Indecomposable motivic cycles on $K3$ surfaces of degree $2$

A degree $2$ $K3$ surface $X$ is a $K3$ surface on which there is a line bundle ${\mathcal{L}}$ such that ${\mathcal{L}}^{2}=({\mathcal{L}}.{\mathcal{L}})=2$. This is equivalent to the $K3$ surface being a double cover of ${\mathds{P}}^{2}$ ramified at a sextic. The linear system $|{\mathcal{L}}|$ gives a map to ${\mathds{P}}^{2}$ which is ramified at a sextic.

Conversely if $S$ is a sextic in ${\mathds{P}}^{2}$ with at most isolated rational double point singularities, then the desingularisation of the double cover ramified at $S$ is a $K3$ surface of degree $2$.

Generically the rank of the Neron-Severi group (the Picard number $\rho(X)$) is $1$ - though there are some interesting special cases when the rank is much larger. If the sextic is a product of six lines, for instance, then the rank is $16$ and the moduli space of such $K3$ surfaces is four dimensional.

On this $4$-fold there are two notable subvarieties. If the six lines are futher tangent to a conic, then the $K3$ surface is a Kummer surface of a principally polarised Abelian surface and the Picard number is $17$ [BW03]. If the six lines degenerate such that three of them are concurrent then Clingher and Malmendier show that the $K3$ surface is the Kummer surface of an Abelian surface with a polarization of type $(1,2)$ [CM17]. Other cases have been studied by Yu and Zheng [YZ23].

In this paper we construct elements in the motivic cohomology group $H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))$ where $X$ is a degree $2$ $K3$ surface. The group $H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))$ has many avatars - it is usually defined as a graded piece of $K_{1}(X)$ with respect to the Adams filtration, but for us we will use the fact that it is the same as the Bloch higher Chow group $CH^{2}(X,1)$.

The cycles we construct are generically indecomposable – namely they are non-trivial elements of the quotient of the motivic cohomology group by subgroup of cycles coming from lower graded pieces of the filtration. In a sense they are ‘new’ cycles.

In general it is not so easy to find indecomposable cycles and the discovery of them is a minor cause for celebration. They have a number of applications - from algebraicity of values of Greens functions [Mel08],[Sre22],[Sre24], [Ker13] to the Hodge-${\mathcal{D}}$-conjecture [CL05], [Sre08],[Sre14] to torsion in co-dimension $2$ [Mil92],[Spi99]. For varieties over number fields, the existence of indecomposable cycles is predicted by the Beilinson conjectures though there are only a few cases when these have been realised.

The method of construction is similar to that in [Sre22] where we studied the case of Abelian surfaces in some detail. The idea is to show the existence of certain rational curves on ${\mathds{P}}^{2}$. In this paper we show that the same construction can be generalised to this much larger class of varieties, The Kummer surfaces of principally polarised Abelian surfaces are special cases of $K3$ surfaces of degree $2$.

As this paper was nearing completion we became aware of the preprint of Ma and Sato [MS23] where they also construct indecomposable motovic cycles in this case. Their methods of proof are quite different though. Their construction can be viewed as a special case when the rational curve is a line.

The history of this article is linked with the online seminar on $K3$ surfaces which has resulted in these proceedings. I spoke in the seminar on the construction in the article [Sre22] which was about Kummer surfaces of principally polarised Abelian surfaces, Prof. Armand Brumer was in the audience and he asked about the non-principally polarised case. When I dug into the literature I became aware of the work of Clingher and Malmendier [CM17] on the $(1,2)$ case and it was clear that the argument extends to that case. It soon became clear that the argument extends to the general degree $2$ case as well. So I would first like to thank Devendra Tiwari and the rest of the organisers for inviting me to speak and second would like to thank Armand Brumer for his comment. I would also like to thank Shouhei Ma, Ken Sato and Subham Sarkar for their comments. Tom Graber and Ritwik Mukherjee for their help with questions on enumerative geometry and Arvind Nair for his remarks on orthogonal Shimura varieties. I would also like to thank the referee for their numerous comments and suggestions. Finally I would like to thank the Indian Statistical Institute for their support.

Let $X$ be a $K3$ surface with a line bundle ${\mathcal{L}}$ such that ${\mathcal{L}}^{2}=2$. The linear system $|{\mathcal{L}}|$ has no fixed points and maps to ${\mathds{P}}^{2}$. This is a double cover ramified at a sextic $S=S_{X}$. We call $S_{X}$ the associated sextic of $X$. The hyperplane section on ${\mathds{P}}^{2}$ pulls back to a double cover of a line ramified at six points - hence is a genus $2$ curve in $|{\mathcal{L}}|$ [May72]. On the complement of an infinite union of closed subvarieties, the Picard number $\rho(X)$ of such $K3$ surfaces is $1$ generated by this curve [Sha80]. Such $K3$ surfaces are called generic.

Conversely suppose $S$ is a sextic in ${\mathds{P}}^{2}$ with only isolated double point singularities. Then the minimal desingularisation of the double cover ramified at $S$ is a $K3$ surface of degree $2$. If there are double points on the sextic the Picard number of the $K3$ surfaces is larger as the exceptional cycles in the desingularization give new elements of the Neron-Severi group.

The compactification of the moduli space of smooth sextics was first studied by Shah [Sha80] and more recently Yu and Zheng [YZ23] have studied the singular sextic cases.

In this section we state some results about the moduli space of $K3$ surfaces of degree $2$. These results can be found in Bruinier [Bru08]. We have followed Peterson [Pet15].

If $X$ is a $K3$ surface, the group $H^{2}(X,{\mathds{Z}})$ is of rank $22$. As a lattice it is

where $U$ is a unimodular lattice of rank $2$ and signature $(1,1)$ and $E_{8}(-1)$ is the root lattice of the Lie algebra $E_{8}$ with the $(-1)$ indicating that we take the negative of the standard positive definite quadratic form. Hence this is a lattice of signature $(3,19)$

In $H^{2}(X,{\mathds{Z}})$ the class of ${\mathcal{L}}$ gives us an element ${\mathcal{L}}$ of square $2$. Let $L_{2}$ be the orthogonal complement ${\mathcal{L}}^{\perp}$ of ${\mathcal{L}}$ in $H^{2}(X,{\mathds{Z}})$. This is a lattice of signature $(2,19)$

where $<-2>$ is the rank 1 lattice generated by an element $w$ such that $(w,w)=-2$.

The moduli of $K3$ surfaces of degree $2$ can be realised as an orthogonal Shimura variety associated to the lattice $L_{2}$ as follows.

For a lattice $L$ let $O(L)$ denote the group of isometries of $L$. Let ${\hat{L}}$ denote the dual lattice

The discriminant group is $D_{L}={\hat{L}}/L$. Let ${\tilde{O}}({\hat{L}})=\operatorname{Ker}\{O(L)\rightarrow O(D_{L})\}$. Let ${O^{+}}(L)$ denote the subgroup of $O(L)$ of spinor norm $1$ and ${\tilde{O}^{+}}(L)={\tilde{O}}(L)\cap{O^{+}}(L)$.

Let

Choose a component ${\mathcal{D}}$. The moduli space of $K3$ surfaces of degree $2$ with transcendental lattice $L$ is

An alternative description is as a quotient of the Hermitian symmetric space of $O(L)$. If $L$ is of signature $(2,n)$ this is

The moduli space ${\mathcal{M}}$ is

When $L$ is the lattice $L_{2}$ this moduli space ${\mathcal{M}}_{2}$ is a $19$ dimensional variety. If one considers singular sextics, the transcendental lattice becomes smaller as the exceptional fibres are elements of the Neron-Severi group. One interesting submoduli is when the sextic $S$ degenerates in to six lines. Here the Picard number is $16$ as the Neron-Severi is generated by the class of ${\mathcal{L}}$ and the $15$ exceptional fibres over the nodes of $S$. The transcendental lattice is hence of signature $(2,4)$ and the moduli space is $4$ dimensional.

In this there is a particularly interesting divisor – if the six lines are tangent to a conic. Such $K3$ surfaces have Picard number $17$ and the transcendental lattice is of signature $(2,3)$. The new cycle is obtained as a component of the double cover of the conic. These $K3$ surfaces are the Kummer surfaces of principaly polarised abelian surfaces and the moduli is nothing but an alternative description of the Siegel modular threefold.

There are several other moduli corresponding to nodal sextics. These are described in Yu and Zheng. [YZ23]. Some of them correspond to $K3$ double covers of del Pezzo surfaces.

As mentioned above, when the sextic has nodal singulaties, the corresponding $K3$ surface has additional elements in the Neron-Severi group coming from the exceptional fibres. In this section we will determine another condition under which there are new elements in the Neron-Severi group. In what follows we use ${\mathcal{L}}$ to denote the class of the line bundle ${\mathcal{L}}$ in the Neron-Severi group. Let $D$ be an element of $NS(X)$ which is not a multiple of ${\mathcal{L}}$. The moduli of those $K3$ surfaces of degree $2$ where the Neron-Severi group is generated by ${\mathcal{L}}$ and $D$ is a divisor on ${\mathcal{M}}_{2}$.

Recall that $({\mathcal{L}},{\mathcal{L}})=2$. Consider the intersection matrix of the lattice generated by ${\mathcal{L}}$ and $D$,

Let $\Delta=\Delta(D)=-\det(A)=({\mathcal{L}}.D)^{2}-2D^{2}$ and $\delta=(D.{\mathcal{L}})\operatorname{mod}2$, so $\delta=\{0,1\}$. From the Hodge Index theorem we have that $\Delta(D)\geq 0$ and $\Delta>0$ if and only if $D$ is not a multiple of ${\mathcal{L}}$.

The irreducible Noether-Lefschetz divisor ${{\mathcal{P}}}_{\Delta,\delta}$ is the closure of the set of polarised $K3$ surfaces where the Picard lattice is a rank $2$ lattice with discriminant $\Delta(D)=\Delta$ and class $\delta$.

For number theoretic applications one considers Heegner divisors which are sums of these ${{\mathcal{P}}}_{\Delta,\delta}$ for certain choices of $\Delta$. [Pet15]. Numbers determined by them are known to appear as coefficients of modular forms.

It is known that the Picard number of a Abelian surface with real multiplication is at least $2$ which translates to the corresponding Kummer surface having Picard number at least $18$. Birkenhake-Wilhelm [BW03] showed, generalizing Humbert, that this corresponds to certain special rational curves on the associated ${\mathds{P}}^{2}$. Conversely, if one has special rational curves on ${\mathds{P}}^{2}$ there are additional cycles in the corresponding $K3$ surface.

Along the lines of their argument, we have the following.

In fact the theorem likely holds for higher genus curves as well [BW03], Proposition 6.4. We will only be concerned with the case when the curve is rational in what follows and hence will restrict ourselves to that case.

The converse need not be true. Namely there exists ${{\mathcal{P}}}_{\Delta,\delta}$ such that the extra cycle need not be represented by a rational curve.

A rational curve $Q$ which meets the sextic $S$ at points of even multiplicity determines some nodes and orders of tangency, Let ${{\mathcal{P}}}_{Q}$ denote the component of ${{\mathcal{P}}}_{\Delta(Q),\delta}$ where, for a $K3$ surface $X$ corresponding to a point on ${{\mathcal{P}}}_{\Delta(Q),\delta}$, there is a rational curve $Q_{X}$ which meets $S_{X}$ is exactly the same manner.

If $S_{X}$ is smooth then the only possibility is that $Q_{X}$ is tangent to $S_{X}$ at some points. However, if $S_{X}$ has nodes and components, $P_{\Delta(D),\delta}$ could have more components. For instance, as mentioned above, nodal sextics correspond to interesting submoduli of the moduli of degree $2$ $K3$ surfaces. In the case when the sextic is a union of six lines $\ell_{i}$ tangent to a conic it is known that the $K3$ surface $X$ is the Kummer surface of a principally polarized Abelian surface. Here the sextic has $15$ nodes $q_{ij}=\ell_{i}\cap\ell_{j}$ corresponding to the points of intersection of the lines. In this case $\Delta$ is called the Humbert invariant and corresponds to the moduli of Abelian surfaces with endomorphism ring containing ${\mathds{Z}}[\sqrt{\Delta}]$. For a generic Kummer surface the Picard number is $17$ and on these Humbert surfaces the Picard number is at least $18$.

As an example, if there is a second conic passing thought $5$ of the $15$ nodes - hence it meets five of the lines at these points and suppose it is tangent to the remaining line, then $\Delta=5$ and ${{\mathcal{P}}}_{\Delta(Q),\delta}$ is the moduli of Kummer surfaces of Abelian surfaces with real multiplication by ${\mathds{Z}}({\sqrt{5}})$. The conics passing through the five points $q_{12},q_{23},q_{34},q_{45}$ and $q_{51}$ and tangent to $\ell_{6}$ determine a component of ${{\mathcal{P}}}_{\Delta(Q),\delta}$ and similarly, the conics passing through $q_{12},q_{23},q_{34},q_{46},q_{61}$ and tangent to $\ell_{5}$ will determine a different component. This special case of Kummer surfaces of simple Abelian surfaces was studied in some detail in [Sre22].

In the case when the sextic degenerates to six lines three of which are concurrent the $K3$ surface is the Kummer surface of an Abelian surface with polarization of type $(1,2)$ [CM17]. Once again the ${{\mathcal{P}}}_{\Delta,\delta}$ will correspond to Abelian surfaces with extra endomorphisms.

Let $X$ be a surface over a field $K$. The group $H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))$ has the following presentation. It is generated by sums of the form

where $C_{i}$ are curves on $X$ and $f_{i}$ functions on them satisfying the co-cycle condition

Equivalently these can be though of as codimensional $2$ subvarieties $Z$ of $X\times{\mathds{P}}^{1}$ such that

The cycle given by the sum of the graphs of $f_{i}$ in $X\times{\mathds{P}}^{1}$ has this property.

Relations are given by the Tame symbol of a pair of functions. If $f$ and $g$ are two functions on $X$, the Tame symbol of the pair

satisfies the co-cycle condition and such cycles are defined to be $0$ is the group. Here $X^{1}$ is the set of irreducible codimensional one subvarieties.

One example of an element of this group is a cycle of the form $(C,a)$ where $a$ is a nonzero constant function and $C$ is a curve on $X$. This trivially satisfies the cocycle condition. More generally, if $L/K$ is a finite extension and $X_{L}=X\times_{\operatorname{Spec}(K)}\operatorname{Spec}(L)$ then one has a product map

The image of this is the group of decomposable cyles $H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))_{dec}$. The group of indecomposable cycles is the quotient group

In general it is not clear how to construct non-trivial elements of this group.

A way of constructing cycles in this group is as follows. Since we use it in the main construction we label it a proposition.

A priori it is not clear that if this cycle is indecomposable. We will show that is the case in many instances. For this we need to use the localization sequence in motivic cohomology.

If ${\mathcal{X}}\rightarrow S$ is a family of varieties over a base $S$ with $X_{\eta}$ the generic fibre and $X_{s}$ the fibre over a closed subvariety $s$ then one has a localization sequence relating the three.

where $S^{1}$ denotes the set of irreducible closed subvarieties of codimension $1$. The boundary map of an element $Z=\sum_{i}(C_{\eta,i},f_{\eta,i})$ is given as follows. Let ${\mathcal{C}}$ denote the closure of $C_{\eta}$ in ${\mathcal{X}}$. Then

The cocycle condition $\sum_{i}\operatorname{div}_{C_{\eta,i}}(f_{\eta,i})=0$ implies that the ‘horizontal’ divisors cancel out and only the components in some of the fibres survive.

Let ${\mathcal{X}}$ be a family of $K3$ surfaces. To check for indecomposability one can use the boundary map $\partial$. Over the algebraic closure of the base field, a decomposable cycle in $H^{3}_{{\mathcal{M}}}(X_{\eta},{\mathds{Q}}(2))$ is given by $(C_{\eta},a_{\eta})$ where $C_{\eta}$ is a curve on $X_{\eta}$ and $a_{\eta}$ a constant (that is, a function on the base $S$).

The boundary map is particularly simple to compute in the case of a decomposable element

where $C_{s}$ is the restriction of the closure ${\mathcal{C}}$ of $C_{\eta}$ to the fibre over $s$. Hence in the boundary of decomposable elements one can obtain only those cycles in the special fibres which are the restrictions of cycles in the generic fibre. In particular, if $Z_{\eta}$ is a motivic cycle and the boundary contains cycles which are not the restriction of cyces in the generic fibre then the motivic cycle is necessarily indecomposable. We will show that that is the case for our cycles.

An alternate way, as is done in Chen-Lewis [CL05], Ma-Sato [MS23] or Sato [Sat23], is to compute the Beilinson regulator and show that, when evaluated against a particular transcendental $(1,1)$ form, this is non-zero. In the case when the base is the ring of integers of a glabal field or a local field, the boundary map can be thought of as a non-Archimedean regulator map and the idea is essentially the same. The transcendental form is represented by a new algebraic cycle in the special fibre. Evaluation against this form corresponds to intersecting with the new cycle. That being non-zero implies that the component of the boundary in the direction of the new cycle is non-zero. This also fits in to the philosophy espoused by Manin [Man91] that the Archimedean component of an arithmetic variety can be viewed as the ‘special fibre at $\infty$’.

There is a well known theorem in enumerative geometry [Zin05] which states the following:

The numbers $N_{1}=1$, $N_{2}=1$, $N_{3}=12$ are classical. The precise number $N_{d}$ is the celebrated theorem of Kontsevich-Manin [KM94] and Ruan-Tian [RT95].

Recall that ${{\mathcal{P}}}_{Q}$ is the submoduli of the moduli of $K3$ surfaces $X$ of degree $2$ where there is a rational curve $Q$ of degree $d$ which meets the sextic $S_{X}$ at points of even multiplicity. Since $Q$ is of degree $d$ it meets $S_{X}$ at $6d$ points. However, since they meet at either ordinary double points of the sextic or points of even multiplicity, there are at most $3d$ distinct points of intersection.

If we discard one of the double points of $Q\cap S_{X}$, we determine $3d-1$ points. Hence the theorem above suggests that perhaps for any $S_{X}$ we can find a rational curve of degree $d$ passing through these $3d-1$ points.

In general it is not clear if ${{\mathcal{P}}}_{Q}$ is non-empty but we will give several examples when such submoduli exist.

We need some definitions. Let $\Sigma_{6}$ be the space of sextic curves in ${\mathds{P}}^{2}$ and let $\Sigma_{6}(n)$ be the subspace with $n$ nodes. A singular type $\Sigma_{T}$ [YZ23], Section 3, is an irreducible component of $\Sigma_{6}(n)$. A sextic in $\Sigma_{T}$ is said to be of type $T$. For $z\in\Sigma_{T}$ there are $n$ nodes $p_{1}(z),\dots,p_{n}(z)$ smoothly varying with $z$.

We have the following theorem, the proof of which was suggested to me by Tom Graber. The proof is easier as we assume that for some $X$ there exists a rational curve meeting $S_{X}$ at the correct number of points.

In general it is not clear if there exists pairs $(S_{t_{0}},Q_{t_{0}})$ of the type we require, though in several special cases they are known to exist. For instance if $k=0$ and $d=2$, so generically $S$ is a smooth sextic and $Q$ is a conic. We know that for the Kummer surface of a principally polarised Abelian surface there is a conic tangent to the sextic (which in this case is the union of six lines) at the six lines. In particular, at five lines. Hence from the theorem above, it says that in a Zariski neigbourhood of the moduli point of the Kummer surface of an Abelian surface there exists a conic tangent to a sextic at five points. In fact, it is a theorem of Gathmann [Gat05] (Corollary 3.6) that there are $70956$ conics tangent to a sextic at five points

If $S$ has nodes, for instance if $S$ is the union of six lines and has $15$ nodes, then the theorems of Birkenhake and Lange ([BW03], Theorems 7.1-7.4) show that for $\Delta$ of the form $2d^{2}-2k+7$ or $2d^{2}-2k+8$ there exists rational curves of degree $d$ passing through $k$ nodes and meeting the remaining lines at points of even multiplicity. In particular, there exists a rational curve of the type required passing through $k$ nodes and $3d-1-k$ points of even multiplicity.

If $S$ has $n$-nodes with $n\leq 8$ then Yu and Zheng [YZ23] show that the desingularization of the double cover of ${\mathds{P}}^{2}$ branched at $S$ in $\Sigma_{T}$ is a del Pezzo surface. So this argument extends to certain families of $K3$ double covers of del Pezzo surfaces.

A related case when there is a similar theorem is ${\mathds{P}}^{1}\times{\mathds{P}}^{1}$. Here minimal desingularization of the double cover of ${{\mathds{P}}\times{\mathds{P}}}$ ramified at a curve of degree $(4,4)$ is also a $K3$ surface. Here the analogous statement would be that if there existed a rational curve $Q$ of degree $(1,n)$ or $(n,1)$ meeting the curve at $2n+1$ double points then in a Zariski open set there exists such a curve.

In the special case that the curve is consists of two pairs of four parallel lines, the double cover is the Kummer surface of a product of elliptic curves, say $E_{1}$ and $E_{2}$. If $E_{1}$ and $E_{2}$ are isogenous by an isogeny of degree $n$ then image of the graph of the isogeny provides a example of the special curve [Sre24].