aa r X i v : . [ m a t h . AG ] F e b SOME NON-SPECIAL CUBIC FOURFOLDS
NICOLAS ADDINGTON AND ASHER AUEL
Abstract.
In [20], Ranestad and Voisin showed, quite surprisingly,that the divisor in the moduli space of cubic fourfolds consisting ofcubics “apolar to a Veronese surface” is not a Noether–Lefschetz divi-sor. We give an independent proof of this by exhibiting an explicit cubicfourfold X in the divisor and using point counting methods over finitefields to show X is Noether–Lefschetz general. We also show that twoother divisors considered in [20] are not Noether–Lefschetz divisors. Introduction
In [20], Ranestad and Voisin introduced some new divisors in the mod-uli space of smooth complex cubic fourfolds, quite different from Hassett’sNoether–Lefschetz divisors [14]. A cubic X ⊂ P is called special if H , ( X, Z ) := H ( X, Z ) ∩ H , ( X )is non-zero, or equivalently if X contains a surface not homologous to acomplete intersection. The locus of special cubic fourfolds is a countableunion of irreducible divisors in the moduli space, called Noether–Lefschetzdivisors. Special cubic fourfolds often have rich connections to K3 surfaces,and it is expected that all rational cubic fourfolds are special; see [15] for arecent survey of the topic.Ranestad and Voisin’s divisors are constructed in a much more algebraicway, using apolarity. Briefly, a cubic fourfold X cut out by a polynomial f ( y , . . . , y ) is said to be apolar to an ideal generated by quadrics, I = h q , . . . , q m i ⊂ C [ y , . . . , y ] , if, writing q i = P a ijk y j y k , we have X a ijk ∂ j ∂ k f = 0 for all i. Ranestad and Voisin showed that the following loci are irreducible divisorsin the moduli space of cubic fourfolds: D V-ap , the set of cubics apolar toa Veronese surface; D IR , the set of cubics apolar to a quartic scroll; and D rk3 , the closure of the set of cubics apolar to the union of a plane and adisjoint hyperplane. They showed that D V-ap is not a Noether–Lefschetzdivisor, by carefully analyzing its singularities. From this they deduced thatfor a generic cubic X , the “varieties of sums of powers” of the polynomial f , which is a hyperk¨ahler fourfold, is not Hodge-theoretically related tothe Fano variety of lines on X , a better-known hyperk¨ahler fourfold. They remarked that D rk3 is “presumably” not a Noether–Lefschetz divisor, andthat if one could prove that D IR is not a Noether–Lefschetz divisor then itwould give another approach to proving their main theorem.We were very surprised to learn that D V-ap is not a Noether–Lefschetzdivisor: we would have guessed that it was Hassett’s divisor C , for thefollowing reason. Cubic fourfolds in C , which are conjectured to be ratio-nal, have associated K3 surfaces of degree 38. Mukai [17] observed that thegeneric such K3 surface S can be described as the variety of sums of pow-ers of a plane sextic g ( x , x , x ); see [19, Thm. 1.7(iii)] for a more detailedaccount. A natural way to construct a cubic fourfold from g is to considerthe multiplication map m : Sym Sym C → Sym C and its transpose m ∨ : Sym C ∨ → Sym Sym C ∨ . Then m ∨ ( g ) cuts out a cubic X ⊂ P (Sym C ∨ ) = P , typically smooth.By [20, Lem. 1.7], the cubics obtained this way are exactly those in D 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 resultimplies that it cannot be.Since the result is so surprising, and the proof quite difficult, at least toour eyes, we thought it worthwhile to seek experimental confirmation. Inthis note, we give a computer-aided proof of the following result, and inparticular a more direct proof of Ranestad and Voisin’s result: Theorem 1.
There is an explicit sextic polynomial g , defined over Q ,such that the cubic fourfold X cut out by m ∨ ( g ) is smooth and satisfies H , ( X, Z ) = 0 . In particular, X ∈ D V-ap , but X is not in any Noether–Lefschetz divisor. We also confirm Ranestad and Voisin’s expectations for the other twodivisors mentioned above: Theorem 2.
There is an explicit cubic fourfold X ∈ D IR , defined over Q ,with H , ( X, Z ) = 0 . In particular, D IR is not a Noether–Lefschetz divisor. Theorem 3.
There is an explicit cubic fourfold X ∈ D rk3 , defined over Q ,with H , ( X, Z ) = 0 . In particular, D rk3 is not a Noether–Lefschetz divisor. Thus it seems that apolarity tends to produce cubic fourfolds of a differentcharacter than those considered by Hassett. It would be very interesting toknow if there is any connection with rationality. They also studied a fourth divisor D copl , not defined using apolarity, but we were un-able to find a suitable cubic in that divisor using the technique described below. Probablyone could be found by working modulo 5, but that would forfeit many of the computationaladvantages of working modulo 2. OME NON-SPECIAL CUBIC FOURFOLDS 3
We follow a strategy developed by van Luijk [23] and refined by Elsen-hans and Jahnel [9, 10], for producing explicit K3 surfaces of Picard rank 1.We find an explicit cubic fourfold with good reduction modulo 2, thencount points over F m for m = 1 , , . . . ,
11 to determine the eigenvaluesof Frobenius acting on H ( X F , Q ℓ (2)), which give a bound on the rankof H , ( X, Z ). In §
2, we give the details of adapting van Luijk’s method tocubic fourfolds.On the one hand, our task is simpler than van Luijk’s: since the geometricPicard rank of a K3 surface over a finite field is necessarily even, to show thata K3 surface has Picard rank 1, van Luijk had to work modulo two differentprimes and compare intersection forms; but here we need only work moduloone prime. On the other hand, a fourfold is much bigger than a surface,and it is infeasible to count points naively by iterating over P . Nor can wecontrol the cohomology of X by counting points on an associated K3 surfaceas in [2] or [16], since there is none. In § X along a line, so that to countpoints we only need to iterate over P , and with a little more work, onlyover P . The same idea was used to count points on cubic three folds byDebarre, Laface, and Roulleau [7, § F m is so fast.We do not use the p -adic cohomology methods of Kedlaya, Harvey, andothers [1, 13, 6]. While these methods are surely the way of the future,they are much harder to implement than our algorithm, and the availableimplementations are not quite ready to handle cubic fourfolds.In §
4, we give the explicit polynomials and the point counts needed toprove Theorems 1, 2, and 3. In §
5, we conclude with some remarks aboutcomputer implementation and verification.The existence of Noether–Lefschetz general cubic fourfolds (and othercomplete intersections) defined over Q was first proved by Terasoma [21],although his proof is not constructive. Elsenhans and Jahnel gave an explicitexample in [10, Example 3.15], also using point-counting methods. Butthe existence of Noether–Lefschetz general cubic fourfolds with specifiedalgebraic properties is far from clear a priori . Acknowledgements.
We thank J.-L. Colliot-Th´el`ene, E. Costa, D. Har-vey, B. Hassett, A. Kuznetsov, K. Ranestad, R.P. Thomas, A. V´arilly-Alvarado, and B. Viray for helpful conversations, and the organizers of theFall 2016 AGNES workshop at UMass Amherst for their hospitality. Inthe course of this project we used the computer algebra system Macaulay2[12] extensively, and Magma [4] to a lesser extent. The second author waspartially supported by NSA Young Investigator Grant H98230-16-1-0321.
N. ADDINGTON AND A. AUEL Adaptation of van Luijk’s method
In this section we adapt the method developed in [23] from K3 surfacesto cubic fourfolds. We begin with the following proposition, which is similarto [22, Cor. 6.3]. Note that due to our choice of Tate twist, our Frobeniuseigenvalues have absolute value 1 rather than q i . Proposition 2.1.
Let R be a discrete valuation ring of a number field L withresidue field k ∼ = F q for q = p r , and let X be a smooth projective schemeover R . Let X an denote the complex manifold associated to the complexvariety X C . Let Φ : X k → X k be the r -th power absolute Frobenius, let ℓ bea prime different from p , and let Φ ∗ be the automorphism of H i ´et ( X ¯ k , Q ℓ ( i )) induced by Φ × on X k × ¯ k .Then the rank of the image of the cycle class map CH i ( X C ) cl −−→ H i ( X an , Z ( i )) (1) is less than or equal to the number of eigenvalues of Φ ∗ , counted with mul-tiplicity, that are roots of unity.In particular, if the Hodge conjecture holds for codimension- i cycles on X , then the rank of H i ( X an , Z ) ∩ H i,i ( X an ) is bounded above by the numberof such eigenvalues.Proof. The rank of the image of (1) agrees with the rank of the image ofCH i ( X C ) cl −−→ H i ( X an , Z ℓ ( i )) . By the comparison theorem between singular and ℓ -adic cohomology, thisagrees with the rank of the image ofCH i ( X C ) cl −−→ H i ´et ( X C , Z ℓ ( i )) . Now let K be the field of fractions of the completion b R , and consider thecommutative diagram CH i ( X C ) cl / / H i ´et ( X C , Z ℓ ( i ))CH i ( X ¯ L ) O O (cid:15) (cid:15) cl / / H i ´et ( X ¯ L , Z ℓ ( i )) ∼ = O O ∼ = (cid:15) (cid:15) CH i ( X ¯ K ) cl / / H i ´et ( X ¯ K , Z ℓ ( i )) . The right-hand vertical maps are isomorphisms by smooth base change,and while the left-hand vertical maps are typically not isomorphisms, the Alternatively we could have embedded ¯
K ֒ → C , but we preferred to use the morenatural embeddings C ← ֓ ¯ L ֒ → ¯ K . OME NON-SPECIAL CUBIC FOURFOLDS 5 images of the three horizontal maps agree thanks to the existence of Hilbertschemes, as remarked in [5, Rem. 46].Next we have a commutative squareCH i ( X ¯ K ) σ (cid:15) (cid:15) cl / / H i ´et ( X ¯ K , Z ℓ ( i )) ∼ = (cid:15) (cid:15) CH i ( X ¯ k ) cl / / H i ´et ( X ¯ k , Z ℓ ( i )) , 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. Thusthe rank of the image of the top horizontal map is less than or equal to thatof the bottom one.Finally we consider the cycle class map after tensoring with Q ℓ CH i ( X ¯ k ) ⊗ Q ℓ cl −−→ H i ( X ¯ k , Q ℓ ( i ))and recall that cycles on X ¯ k are defined over some finite extension of k , henceare fixed by some power of Frobenius, hence their classes in cohomologyare eigenvectors with eigenvalues a root of unity as in the proof of [22,Cor. 6.3]. (cid:3) In our application, we will take R = Z (2) , so L = Q , q = p = 2, and K = Q .Now specialize to the case where X is a cubic fourfold. The Hodge conjec-ture holds for cubic fourfolds [26, 18, 25], so to show that H , ( X, Z ) = 0it is enough to show that no eigenvalue of Φ ∗ acting on V := H , prim ( X ¯ k , Q ℓ (2)) ∼ = Q ℓ is a root of unity, or equivalently that the characteristic polynomial χ ( t ) := det( t · Id V − Φ ∗ | V )has no cyclotomic factor. For this it is enough to show that χ is irreducibleover Q and that not all its coefficients are integers.The cohomology of X is H i ´et ( X ¯ k , Q ℓ ( i )) = Q ℓ i = 0 , Q ℓ · h i = 2 Q ℓ · h ⊕ V i = 4 Q ℓ · h i = 6 Q ℓ · h i = 80 otherwise,where h is the hyperplane class, so by the Lefschetz trace formula we have X ( F q m ) = 1 + q m + q m (cid:16) ∗ m | V ) (cid:17) + q m + q m . (2) N. ADDINGTON AND A. AUEL
The method of passing from traces of powers of Φ ∗ | V to the characteristicpolynomial using Newton’s identities is discussed in [23, § § § χ ( t ) = ± t χ ( t − ) it is usuallyenough 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 , , . . . ,
11? As we said in the introduction, it is not feasibleto iterate over P ( F m ), evaluating our cubic polynomial at every point: inMagma this would take many years, and in a program written optimizedspecially for the purpose it would take months, or at best weeks. Insteadwe project from a line to obtain a conic fibration.Continue to work with a smooth cubic X defined over an arbitrary F q .Choose a line l ⊂ X defined over F q ; by [7] such a line always exists for q = 2 or q ≥
5, and probably for q = 3 or 4 as well. Change variables sothat l is given by y = y = y = y = 0. Then we can write the equation of X as Ay + By y + Cy + Dy + Ey + F, where A , B , and C are linear in y , . . . , y , C and D are quadratic, and F is cubic. If A, . . . , F vanish simultaneously at some point of P then X contains a plane, contributing an unwanted Frobenius eigenvalue, so westop. Otherwise we obtain a flat conic bundleBl l ( X ) −→ P y : ... : y ) with fibers given by the homogenization of the quadratic form above. Nowwe use the following. Proposition 3.1.
Let Z be an F q -scheme of finite type, let π : Y → Z bea flat conic bundle, let ∆ ⊂ Z be the locus parametrizing degenerate conics,and let ˜∆ be the (possibly branched) double cover of ∆ parametrizing linesin the fibers of π . Then Y ( F q ) = ( q + 1) · Z + q · ( − . (3) Proof.
A smooth conic over F q is isomorphic to P , hence has q + 1 points.For a singular conic, there are three possibilities: • a pair of lines defined over F q , contributing 2 q + 1 points; • a pair of conjugate lines defined over F q , contributing only one F q -point; • a double line, contributing q + 1 points.The fiber of ˜∆ over the relevant point of ∆ consists of 2, 0, or 1 pointsrespectively. Thus we have Y ( F q ) = ( q + 1) · ( Z − | {z } from smooth conics + ( q · | {z } from singular conics , which simplifies to give (3). (cid:3) OME NON-SPECIAL CUBIC FOURFOLDS 7
In our case, with Y = Bl l ( X ) and Z = P , this yields X ( F q ) = q + q + q ( − q + 1 . The discriminant locus ∆ ⊂ P is cut out by the quintic polynomial AE + B F + CD − BDE − ACF. (4)This formula remains valid in characteristic 2, although of course the lastterm vanishes. The double cover ˜∆ can also be described as the variety oflines on X that meet l . So we can iterate over P and count points on ∆ and ˜∆. To count pointson ˜∆ in characteristic 2, we note that if B = D = E = 0 then the conicis a double line; otherwise we compute an Arf invariant: if B = 0 (resp. D = 0 or E = 0), then the conic has 2 q + 1 points if AC/B (resp. AF/D or CF/E ) is of the form a + a for some a ∈ F q , and 1 point if it is not.This algorithm runs up to q = 2 in about half a minute on the first au-thor’s laptop. But to find the explicit cubics below we had to search throughdozens of candidates, so it was worthwhile to make a further optimization,iterating only over ∆ rather than all of P , as follows.The quintic ∆ is not smooth; in characteristic 2, it is singular at leastalong the locus where B = D = E = 0 , which has expected dimension 0 and degree 4. Suppose this locus contains an F -point y . Projecting from y , the quintic ∆ becomes a 3-to-1 cover of P ,so we can iterate over P and find the three (or fewer) sheets of the cover ateach point with a suitable version of Cardano’s formula [8, Exercise 14.7.15].With this improvement the algorithm runs up to q = 2 in less than asecond, and up to q = 2 in a little more than a minute. In § The explicit cubics
Proof of Theorem 1.
Let us begin by discussing the map m ∨ from the introduction in veryconcrete terms, embracing the monomial basis for the polynomial ring ratherthan working invariantly, and staying in characteristic 0 as long as possibleto avoid discussing divided powers. The topology of ˜∆ over C has been studied in [24, §
3, Lemmas 1–3]. For a generic l ⊂ X , it is a smooth surface with Hodge diamond
10 05 50 5.0 01 In practice this usually happens, although not always. That is, there exist smoothcubics X and F -lines l ⊂ X such that ∆ sing has no F -point, but they are relativelyrare. We have not encountered a cubic X such that for every F -line l ⊂ X , ∆ sing has no F -point. We wonder whether any such cubic exists. N. ADDINGTON AND A. AUEL
Let R = C [ x , . . . , x n ], and let R d ⊂ R be the subspace of homogeneouspolynomials of degree d . We identify R with its dual via the pairing h x i , x j i = ∂∂x i x j = δ ij , and extend this to a pairing R k ⊗ R d → R d − k for positive integers k ≤ d , again by differentiation. If k = d this is a perfect,symmetric pairing. We have, for example, h x x , x x i = ∂∂x ∂∂x x x = 1 , but h x , x i = ∂∂x ∂∂x x = 2 , so the monomials form an orthogonal basis for R d but not an orthonormalbasis. For k > d we set h R k , R d i = 0.Now with a view toward Theorem 1, let R = C [ x , x , x ] and S = C [ y , . . . , y ]. The isomorphism m : S → R given by y x y x x y x x y x y x x y x . induces a map m : S d → R d for all d .Let g ∈ R be given by g = 130 x x + 16 x x + 16 x x + 130 x x + 1120 x + 43 x x x + 23 x x x + 16 x x + 2 x x x + 13 x x x + 112 x x + 23 x x x + 16 x x + 13 x x + 115 x x , and let f ∈ S be given by f = 2 y y + 4 y y + 8 y y + 4 y y + 4 y y + 4 y y + 16 y y y + 8 y y y + 8 y y + 4 y y + 2 y y + y + 16 y y y + 4 y y + 16 y y y + 4 y y + 8 y y y + 4 y y y + 8 y y + 2 y y + 2 y y + y + 4 y y + 8 y y + 8 y y y + 8 y y + 16 y y y + 4 y y y + 2 y y + 8 y y y + 6 y y y + 8 y y + 4 y y . OME NON-SPECIAL CUBIC FOURFOLDS 9
We claim that f = m ∨ ( g ), i.e. that h h, f i = h m ( h ) , g i for all h ∈ S . This can be checked tediously by hand, or with the Macaulay2code given in the ancillary file thm1.m2 .Let X ⊂ P be the hypersurface cut out by f . After substituting y y , y y , y y , we obtain a model of X with good reduction modulo 2. Its reduction con-tains the line y + y = y = y + y = y = 0 . The point counts of X over F m are given in Table 1. Thus the characteristicpolynomial of Φ ∗ acting on H , prim ( X ¯ k , Q ℓ (2)) is χ ( t ) = t − t + 32 t − t + 12 t + 12 t − t + 32 t − t + 12 t + 12 t − t + 32 t − t + 1 , which is irreducible over Q . By our discussion in §
2, this proves Theorem 1.4.2.
Proof of Theorem 2.
Continue to let S = C [ y , . . . , y ]. A homogeneous polynomial f ∈ S issaid to be apolar to a homogeneous ideal I ⊂ S if h i, f i = 0 for all i ∈ I. 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 imageof m ∨ : R → S if and only if it is apolar to ideal generated by the 2 × y y y y y y y y y , which cuts out a Veronese surface. This is checked for the previous section’scubic in thm1.m2 .For Theorem 2, we take f = y + 2 y y + y + y y + 2 y y y + 8 y y y + y y + 4 y y + 8 y y y + y y + 4 y y y + y y + 2 y y + y y + y + 8 y y y + 2 y y y + 4 y y y + 2 y y y + 4 y y y + 2 y y y + 6 y y y + y y + y y + y y + y y + y y . This is apolar to the ideal generated by the 2 × (cid:18) y y y y y y y y (cid:19) , which cuts out a quartic scroll. Apolarity can be checked by hand or with thm2.m2 . Let X ⊂ P be the hypersurface cut out by f . After substituting y y we obtain a model of X with good reduction modulo 2. It contains the line y = y = y = y = 0 . The point counts of X over F m are given in Table 1. Thus the characteristicpolynomial of Φ ∗ acting on H , prim ( X ¯ k , Q ℓ (2)) is χ ( t ) = t + t + 12 t + 12 t + 12 t − t − t − t − t − t − t − t + 12 t + 12 t + 12 t + t + 1 , which is irreducible over Q . By our discussion in §
2, this proves Theorem 2.4.3.
Proof of Theorem 3.
The cubic fourfold X cut out by f = y y + y y + y y y + y y + y + y y + y y y + y y + y y + y y y + y y y + y y y + y y + y y y + y y y + y y y + y y + y y + y + y y + y y y + y y + y y + y has good reduction modulo 2. The polynomial f is apolar to the ideal h y y , y y , y y i , as can be checked by hand or with thm3.m2 . We do not review the definitionof D rk3 , but only refer to the proof of [20, Lem. 2.1] for the fact that thisimplies X ∈ D rk3 .The reduction of X contains the line y = y + y = y = y + y = 0 . The point counts of X over F m are given in Table 1. Thus the characteristicpolynomial of Φ ∗ acting on H , prim ( X ¯ k , Q ℓ (2)) is χ ( t ) = t − t + 32 t − t − t + 12 t − t + 12 t + 12 t + 12 t + 12 t + 12 t − t + 12 t − t − t + 32 t − t + 1 . which is irreducible over Q . By our discussion in §
2, this proves Theorem 3. Ranestad and Voisin gave a different definition of D IR and proved that a cubic ofWaring rank 10 (the maximum possible) is in D IR if and only if it is apolar to a quarticscroll [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 givenquartic scroll form a linear space, in which the general one has rank 10, so D IR consists ofall cubics apolar to a quartic scroll, with no restriction on rank. We thank K. Ranestadfor explaining this to us. OME NON-SPECIAL CUBIC FOURFOLDS 11 m X ( F m )Theorem 1 Theorem 2 Theorem 31 31 31 332 389 309 2973 4 681 4 585 4 6414 69 521 69 905 70 9455 1 082 401 1 082 401 1 084 0336 17 040 449 17 050 689 17 057 4097 270 491 777 270 577 793 270 525 9538 4 311 818 497 4 312 006 913 4 311 720 4499 68 854 546 945 68 854 448 641 68 853 843 96910 1 100 584 649 729 1 100 596 118 529 1 100 585 936 89711 17 600 762 873 857 17 600 774 408 193 17 600 759 586 817 Table 1.
Point counts.5.
Verification and implementation
Our implementation of the algorithm described in § count.cpp . We double-checked its output very thoroughly: • For small m , we checked the counts over F m using the naive algorithmdiscussed at the beginning of § • 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 forall m . We checked these up to m = 14, and even m = 15 on a computerwith much more memory than the first author’s laptop. • The characteristic polynomial of Φ ∗ acting on H , prim ( X ¯ F , Q ℓ ) , with no Tate twist, is 4 χ ( t/ H , prim ( X ¯ F , Q ℓ (1)) ∼ = H , prim ( F ¯ F , Q ℓ ) , where F is the Fano variety of lines on X , so 2 χ ( t/
2) must have integercoefficients. 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 F m as unsigned integers, interpreting thebits as coefficients of a polynomial in F [ x ] modulo a fixed irreducible polynomial of degree m . Thus addition is given by “xor” and multipli-cation by a well-known algorithm. • We stored multiplication in a lookup table, which sped up the programby an order of magnitude. • We also stored division in a lookup table, as well as roots of quadraticand depressed cubic polynomials, which saved us the trouble of writingthose algorithms. This did not start to use an unreasonable amount ofmemory until m = 14. • Following [9, Alg. 15] and [16, § F m , and then touched each Galoisorbit of P only once, which sped up the program by a factor of m . • We did not bother with parallelization, although this problem is ideallysuited to it.
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Nicolas Addington, Department of Mathematics, University of Oregon,Eugene, Oregon 97403, United States
E-mail address : [email protected] Asher Auel, Department of Mathematics, Yale University, New Haven,Connecticut 06511, United States
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