Some non-special cubic fourfolds

TL;DR

This paper provides an independent proof that certain divisors in the moduli space of cubic fourfolds are not Noether-Lefschetz divisors, using explicit examples and point counting over finite fields.

Contribution

It offers a new proof and explicit examples demonstrating that some divisors previously thought to be special are actually not Noether-Lefschetz divisors.

Findings

Explicit cubic fourfold in the divisor is Noether-Lefschetz general

Two other divisors are shown not to be Noether-Lefschetz divisors

Point counting methods over finite fields are used for proof

Abstract

In [1309.1899], Ranestad and Voisin showed, quite surprisingly, that the divisor in the moduli space of cubic fourfolds consisting of cubics "apolar to a Veronese surface" is not a Noether-Lefschetz divisor. We give an independent proof of this by exhibiting an explicit cubic fourfold X in the divisor and using point counting methods over finite fields to show X is Noether-Lefschetz general. We also show that two other divisors considered in [ibid.] are not Noether-Lefschetz divisors.

Tables

Table 1. Table 1. Point counts.

Equations

$$H_{\text{prim}}^{2,2}(X,\mathbb{Z}):=H_{\text{prim}}^4(X,\mathbb{Z})\cap H^{2,2}(X)$$

$$I=⟨q_1,\dots ,q_m⟩\subset \mathbb{C}[y_0,\dots ,y_5],$$

$$\sum a_{ijk}{\partial }_j{\partial }_{k}f=0\text{ for all }i.$$

$$m:{\mathrm{Sym}}^3{\mathrm{Sym}}^2{\mathbb{C}}^3\to {\mathrm{Sym}}^6{\mathbb{C}}^3$$

$$m^{\vee }:{\mathrm{Sym}}^6{\mathbb{C}}^{3\vee }\to {\mathrm{Sym}}^3{\mathrm{Sym}}^2{\mathbb{C}}^{3\vee }.$$

$$H_{\text{eˊt}}^{2i}(X_{\bar{k}},{\mathbb{Q}}_{\ell }(i))$$

$${\mathrm{CH}}^i(X_{\mathbb{C}})\stackrel{\mathrm{cl}}{\to }{H}^{2i}(X^{\text{an}},\mathbb{Z}(i))$$

$${\mathrm{CH}}^i(X_{\mathbb{C}})\stackrel{\mathrm{cl}}{\to }{H}^{2i}(X^{\text{an}},{\mathbb{Z}}_{\ell }(i)).$$

$${\mathrm{CH}}^i(X_{\mathbb{C}})\stackrel{\mathrm{cl}}{\to }{H}_{\text{eˊt}}^{2i}(X_{\mathbb{C}},{\mathbb{Z}}_{\ell }(i)).$$

$${\mathrm{CH}}^i(X_{\mathbb{C}})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{CH}}^i(X_{\bar{K}})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{CH}}^i(X_{\bar{k}})\otimes {\mathbb{Q}}_{\ell }\stackrel{\mathrm{cl}}{\to }{H}^{2i}(X_{\bar{k}},{\mathbb{Q}}_{\ell }(i))$$

$$V:=H_{\text{eˊt},\text{prim}}^4(X_{\bar{k}},{\mathbb{Q}}_{\ell }(2))\cong {\mathbb{Q}}_{\ell }^{22}$$

$$\chi (t):=\det (t\cdot {\mathrm{Id}}_V-{\Phi }^{\ast }{|}_V)$$

$$H_{\text{eˊt}}^i(X_{\bar{k}},{\mathbb{Q}}_{\ell }(i))=\{\begin{array}{ll}{\mathbb{Q}}_{\ell }& i=0,\\ {\mathbb{Q}}_{\ell }\cdot h& i=2\\ {\mathbb{Q}}_{\ell }\cdot h^2\oplus V& i=4\\ {\mathbb{Q}}_{\ell }\cdot h^3& i=6\\ {\mathbb{Q}}_{\ell }\cdot h^4& i=8\\ 0& \text{otherwise,}\end{array}$$

$$\#X(\mathbb{F}_{q^{m}})=1+q^{m}+q^{2m}\Bigl{(}1+\operatorname{tr}(\Phi^{*m}|_{V})\Bigr{)}+q^{3m}+q^{4m}.$$

$$A{y}_4^2+B{y}_4{y}_5+C{y}_5^2+D{y}_4+E{y}_5+F,$$

$${\mathrm{Bl}}_l(X)⟶{\mathbb{P}}_{(y_0:\dots :y_3)}^3$$

$$\#Y({\mathbb{F}}_q)=(q+1)\cdot \#Z+q\cdot (\#\tilde{\Delta }-\#\Delta ).$$

$$\#Y({\mathbb{F}}_q)=\underset{\text{from smooth conics}}{\underbrace{(q+1)\cdot (\#Z-\#\Delta )}}+\underset{\text{from singular conics}}{\underbrace{(q\cdot \#\tilde{\Delta }+\#\Delta )}},$$

$$\#X({\mathbb{F}}_q)=q^4+q^3+q(\#\tilde{\Delta }-\#\Delta )+q+1.$$

$$A{E}^2+B^{2}F+C{D}^2-BDE-4ACF.$$

$$B=D=E=0,$$

$$⟨x_i,x_j⟩=\frac{\partial }{\partial x_i}{x}_j={\delta }_{ij},$$

$$R_k\otimes R_d\to R_{d-k}$$

$$⟨x_0{x}_1,x_0{x}_1⟩=\frac{\partial }{\partial x_0}\frac{\partial }{\partial x_1}{x}_0{x}_1=1,$$

$$⟨x_0^2,x_0^2⟩=\frac{\partial }{\partial x_0}\frac{\partial }{\partial x_0}{x}_0^2=2,$$

$$m:S_1\to R_2$$

$$y_0$$

$$y_3$$

$$m:S_d\to R_{2d}$$

Full text

1. Introduction

In [20], Ranestad and Voisin introduced some new divisors in the moduli space of smooth complex cubic fourfolds, quite different from Hassett’s Noether–Lefschetz divisors [14]. A cubic $X\subset\mathbb{P}^{5}$ is called special if

[TABLE]

is non-zero, or equivalently if $X$ contains a surface not homologous to a complete intersection. The locus of special cubic fourfolds is a countable union of irreducible divisors in the moduli space, called Noether–Lefschetz divisors. Special cubic fourfolds often have rich connections to K3 surfaces, and it is expected that all rational cubic fourfolds are special; see [15] for a recent survey of the topic.

Ranestad and Voisin’s divisors are constructed in a much more algebraic way, using apolarity. Briefly, a cubic fourfold $X$ cut out by a polynomial $f(y_{0},\dotsc,y_{5})$ is said to be apolar to an ideal generated by quadrics,

[TABLE]

if, writing $q_{i}=\sum a_{ijk}\,y_{j}y_{k}$, we have

[TABLE]

Ranestad and Voisin showed that the following loci are irreducible divisors in the moduli space of cubic fourfolds: $D_{\textit{V-ap}}$, the set of cubics apolar to a Veronese surface; $D_{\textit{IR}}$, the set of cubics apolar to a quartic scroll; and $D_{\textit{rk3}}$, the closure of the set of cubics apolar to the union of a plane and a disjoint hyperplane. They showed that $D_{\textit{V-ap}}$ is not a Noether–Lefschetz divisor, by carefully analyzing its singularities. From this they deduced that for a generic cubic $X$, the “varieties of sums of powers” of the polynomial $f$, which is a hyperkähler fourfold, is not Hodge-theoretically related to the Fano variety of lines on $X$, a better-known hyperkähler fourfold. They remarked that $D_{\textit{rk3}}$ is “presumably” not a Noether–Lefschetz divisor, and that if one could prove that $D_{\textit{IR}}$ is not a Noether–Lefschetz divisor then it would give another approach to proving their main theorem.

We were very surprised to learn that $D_{\textit{V-ap}}$ is not a Noether–Lefschetz divisor: we would have guessed that it was Hassett’s divisor $\mathcal{C}_{38}$, for the following reason. Cubic fourfolds in $\mathcal{C}_{38}$, which are conjectured to be rational, have associated K3 surfaces of degree 38. Mukai [17] observed that the generic such K3 surface $S$ can be described as the variety of sums of powers of a plane sextic $g(x_{0},x_{1},x_{2})$; see [19, Thm. 1.7(iii)] for a more detailed account. A natural way to construct a cubic fourfold from $g$ is to consider the multiplication map

[TABLE]

and its transpose

[TABLE]

Then $m^{\vee}(g)$ cuts out a cubic $X\subset\mathbb{P}(\operatorname{Sym}^{2}\mathbb{C}^{3\vee})=\mathbb{P}^{5}$, typically smooth. By [20, Lem. 1.7], the cubics obtained this way are exactly those in $D_{\textit{V-ap}}$. Though it seemed reasonable to expect that the cubic $X$ would be Hodge-theoretically associated with the K3 surface $S$, Ranestad and Voisin’s result implies that it cannot be.

Since the result is so surprising, and the proof quite difficult, at least to our eyes, we thought it worthwhile to seek experimental confirmation. In this note, we give a computer-aided proof of the following result, and in particular a more direct proof of Ranestad and Voisin’s result:

Theorem 1**.**

There is an explicit sextic polynomial $g$, defined over $\mathbb{Q}$, such that the cubic fourfold $X$ cut out by $m^{\vee}(g)$ is smooth and satisfies $H^{2,2}_{\textup{prim}}(X,\mathbb{Z})=0$. In particular, $X\in D_{\textit{V-ap}}$, but $X$ is not in any Noether–Lefschetz divisor.

We also confirm Ranestad and Voisin’s expectations for the other two divisors mentioned above:111They also studied a fourth divisor $D_{\textit{copl}}$, not defined using apolarity, but we were unable to find a suitable cubic in that divisor using the technique described below. Probably one could be found by working modulo 5, but that would forfeit many of the computational advantages of working modulo 2.

Theorem 2**.**

There is an explicit cubic fourfold $X\in D_{\textit{IR}}$, defined over $\mathbb{Q}$, with $H^{2,2}_{\textup{prim}}(X,\mathbb{Z})=0$. In particular, $D_{\textit{IR}}$ is not a Noether–Lefschetz divisor.

Theorem 3**.**

There is an explicit cubic fourfold $X\in D_{\textit{rk3}}$, defined over $\mathbb{Q}$, with $H^{2,2}_{\textup{prim}}(X,\mathbb{Z})=0$. In particular, $D_{\textit{rk3}}$ is not a Noether–Lefschetz divisor.

Thus it seems that apolarity tends to produce cubic fourfolds of a different character than those considered by Hassett. It would be very interesting to know if there is any connection with rationality.

We follow a strategy developed by van Luijk [23] and refined by Elsenhans and Jahnel [9, 10], for producing explicit K3 surfaces of Picard rank 1. We find an explicit cubic fourfold with good reduction modulo 2, then count points over $\mathbb{F}_{2^{m}}$ for $m=1,2,\dotsc,11$ to determine the eigenvalues of Frobenius acting on $H^{4}_{\textup{prim}}(X_{\overline{\mathbb{F}}_{2}},\mathbb{Q}_{\ell}(2))$, which give a bound on the rank of $H^{2,2}_{\textup{prim}}(X,\mathbb{Z})$. In §2, we give the details of adapting van Luijk’s method to cubic fourfolds.

On the one hand, our task is simpler than van Luijk’s: since the geometric Picard rank of a K3 surface over a finite field is necessarily even, to show that a K3 surface has Picard rank 1, van Luijk had to work modulo two different primes and compare intersection forms; but here we need only work modulo one prime. On the other hand, a fourfold is much bigger than a surface, and it is infeasible to count points naively by iterating over $\mathbb{P}^{5}$. Nor can we control the cohomology of $X$ by counting points on an associated K3 surface as in [2] or [16], since there is none. In §3 we explain how to exploit the conic bundle structure on the blow-up of $X$ along a line, so that to count points we only need to iterate over $\mathbb{P}^{3}$, and with a little more work, only over $\mathbb{P}^{2}$. The same idea was used to count points on cubic threefolds by Debarre, Laface, and Roulleau [7, §4.3], who trace it back to Bombieri and Swinnerton–Dyer [3]. Whereas those papers restrict to odd characteristic, we find that the hassle of working with conics in characteristic 2 is more than repaid by the fact that computation in $\mathbb{F}_{2^{m}}$ is so fast.

We do not use the $p$-adic cohomology methods of Kedlaya, Harvey, and others [1, 13, 6]. While these methods are surely the way of the future, they are much harder to implement than our algorithm, and the available implementations are not quite ready to handle cubic fourfolds.

In §4, we give the explicit polynomials and the point counts needed to prove Theorems 1, 2, and 3. In §5, we conclude with some remarks about computer implementation and verification.

The existence of Noether–Lefschetz general cubic fourfolds (and other complete intersections) defined over $\mathbb{Q}$ was first proved by Terasoma [21], although his proof is not constructive. Elsenhans and Jahnel gave an explicit example in [10, Example 3.15], also using point-counting methods. But the existence of Noether–Lefschetz general cubic fourfolds with specified algebraic properties is far from clear a priori.

Acknowledgements

We thank J.-L. Colliot-Thélène, E. Costa, D. Harvey, B. Hassett, A. Kuznetsov, K. Ranestad, R.P. Thomas, A. Várilly-Alvarado, and B. Viray for helpful conversations, and the organizers of the Fall 2016 AGNES workshop at UMass Amherst for their hospitality. In the course of this project we used the computer algebra system Macaulay2 [12] extensively, and Magma [4] to a lesser extent. The second author was partially supported by NSA Young Investigator Grant H98230-16-1-0321.

2. Adaptation of van Luijk’s method

In this section we adapt the method developed in [23] from K3 surfaces to cubic fourfolds. We begin with the following proposition, which is similar to [22, Cor. 6.3]. Note that due to our choice of Tate twist, our Frobenius eigenvalues have absolute value 1 rather than $q^{i}$.

Proposition 2.1**.**

Let $R$ be a discrete valuation ring of a number field $L$ with residue field $k\cong\mathbb{F}_{q}$ for $q=p^{r}$, and let $X$ be a smooth projective scheme over $R$. Let $X^{\textup{an}}$ denote the complex manifold associated to the complex variety $X_{\mathbb{C}}$. Let $\Phi\colon X_{k}\to X_{k}$ be the $r$-th power absolute Frobenius, let $\ell$ be a prime different from $p$, and let $\Phi^{*}$ be the automorphism of

[TABLE]

induced by $\Phi\times 1$ on $X_{k}\times\bar{k}$.

Then the rank of the image of the cycle class map

[TABLE]

is less than or equal to the number of eigenvalues of $\Phi^{*}$, counted with multiplicity, that are roots of unity.

In particular, if the Hodge conjecture holds for codimension-$i$ cycles on $X$, then the rank of $H^{2i}(X^{\textup{an}},\mathbb{Z})\cap H^{i,i}(X^{\textup{an}})$ is bounded above by the number of such eigenvalues.

Proof.

The rank of the image of (1) agrees with the rank of the image of

[TABLE]

By the comparison theorem between singular and $\ell$-adic cohomology, this agrees with the rank of the image of

[TABLE]

Now let $K$ be the field of fractions of the completion $\widehat{R}$, and consider the commutative diagram222Alternatively we could have embedded $\bar{K}\hookrightarrow\mathbb{C}$, but we preferred to use the more natural embeddings $\mathbb{C}\hookleftarrow\bar{L}\hookrightarrow\bar{K}$.

[TABLE]

The right-hand vertical maps are isomorphisms by smooth base change, and while the left-hand vertical maps are typically not isomorphisms, the images of the three horizontal maps agree thanks to the existence of Hilbert schemes, as remarked in [5, Rem. 46].

Next we have a commutative square

[TABLE]

where the left-hand vertical map is the specialization map for Chow groups; see Fulton [11, Example 20.3.5] for the commutativity of the square. Thus the rank of the image of the top horizontal map is less than or equal to that of the bottom one.

Finally we consider the cycle class map after tensoring with $\mathbb{Q}_{\ell}$

[TABLE]

and recall that cycles on $X_{\bar{k}}$ are defined over some finite extension of $k$, hence are fixed by some power of Frobenius, hence their classes in cohomology are eigenvectors with eigenvalues a root of unity as in the proof of [22, Cor. 6.3]. ∎

In our application, we will take $R=\mathbb{Z}_{(2)}$, so $L=\mathbb{Q}$, $q=p=2$, and $K=\mathbb{Q}_{2}$.

Now specialize to the case where $X$ is a cubic fourfold. The Hodge conjecture holds for cubic fourfolds [26, 18, 25], so to show that $H^{2,2}_{\textup{prim}}(X,\mathbb{Z})=0$ it is enough to show that no eigenvalue of $\Phi^{*}$ acting on

[TABLE]

is a root of unity, or equivalently that the characteristic polynomial

[TABLE]

has no cyclotomic factor. For this it is enough to show that $\chi$ is irreducible over $\mathbb{Q}$ and that not all its coefficients are integers.

The cohomology of $X$ is

[TABLE]

where $h$ is the hyperplane class, so by the Lefschetz trace formula we have

[TABLE]

The method of passing from traces of powers of $\Phi^{*}|_{V}$ to the characteristic polynomial using Newton’s identities is discussed in [23, §3], [9, §3], or [16, §6.1]. Thanks to the functional equation $\chi(t)=\pm t^{22}\chi(t^{-1})$ it is usually enough to count up to $m=11$.

3. The algorithm using conic bundles

How then can we compute the point counts (2) for an explicit cubic with $q=2$ and $m=1,2,\dotsc,11$? As we said in the introduction, it is not feasible to iterate over $\mathbb{P}^{5}(\mathbb{F}_{2^{m}})$, evaluating our cubic polynomial at every point: in Magma this would take many years, and in a program written optimized specially for the purpose it would take months, or at best weeks. Instead we project from a line to obtain a conic fibration.

Continue to work with a smooth cubic $X$ defined over an arbitrary $\mathbb{F}_{q}$. Choose a line $l\subset X$ defined over $\mathbb{F}_{q}$; by [7] such a line always exists for $q=2$ or $q\geq 5$, and probably for $q=3$ or 4 as well. Change variables so that $l$ is given by $y_{0}=y_{1}=y_{2}=y_{3}=0$. Then we can write the equation of $X$ as

[TABLE]

where $A$, $B$, and $C$ are linear in $y_{0},\dotsc,y_{3}$, $C$ and $D$ are quadratic, and $F$ is cubic. If $A,\dotsc,F$ vanish simultaneously at some point of $\mathbb{P}^{3}$ then $X$ contains a plane, contributing an unwanted Frobenius eigenvalue, so we stop. Otherwise we obtain a flat conic bundle

[TABLE]

with fibers given by the homogenization of the quadratic form above. Now we use the following.

Proposition 3.1**.**

Let $Z$ be an $\mathbb{F}_{q}$-scheme of finite type, let $\pi\colon Y\to Z$ be a flat conic bundle, let $\Delta\subset Z$ be the locus parametrizing degenerate conics, and let $\tilde{\Delta}$ be the (possibly branched) double cover of $\Delta$ parametrizing lines in the fibers of $\pi$. Then

[TABLE]

Proof.

A smooth conic over $\mathbb{F}_{q}$ is isomorphic to $\mathbb{P}^{1}$, hence has $q+1$ points. For a singular conic, there are three possibilities:

• •

a pair of lines defined over $\mathbb{F}_{q}$, contributing $2q+1$ points;

• •

a pair of conjugate lines defined over $\mathbb{F}_{q^{2}}$, contributing only one $\mathbb{F}_{q}$-point;

• •

a double line, contributing $q+1$ points.

The fiber of $\tilde{\Delta}$ over the relevant point of $\Delta$ consists of 2, 0, or 1 points respectively. Thus we have

[TABLE]

which simplifies to give (3). ∎

In our case, with $Y=\operatorname{Bl}_{l}(X)$ and $Z=\mathbb{P}^{3}$, this yields

[TABLE]

The discriminant locus $\Delta\subset\mathbb{P}^{3}$ is cut out by the quintic polynomial

[TABLE]

This formula remains valid in characteristic 2, although of course the last term vanishes. The double cover $\tilde{\Delta}$ can also be described as the variety of lines on $X$ that meet $l$.333The topology of $\tilde{\Delta}$ over $\mathbb{C}$ has been studied in [24, §3, Lemmas 1–3]. For a generic $l\subset X$, it is a smooth surface with Hodge diamond

1

0

0

5

50

0

0

1

So we can iterate over $\mathbb{P}^{3}$ and count points on $\Delta$ and $\tilde{\Delta}$. To count points on $\tilde{\Delta}$ in characteristic 2, we note that if $B=D=E=0$ then the conic is a double line; otherwise we compute an Arf invariant: if $B\neq 0$ (resp. $D\neq 0$ or $E\neq 0$), then the conic has $2q+1$ points if $AC/B^{2}$ (resp. $AF/D^{2}$ or $CF/E^{2}$) is of the form $a^{2}+a$ for some $a\in\mathbb{F}_{q}$, and 1 point if it is not.

This algorithm runs up to $q=2^{11}$ in about half a minute on the first author’s laptop. But to find the explicit cubics below we had to search through dozens of candidates, so it was worthwhile to make a further optimization, iterating only over $\Delta$ rather than all of $\mathbb{P}^{3}$, as follows.

The quintic $\Delta$ is not smooth; in characteristic 2, it is singular at least along the locus where

[TABLE]

which has expected dimension 0 and degree 4. Suppose this locus contains an $\mathbb{F}_{2}$-point $y$.444In practice this usually happens, although not always. That is, there exist smooth cubics $X$ and $\mathbb{F}_{2}$-lines $l\subset X$ such that $\Delta_{\textup{sing}}$ has no $\mathbb{F}_{2}$-point, but they are relatively rare. We have not encountered a cubic $X$ such that for every $\mathbb{F}_{2}$-line $l\subset X$, $\Delta_{\textup{sing}}$ has no $\mathbb{F}_{2}$-point. We wonder whether any such cubic exists. Projecting from $y$, the quintic $\Delta$ becomes a 3-to-1 cover of $\mathbb{P}^{2}$, so we can iterate over $\mathbb{P}^{2}$ and find the three (or fewer) sheets of the cover at each point with a suitable version of Cardano’s formula [8, Exercise 14.7.15].

With this improvement the algorithm runs up to $q=2^{11}$ in less than a second, and up to $q=2^{14}$ in a little more than a minute. In §5 we make some practical comments about our implementation of the algorithm, and sanity checks on the output.

4. The explicit cubics

4.1. Proof of Theorem 1

Let us begin by discussing the map $m^{\vee}$ from the introduction in very concrete terms, embracing the monomial basis for the polynomial ring rather than working invariantly, and staying in characteristic 0 as long as possible to avoid discussing divided powers.

Let $R=\mathbb{C}[x_{0},\dotsc,x_{n}]$, and let $R_{d}\subset R$ be the subspace of homogeneous polynomials of degree $d$. We identify $R_{1}$ with its dual via the pairing

[TABLE]

and extend this to a pairing

[TABLE]

for positive integers $k\leq d$, again by differentiation. If $k=d$ this is a perfect, symmetric pairing. We have, for example,

[TABLE]

but

[TABLE]

so the monomials form an orthogonal basis for $R_{d}$ but not an orthonormal basis. For $k>d$ we set $\langle R_{k},R_{d}\rangle=0$.

Now with a view toward Theorem 1, let $R=\mathbb{C}[x_{0},x_{1},x_{2}]$ and $S=\mathbb{C}[y_{0},\dotsc,y_{5}]$. The isomorphism

[TABLE]

given by

[TABLE]

induces a map

[TABLE]

for all $d$.

Let $g\in R_{6}$ be given by

[TABLE]

and let $f\in S_{3}$ be given by

[TABLE]

We claim that $f=m^{\vee}(g)$, i.e. that

[TABLE]

for all $h\in S_{3}$. This can be checked tediously by hand, or with the Macaulay2 code given in the ancillary file thm1.m2.

Let $X\subset\mathbb{P}^{5}$ be the hypersurface cut out by $f$. After substituting

[TABLE]

we obtain a model of $X$ with good reduction modulo 2. Its reduction contains the line

[TABLE]

The point counts of $X$ over $\mathbb{F}_{2^{m}}$ are given in Table 1. Thus the characteristic polynomial of $\Phi^{*}$ acting on $H^{4}_{\textup{\'{e}t},\textup{prim}}(X_{\bar{k}},\mathbb{Q}_{\ell}(2))$ is

[TABLE]

which is irreducible over $\mathbb{Q}$. By our discussion in §2, this proves Theorem 1.

4.2. Proof of Theorem 2

Continue to let $S=\mathbb{C}[y_{0},\dotsc,y_{5}]$. A homogeneous polynomial $f\in S$ is said to be apolar to a homogeneous ideal $I\subset S$ if

[TABLE]

It is enough to check this on a set of generators for $I$.

Ranestad and Voisin observe [20, Lem. 1.7] that a cubic is in the image of $m^{\vee}\colon R_{6}\to S_{3}$ if and only if it is apolar to ideal generated by the $2\times 2$ minors of

[TABLE]

which cuts out a Veronese surface. This is checked for the previous section’s cubic in thm1.m2.

For Theorem 2, we take

[TABLE]

This is apolar to the ideal generated by the $2\times 2$ minors of the matrix

[TABLE]

which cuts out a quartic scroll. Apolarity can be checked by hand or with thm2.m2.555Ranestad and Voisin gave a different definition of $D_{\textit{IR}}$ and proved that a cubic of Waring rank 10 (the maximum possible) is in $D_{\textit{IR}}$ if and only if it is apolar to a quartic scroll [20, Lem. 2.4]. Our cubic does have rank 10, as can be checked using [20, Lem. 3.18]. But in fact the rank condition can be ignored: the cubic forms that are apolar to a given quartic scroll form a linear space, in which the general one has rank 10, so $D_{\textit{IR}}$ consists of all cubics apolar to a quartic scroll, with no restriction on rank. We thank K. Ranestad for explaining this to us.

Let $X\subset\mathbb{P}^{5}$ be the hypersurface cut out by $f$. After substituting $y_{1}\mapsto\frac{1}{2}y_{1}$ we obtain a model of $X$ with good reduction modulo 2. It contains the line

[TABLE]

The point counts of $X$ over $\mathbb{F}_{2^{m}}$ are given in Table 1. Thus the characteristic polynomial of $\Phi^{*}$ acting on $H^{4}_{\textup{\'{e}t},\textup{prim}}(X_{\bar{k}},\mathbb{Q}_{\ell}(2))$ is

[TABLE]

which is irreducible over $\mathbb{Q}$. By our discussion in §2, this proves Theorem 2.

4.3. Proof of Theorem 3

The cubic fourfold $X$ cut out by

[TABLE]

has good reduction modulo 2. The polynomial $f$ is apolar to the ideal

[TABLE]

as can be checked by hand or with thm3.m2. We do not review the definition of $D_{\textit{rk3}}$, but only refer to the proof of [20, Lem. 2.1] for the fact that this implies $X\in D_{\textit{rk3}}$.

The reduction of $X$ contains the line

[TABLE]

The point counts of $X$ over $\mathbb{F}_{2^{m}}$ are given in Table 1. Thus the characteristic polynomial of $\Phi^{*}$ acting on $H^{4}_{\textup{\'{e}t},\textup{prim}}(X_{\bar{k}},\mathbb{Q}_{\ell}(2))$ is

[TABLE]

which is irreducible over $\mathbb{Q}$. By our discussion in §2, this proves Theorem 3.

5. Verification and implementation

Our implementation of the algorithm described in §3 is included as an ancillary file count.cpp. We double-checked its output very thoroughly:

• •

For small $m$, we checked the counts over $\mathbb{F}_{2^{m}}$ using the naive algorithm discussed at the beginning of §3.

• •

We checked the counts up to about $m=9$ with a “semi-sophisticated” algorithm that projects from a point rather than a line.

• •

We projected from several different lines and got the same counts.

• •

After finding the characteristic polynomial one can predict the counts for all $m$. We checked these up to $m=14$, and even $m=15$ on a computer with much more memory than the first author’s laptop.

• •

The characteristic polynomial of $\Phi^{*}$ acting on

[TABLE]

with no Tate twist, is $4^{22}\chi(t/4)$, and this must have integer coefficients. But even stronger, we have

[TABLE]

where $F$ is the Fano variety of lines on $X$, so $2^{22}\chi(t/2)$ must have integer coefficients. We verified this.

• •

We used our program to count points on Elsenhans and Jahnel’s cubic [10, Example 3.15], and our numbers agreed with theirs.

We conclude with a few practical comments about our implementation:

• •

We represented elements of $\mathbb{F}_{2^{m}}$ as unsigned integers, interpreting the bits as coefficients of a polynomial in $\mathbb{F}_{2}[x]$ modulo a fixed irreducible polynomial of degree $m$. Thus addition is given by “xor” and multiplication by a well-known algorithm.

• •

We stored multiplication in a lookup table, which sped up the program by an order of magnitude.

• •

We also stored division in a lookup table, as well as roots of quadratic and depressed cubic polynomials, which saved us the trouble of writing those algorithms. This did not start to use an unreasonable amount of memory until $m=14$.

• •

Following [9, Alg. 15] and [16, §8], we pre-computed a list of Galois orbit representatives (and orbit sizes) in $\mathbb{F}_{2^{m}}$, and then touched each Galois orbit of $\mathbb{P}^{2}$ only once, which sped up the program by a factor of $m$.

• •

We did not bother with parallelization, although this problem is ideally suited to it.