{-1}=\mathcal{O}_{P}(2,n+1)$. We will always take $V=H^{0}(E)$.

The polarization $L$ on our K3 surface $S$ will be the restriction of the very
ample line bundle $\mathcal{O}_{P}(a,1)$ on $P$, for $a\geq 1$. Let
$p,h\in\operatorname{CH}^{1}(P)$ be the pull back of the class of a point in
$\mathbb{P}^{1}$ and of the hyperplane class in $\mathbb{P}^{n}$. Then

$2g-2=(L)^{2}=(ap+h)^{2}\cdot[S]=\bigl{(}2a(p\cdot
h)+h^{2}\bigr{)}\cdot[S]\,.$

Case I : $n=2$, $E=\mathcal{O}_{P}(2,3)$, hence

$2g-2=\bigl{(}2a(p\cdot h)+h^{2}\bigr{)}\cdot(2p+3h)=2(3a+1)\,.$

Case II : $n=3$, $E=\mathcal{O}_{P}(1,1)\oplus\mathcal{O}_{P}(1,3)$, hence

$2g-2=\bigl{(}2a(p\cdot h)+h^{2}\bigr{)}\cdot(p+h)(p+3h)=2(3a+2)\,.$

Case III : $n=4$,
$E=\mathcal{O}_{P}(0,3)\oplus\mathcal{O}_{P}(1,1)\oplus\mathcal{O}_{P}(1,1)$,
hence

$2g-2=\bigl{(}2a(p\cdot h)+h^{2}\bigr{)}\cdot 3h(p+h)^{2}=2(3a+3)\,.$

Thus we get all values of $g\geq 8$.

It remains to prove that the three families just constructed are
$18$-dimensional. The exact sequence

$0\rightarrow T_{S}\rightarrow T_{P|S}\rightarrow N_{S/P}\rightarrow 0$

gives rise to an exact sequence

$0\rightarrow H^{0}(T_{P|S})\rightarrow H^{0}(N_{S/P})\xrightarrow{\ \partial\
}H^{1}(S,T_{S})\,;$

the image of $\partial$ describes, inside the space of first order
deformations of $S$, those which come from our family. Thus we want to prove
$\dim\operatorname{Im}\partial=18$, or equivalently
$h^{0}(N_{S/P})-h^{0}(T_{P|S})=18$.

We have
$T_{P}=\operatorname{pr}_{1}^{*}T_{\mathbb{P}^{1}}\oplus\,\operatorname{pr}_{2}^{*}T_{\mathbb{P}^{n}}$;
from the Euler exact sequence we get
$h^{0}((\operatorname{pr}_{1}^{*}T_{\mathbb{P}^{1}})_{|S})=h^{0}(\operatorname{pr}_{1}^{*}T_{\mathbb{P}^{1}})$,
and similarly for $\operatorname{pr}_{2}^{*}T_{\mathbb{P}^{n}}$. Thus
$h^{0}(T_{P|S})=h^{0}(T_{\mathbb{P}^{1}})+h^{0}(T_{\mathbb{P}^{n}})=3+n(n+2)$.

Let us denote by $d_{S}$ the restriction to $S$ of a class
$d\in\operatorname{Pic}(P)$. Using $d_{S}\cdot d^{\prime}_{S}=d\cdot
d^{\prime}\cdot[S]$, we find

$p_{S}^{2}=0\ ,\quad p_{S}.h_{S}=3\ ,\quad h_{S}^{2}=2n-2\ .$

By Riemann-Roch, we have
$h^{0}(\mathcal{O}_{S}(i,j))=2+\dfrac{1}{2}(ip_{S}+jh_{S})^{2}=2+3ij+j^{2}(n-1)$.

Case I : $h^{0}(N_{S/P})=h^{0}(\mathcal{O}_{S}(2,3))=29$, $h^{0}(T_{P|S})=11$.

Case II :
$h^{0}(N_{S/P})=h^{0}(\mathcal{O}_{S}(1,1))+h^{0}(\mathcal{O}_{S}(1,3))=9+29=36$,
$h^{0}(T_{P|S})=18$.

Case III :
$h^{0}(N_{S/P})=2h^{0}(\mathcal{O}_{S}(1,1))+h^{0}(\mathcal{O}_{S}(0,3))=2\cdot
8+29=45$, $h^{0}(T_{P|S})=27$.

In each case we find $h^{0}(N_{S/P})-h^{0}(T_{P|S}=18$ as required.∎

_Remarks_ $.-$ 1) In fact, for $S$ very general in each family,
$\operatorname{Pic}(S)$ is generated by $p_{S}$ and $h_{S}$: this follows from
the Noether-Lefschetz theory, see [V, Thm. 3.33]. Therefore
$\operatorname{Pic}(S)$ is the rank 2 lattice with intersection matrix
$\begin{pmatrix}0&3\\\ 3&2n-2\end{pmatrix}$.

2) Our 3 families admit actually a simple geometric description. In what
follows we consider a general surface $S$ in each family. We fix homogeneous
coordinates $U,V$ on $\mathbb{P}^{1}$.

Case I : $S$ is given by an equation $U^{2}A+2UVB+V^{2}C=0$ in
$P=\mathbb{P}^{1}\times\mathbb{P}^{2}$, with $A,B,C$ cubic forms on
$\mathbb{P}^{2}$. Projecting onto $\mathbb{P}^{2}$ gives a double covering
$S\rightarrow\mathbb{P}^{2}$ branched along the sextic plane curve
$\Gamma:B^{2}-AC=0$. Let $\alpha$ and $\gamma$ be the divisors on $\Gamma$
defined by $A=B=0$ and $C=B=0$; then $2\alpha$, $2\gamma$ and $\alpha+\gamma$
are induced by the cubic curves $A=0$, $C=0$ and $B=0$ respectively, hence
belong to the canonical system $\lvert{K_{\Gamma}}\rvert$. It follows that
$\alpha$ and $\gamma$ are linearly equivalent theta-characteristics, hence
belong to a half-canonical $g^{1}_{9}$, that is, a vanishing thetanull on
$\Gamma$. Conversely, it is easy to see that a smooth plane sextic with a
vanishing thetanull has an equation of the above form. We conclude that _the
surfaces in Case I are the double covers of $\mathbb{P}^{2}$ branched along a
sextic curve with a vanishing thetanull_.

Case II : The equations of $S$ in $P=\mathbb{P}^{1}\times\mathbb{P}^{3}$ have
the form $UL+VM=UA+VB=0$, where $L,M;A,B$ are forms of degree $1$ and $3$ on
$\mathbb{P}^{3}$. The projection $S\rightarrow\mathbb{P}^{3}$ is an
isomorphism onto the quartic surface $LB-MA=0$; this is the equation of a
general quartic containing a line. Thus _the surfaces in Case II are the
quartic surfaces containing a line_.

Case III : The equations of $S$ in $P=\mathbb{P}^{1}\times\mathbb{P}^{4}$ are
of the form $UA+VB=UC+VD=F=0$, where $A,B,C,D;F$ are forms of degree $1$ and
$3$ on $\mathbb{P}^{3}$. The projection $S\rightarrow\mathbb{P}^{4}$ is an
isomorphism onto the surface $AD-BC=F=0$, that is, the intersection of a
quadric cone (with one singular point) and a cubic in $\mathbb{P}^{4}$. Thus
_the surfaces in Case III are the complete intersections of a quadric cone and
a cubic in $\mathbb{P}^{4}$_.

Note that one sees easily from this description that each family depends
indeed on 18 moduli.

3) Our construction shows in particular that _the moduli space
$\mathscr{F}_{g}$ contains a unirational hypersurface for any $g$_, despite
the fact that it is of general type for $g\gg 0$.

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