Separation of periods of quartic surfaces
SSEPARATION OF PERIODS OF QUARTIC SURFACES
PIERRE LAIREZ AND EMRE CAN SERTÖZ
Abstract.
We give an effective lower bound on the distance betweentwo distinct periods of a given quartic surface defined over the algebraicnumbers. The main ingredient is the determination of height bounds oncomponents of the Noether–Lefschetz loci. This makes it possible to studythe Diophantine properties of periods of quartic surfaces and to certify apart of the numerical computation of their Picard groups. Introduction
Periods are a countable set of complex numbers containing all the algebraicnumbers as well as many of the transcendental constants of nature. In light ofthe ubiquity of periods in mathematics and the sciences, Kontsevich and Zagier(2001) ask for the development of an algorithm to check for the equality of twogiven periods. We solve this problem for periods coming from quartic surfaces bygiving a computable separation bound, that is, a lower bound on the minimumdistance between distinct periods.Let f P C r w, x, y, z s be a homogeneous quartic polynomial defining a smoothquartic X f in P p C q . The periods of X f are the integrals of a nowhere vanishingholomorphic -form on X f over integral -cycles in X f . The periods can also begiven in the form of integrals of a rational function(1) πi ¿ γ d x d y d zf p , x, y, z q , where γ is a -cycle in C z X f . The integral (1) depends only on the homologyclass of γ . These periods form a group under addition. The geometry of quarticsurfaces dictates that there are only independent -cycles in C z X f . Thesegive periods α , . . . , α P C such that the integral over any other -cycle isan integer linear combination of these periods.It is possible to compute the periods to high precision (Sertöz 2019), typicallyto thousands of decimal digits, and to deduce from them interesting algebraicinvariants such as the Picard group of X f (Lairez and Sertöz 2019). This pointof view has been fruitful for computing algebraic invariants for algebraic curvesfrom their periods (Booker et al. 2016; Bruin et al. 2019; Costa et al. 2019;van Wamelen 1999). Date : December 9, 2020. a r X i v : . [ m a t h . AG ] D ec P. LAIREZ AND E. C. SERTÖZ
For quartic surfaces, the computation of the Picard group reduces to computingthe lattice in Z of integer relations x α ` ¨ ¨ ¨ ` x α “ , x i P Z . A basisfor this lattice can be guessed from approximate α i ’s using lattice reductionalgorithms. But is it possible to prove that all guessed relations are true relations?Previous work related to this question (Simpson 2008) require explicit constructionof algebraic curves on X f , which becomes challenging very quickly. Instead, wegive a method of proving relations by checking them at a predetermined finiteprecision. At the moment, this is equally challenging, but we conjecture that thenumerical approach can be made asymptotically faster, see §4.5 for details.The Lefschetz theorem on p , q -classes (§2.2) associates a divisor on X f toany integer relation between the periods of X f . In turn, the presence of a divisorimposes algebraic conditions on the coefficients of f . Such algebraic conditionsdefine the Noether–Lefschetz loci on the space of quartic polynomials (§3). Inaddition to the degree computations of Maulik and Pandharipande (2013), wegive height bounds on the polynomial equations defining the Noether–Lefschetzloci (Theorem 14). These lead to our main result (Theorem 17): Assume f hasinteger coefficients, then for x i P Z ,(2) x α ` ¨ ¨ ¨ ` x α “ or | x α ` ¨ ¨ ¨ ` x α | ą ´ c max i | xi | for some constant c ą depending only on f . The constant c is computable inrather simple terms and without prior knowledge of the Picard group of X f . Theresult generalizes to f with algebraic coefficients (Theorem 19).As a consequence of this separation bound, we apply a construction in themanner of Liouville (1851) and prove, for instance, that the number(3) ÿ n ě p Ò n q ´ is not a quotient of two periods of a single quartic surface defined over Q ,where Ò n denotes an exponentiation tower with n twos (Theorem 20,with θ i ` “ θi ). Acknowledgements.
We thank Bjorn Poonen for suggesting the use of heightsof Noether–Lefschetz loci and Gavril Farkas for suggesting the paper by Maulikand Pandharipande. We also thank Alin Bostan and Matthias Schütt for numer-ous helpful comments.ECS was supported by Max Planck Institute for Mathematics in the Sciences,Leibniz University Hannover and Max Planck Institute for Mathematics. PL wassupported by the project
De Rerum Natura
ANR-19-CE40-0018 of the FrenchNational Research Agency (ANR).2.
Periods and deformations
Construction of the period map.
For any non-zero homogeneous poly-nomial f in C r w, x, y, z s , let X f denote the surface in P defined as the zero locus EPARATION OF PERIODS OF QUARTIC SURFACES 3 of f . Let R . “ C r w, x, y, z s and let R Ă R be the subspace of degree elements.Let U Ă R denote the dense open subset of all homogeneous polynomials f ofdegree such that X f is smooth. For our purposes, it will be useful to considernot only the periods of a single quartic surface X f but also the period map tostudy the dependence of periods on f .The topology of X f does not depend on f as long as X f is smooth: given twopolynomials f and g P U , we can connect them by a continuous path in U and thesurface X f deforms continuously along this path, giving a homeomorphism X f » X g , which is uniquely defined up to isotopy. In particular, if we fix a base point b P U , then for every f P r U , where r U is a universal covering of U , we have auniquely determined isomorphism of cohomology groups H p X b , Z q » H p X f , Z q .Let H Z denote the second cohomology group of X b , which is isomorphic to Z (e.g. Huybrechts 2016, §1.3.3).An element of r U can be viewed as a polynomial f P U together with anidentification of H p X f , Z q with H Z . We often work locally around a givenpolynomial f and, in that case, we do not actively distinguish between U andits universal covering.The group H Z is endowed with an even unimodular pairing(4) p x, y q P H Z ˆ H Z Ñ x ¨ y P Z , given by the intersection form on cohomology. Through this pairing, the secondhomology and cohomology groups are canonically identified with one another.For K3 surfaces, such as smooth quartic surfaces in P , the structure of thelattice H Z with its intersection form is explicitly known (ibid., Proposition .3.5).The fundamental class of a generic hyperplane section of X f gives an elementof H Z denoted by h .Furthermore, the complex cohomology group H p X f , C q , which is just H C . “ H Z b C , is isomorphic to the corresponding de Rham cohomology H dR p X f , C q group as follows. Elements of H dR p X f , C q are represented by differential -forms.To a form Ω one associates the element Θ p Ω q of H p X f , C q given by the map(5) Θ p Ω q : r γ s P H p X f , C q ÞÑ ż γ Ω P C . The group H dR p X f , C q has a distinguished element Ω f , a nowhere vanishingholomorphic -form, described below. Every other holomorphic -form on X f isa scalar multiple of Ω f (ibid., Example .1.3). Mapping Ω f to H C gives rise tothe period map (6) P : f P r U ÞÑ ω f . “ Θ p Ω f q P H C . The coordinates of the period vector ω f , in some fixed basis of H Z , generates thegroup of periods of X f .To make the connection clear, we first consider the tube map(7) T : H p X f , Z q Ñ H p P z X f , Z q , P. LAIREZ AND E. C. SERTÖZ constructed as follows (Griffiths 1969, §3). Let ε ą be small enough. For x P X ,the normal ε -circle over x is the set of all points y P P such that d p y, X q “ ε and x is the closest point to y in X (which is unique if ε is small enough), thatis d p x, y q “ ε . The union of all normal ε -circles over the points of an effective -cycle γ P H p X, Z q is a -cycle in P z X , denoted by T p γ q . The map T is asurjective morphism and its kernel is generated by the class of a hyperplanesection of X f . We choose Ω f so that the following identity holds(8) ż γ Ω f “ πi ż T p γ q d x d y d zf p , x, y, z q . Therefore, in view of (5), the coefficients of ω f in a basis of H Z coincides withperiods as defined in (1).The image D of the period map P is called the period domain . It admits asimple description:(9) D . “ P p r U q “ t w P H C z t u | w ¨ h “ , w ¨ w “ , w ¨ w ą u , where “ ¨ ” denotes the intersection form on H Z and h the fundamental class of ahyperplane section, as introduced above (Huybrechts 2016, Chapter 6). Moreover,by the local Torelli theorem for K3 surfaces (ibid., Proposition .2.8), the map P is a submersion; its derivative at any point of r U is surjective.2.2. The Lefschetz (1,1)-theorem.
The linear integer relations between theperiods of a quartic surface X f are in correspondence with formal linear com-binations of algebraic curves in X f . Let C Ă X f be an algebraic curve. Itsfundamental class is the element r C s of H Z obtained as the Poincaré dual of thehomology class of C . The Picard group Pic p X f q of X f is the sublattice of H Z spanned by the fundamental classes of algebraic curves.It follows from the definition that for any class r Ω s P H dR p X f q of a differential -form on X f ,(10) r C s ¨ Θ p Ω q “ ż C Ω . Moreover, if Ω is a holomorphic -form, then ş C Ω “ because the restriction of Ω to the complex -dimensional subvariety C vanishes. In particular r C s ¨ ω f “ .It turns out that this condition characterizes the elements of Pic p X f q .More precisely, let H , p X f q Ă H C denote the space orthogonal to ω f and ω f ,the conjugate of ω f , with respect to the intersection form. This space is a directsummand in the Hodge decomposition of H p X f , C q .The Lefschetz (1,1)-theorem (Griffiths and Harris 1978, p. 163) asserts thatthe lattice of integer relations coincide with the Picard group:(11) Pic p X f q “ H Z X H , p X f q . EPARATION OF PERIODS OF QUARTIC SURFACES 5
Noting that for any γ P H Z , γ “ γ , where γ denotes the complex conjugate, wehave ω f ¨ γ “ ω f ¨ γ , so that (11) becomes(12) Pic p X f q “ t γ P H Z | γ ¨ ω f “ u . A deformation argument.
Let γ , . . . , γ be a basis of H Z . The space H R (resp. H C ) is endowed with the coefficient wise Euclidean (resp. Hermitian)norm(13) ›››› ÿ i “ x i γ i ›››› . “ ÿ i “ | x i | . For γ P H Z , if | γ ¨ ω f | is small enough, then γ is close to being an integerrelation between the periods of X f . We want to argue that, in this case, γ is agenuine integer relation between the periods of X g for some polynomial g P U close to f .Recall f, g P r U means f and g are smooth quartics with second cohomologyidentified with H Z . The space r U inherits a metric from U so that r U Ñ U islocally isometric. The metric on U Ă R » C is induced by an inner product.The choice of an inner product will change the distances but this is absorbedinto the constants in the statements below.Let f P r U be fixed. For any g P R and t P C small enough, the polynomials f ` tg P R lift canonically to r U . For any γ P H C we consider the map(14) φ γ,g p t q . “ γ ¨ P p f ` tg q which is well-defined and analytic in a neighbourhood of in C . Lemma 1.
There is a constant C ą , depending only on f , such that forany γ P H C satisfying γ ¨ h “ and | γ ¨ ω f | } ω f } ď } γ }p ω f ¨ ω f q , there is amonomial m P R for which ˇˇ φ γ,m p q ˇˇ ě C } γ } .Proof. Observe that φ γ,m p q “ γ ¨ d f P p m q . It follows that any constant C satisfying the following inequality would work, provided the infimum is not zero,(15) C ă inf } γ }“ max m | γ ¨ d f P p m q| , with the infimum taken over γ satisfying h ¨ γ “ and | γ ¨ ω f | } ω f } ď p ω f ¨ ω f q .If the infimum is zero, it is realized by some γ of norm one that annihilates d f P p m q for each monomial m . It follows that γ is orthogonal (with respect tothe intersection product) to the tangent space T ω f D of D at ω f . By (9),(16) T ω f D “ t w P H C | w ¨ h “ w ¨ ω f “ u . It follows that γ “ ah ` bω f for some a, b P C . The condition γ ¨ h “ implies a “ (note that ω f ¨ h “ because ω f P D ). Since } γ } “ we have | γ ¨ ω f | “ } ω f } ´ p ω f ¨ ω f q which is a contradiction. (cid:3) The next statement is proved using the following result of Smale (1986). Let φ be an analytic function on a maximal open disc around in C with φ p q ‰ . P. LAIREZ AND E. C. SERTÖZ
We define(17) γ Smale p φ q . “ sup k ě ˇˇˇˇ k ! φ p k q p q φ p q ˇˇˇˇ k ´ and β Smale p φ q . “ ˇˇˇˇ φ p q φ p q ˇˇˇˇ . If β Smale p φ q γ Smale p φ q ď , then there is a t P C such that | t | ď β Smale p φ q and φ p t q “ (Smale 1986; see also Blum et al. 1998, Chapter 8, Theorem 2). Proposition 2.
For any f P r U , there exists C f and ε f ą such that forall ε ă ε f the following holds. For any γ P H R , if γ ¨ h “ and | γ ¨ ω f | ď ε } γ } then there is a monomial m P R and t P C such that | t | ď C f ε and γ ¨ ω f ` tm “ .Proof. Let γ P H R such that γ ¨ h “ and(18) | γ ¨ ω f | ď ˆ ω f ¨ ω f } ω f } ˙ } γ } . Since γ has real coefficients, we have | γ ¨ ω f | “ | γ ¨ ω f | and we may apply Lemma 1to obtain a monomial m and a constant C such that(19) ˇˇ φ γ,m p q ˇˇ ě C } γ } . It follows in particular that(20) β Smale p φ γ,m q ď | γ ¨ ω f | C } γ } . Moreover, for any k ě , and using C ď , ˇˇˇˇˇ k ! φ p k q γ,m p q φ γ,m p q ˇˇˇˇˇ k ´ ď C ´ ˇˇˇˇˇ φ p k q γ,m p q} γ } ˇˇˇˇˇ k ´ “ C ´ ˇˇˇˇ γ } γ } ¨ d kf P p m, . . . , m q ˇˇˇˇ k ´ (21) ď C ´ (cid:23)(cid:23) k ! d kf P (cid:23)(cid:23) k ´ , (22)where ~¨~ is the operator norm defined as(23) (cid:23)(cid:23) k ! d kf P (cid:23)(cid:23) . “ sup γ P H C sup h ,...,h k ˇˇˇ γ ¨ k ! d kf P p h , . . . , h n q ˇˇˇ } γ }} h } ¨ ¨ ¨ } h n } , with supremum taken over h , . . . , h n P C r w, x, y, z s . It follows that(24) γ Smale p φ γ,m q ď C ´ sup k ě (cid:23)(cid:23) k ! d kf P (cid:23)(cid:23) k ´ . Let Γ denote the right-hand side of (24). By Smale’s theorem, together with (20)and (24), if | γ ¨ ω f | ď C Γ ´ } γ } , then there is a t P C such that | t | ď C ´ | γ ¨ ω f | and γ ¨ P p f ` tm q “ . The claim follows with C f . “ C ´ and(25) ε f . “ min ˆ C Γ ´ , ω f ¨ ω f } ω f } ˙ . (cid:3) The constants C f and ε f are actually computable with simple algorithms. Theconstant from Lemma 1 is not hard to get with elementary linear algebra. Itonly remains to compute an upper bound for Γ . We address this issue in §2.4. EPARATION OF PERIODS OF QUARTIC SURFACES 7
Corollary 3.
For any f P r U , any ε ă ε f , and any γ P H Z , if | γ ¨ ω f | ď ε then there exists a monomial m P R and t P C such that | t | ď C f ε and γ P Pic p X f ` tm q .Proof. We may assume that γ ¨ ω f ‰ (otherwise choose any m and t “ ).Let γ “ γ ´ p γ ¨ h q h . Since h ¨ h “ , we have γ ¨ h “ . Moreover γ ¨ ω f “ γ ¨ ω f ‰ . In particular, γ ‰ and since γ P H Z , we have } γ } ě and then(26) ˇˇ γ ¨ ω f ˇˇ ď } γ } | γ ¨ ω f | ď ε } γ } , and Proposition 2 applies. (cid:3) Effective bounds for the higher derivatives of the period map.
Inthe proof of Proposition 2, only the quantity Γ is not clearly computable. We showin this section how to compute an upper bound for Γ using the Griffiths–Dworkreduction. We follow here Griffiths (1969).Firstly, as a variant of (8) avoiding dehomogeneization, we write(27) P p f q “ ˜ πi ż T p γ i q Vol f ¸ ď i ď where Vol is the projective volume form(28)
Vol . “ w d x d y d z ´ x d w d y d z ` y d w d x d z ´ z d w d x d y. For any k ą and a P R k ´ , we denote(29) ż a Vol f k . “ ˜ πi ż T p γ i q a Vol f k ¸ ď i ď P H C . For any h P R close enough to 0, we have the power series expansion(30) ż Vol f ` h “ ÿ k ě p´ q k ´ ż h k ´ Vol f k . Proposition 4.
For any k ě , there is a linear map G k : R k ´ Ñ R suchthat ż af k Vol “ ż G k p a q f Vol . Moreover, there is a computable constant C , which depends only on f , such thatfor any k ě , ~ G k ~ ď C k ´ , where R is endowed with the -norm (55) . Before we begin the proof of proposition, let us show that this is enough tobound Γ . Let A : a P R ÞÑ ş af Vol P H C , then, using (30) we obtain(31) ż Vol f ` h “ ÿ k ě p´ q k ´ A p G k p h k ´ qq , and it follows that(32) k ! d kf P p h , . . . , h k q “ p´ q k A p G k ` p h ¨ ¨ ¨ h k qq . P. LAIREZ AND E. C. SERTÖZ
In particular, ›› k ! d kf P p h , . . . , h k q ›› ď ~ A ~ ~ G k ` ~ } h ¨ ¨ ¨ h n } (33) ď ~ A ~ ~ G k ` ~ } h } ¨ ¨ ¨ } h n } , (34)and therefore (cid:23)(cid:23)(cid:23) k ! d kf P (cid:23)(cid:23)(cid:23) ď ~ A ~ C k ` , and it follows(35) Γ ď C max ` ~ A ~ C , ˘ Proof of Proposition 4.
Let R “ C r w, x, y, z s . We define two families ofmaps for this proof. First, for d ě , a multivariate division map Q d : R d Ñ R d ´ , such that for any a P R d ,(36) a “ ÿ i “ Q d p a q i B i f. Note that such a map exists as soon as d ě by a theorem due to Macaulay(see Lazard 1977, Corollaire, p. 169). The choice of Q d is not unique. We fix Q arbitrarily and define Q d p a q , for d ą and a P R , as follows. Write a “ ř i “ x i a i , in such a way that the terms of the sum have disjoint monomialsupport, and define(37) Q d p a q “ ÿ i “ x i Q d ´ p a i q . It is easy to check that this definition satisfies (36).Second, for k ě , we define G k : R k ´ Ñ R as follows. Begin with G “ id and then define G k for k ě inductively as follows. For a P R k ´ wewrite p b , . . . , b q “ Q k ´ p a q and define(38) G k p a q . “ G k ´ ˆ k ´ pB b ` ¨ ¨ ¨ ` B b q ˙ . This map is the Griffiths–Dwork reduction, and it satisfies(39) ż γ a Ω f k “ ż γ G k p a q Ω f . Lemma 5.
For any d ě , ~ Q d ~ ď ~ Q ~ , where R is endowed with the -norm and R with the norm }p f , . . . , f q} . “ } f } ` ¨ ¨ ¨ ` } f } .Proof. For any a P R d , } Q d p a q} “ ÿ i “ } Q d p a q i } ď ÿ i “ ÿ j “ } x j Q d ´ p a j q i } (40) “ ÿ i “ ÿ j “ } Q d ´ p a j q i } “ ÿ j } Q d ´ p a j q} (41) ď ~ Q d ´ ~ ÿ j } a j } “ ~ Q d ´ ~ } a } , (42) EPARATION OF PERIODS OF QUARTIC SURFACES 9 using, for the last equality, that the terms a j have disjoint monomial support. (cid:3) Lemma 6.
For any k ě , ~ G k ~ ď p ~ Q ~q k ´ , where R is endowed with the -norm.Proof. We proceed by induction on k (the base case k “ is trivial since G “ id ).Let a P R k ´ and p b , . . . , b q “ Q k ´ p a q . By (38), we have } G k p a q} ď ~ G k ´ ~ k ´ p}B b } ` ¨ ¨ ¨ ` }B b } q . (43)By induction hypothesis, ~ G k ´ ~ ď p ~ Q ~q k ´ and moreover }B i b i } ď p k ´ q} b i } , since each b i has degree k ´ . If follows that } G k p a q} ď p ~ Q ~q k ´ k ´ k ´ p} b } ` ¨ ¨ ¨ ` } b } q . (44)Next, we note that } b } `¨ ¨ ¨`} b } “ } Q k ´ p a q} and, by Lemma 5, ~ Q k ´ p a q~ ď~ Q ~ . Therefore(45) } G k p a q} ď p ~ Q ~q k ´ } a } , and the claim follows. (cid:3) The Noether–Lefschetz locus
Basic properties.
We define the Noether–Lefschetz locus for quartic sur-faces and review a few classical properties, especially algebraicity, with a viewtowards Theorem 14 about the degree and the height of the equations definingthe components of the Noether–Lefschetz locus.3.1.1.
Definition.
The Noether–Lefschetz locus of quartics NL is the set ofall f P U such that the rank of Pic p X f q is at least 2. Equivalently, in viewof (12), NL is the set of quartic polynomials f whose primitive periods (1) are Z -linearly dependent.The set NL is locally the union of smooth analytic hypersurfaces in U . Tosee this, let Ą NL be the lift of NL in the universal covering r U of U . Recall P : r U Ñ D is the period map. The Lefschetz (1,1)-theorem implies(46) Ą NL “ ď γ P H Z z Z h P ´ t w P D | w ¨ γ “ u . That is, Ą NL is the pullback of smooth hyperplane sections of D . Since P is asubmersion, Ą NL is the union of smooth analytic hypersurfaces. It follows that NL is locally the union of smooth analytic hypersurfaces.We break NL into algebraic pieces as follows. For any integers d and g ,let NL d,g be the set(47) NL d,g “ t f P U | D γ P Pic p X f qz Z h : γ ¨ h “ d and γ ¨ γ “ g ´ u , By replacing γ by γ ` h or ´ γ , we observe that(48) NL d,g “ NL d ` ,g ` d ` “ NL ´ d,g . In particular, NL d,g is equal to some NL d ,g with d ą and g ě , so that(49) NL “ ď d ą ď g ě NL d,g . For γ P H Z , let ∆ p γ q “ p h ¨ γ q ´ γ ¨ γ . It is the opposite of the discriminantof the lattice generated by h and γ in H Z , with respect to the intersectionproduct (and it is zero if γ P Z h ). It follows from the Hodge index theorem(see Hartshorne 1977, Theorem V.1.9) that for any f P U and any γ P Pic p X f q , ∆ p γ q ě , with equality if and only if γ P Z h . If γ ¨ h “ d and γ ¨ γ “ g ´ ,then ∆ p γ q “ d ´ g ` . We obtain therefore that for any d ą and g ě ,(50) NL d,g “ " (cid:32) f P U ˇˇ D γ P Pic p X f q : γ ¨ h “ d and γ ¨ γ “ g ´ ( if d ą g ´ ∅ otherwise. It is in fact more natural to introduce, for ∆ ą , the following locus NL ∆ . “ t f P U | D γ P Pic p X f q : ∆ p γ q “ ∆ u (51) “ ď d ą d ” ∆ mod NL d, d ´ ∆8 ` . (52)Due to (48), NL ∆ reduces to a single NL d,g . Namely,(53) NL ∆ “ $’’’’&’’’’% NL t, t ` ´ ∆8 , if ∆ ” , NL t ` , t ` t ` ´ ∆8 if ∆ ” , NL t ` , t ` t ` ´ ∆8 , if ∆ ” , ∅ otherwise,where t “ r ? ∆ s . Conversely, each NL d,g “ NL d ´ g ` .3.1.2. Algebraicity.
For any d ą and g ě , the set NL d,g is either emptyor an algebraic hypersurface in U . This is a classical result (e.g. Voisin 2003,Theorem 3.32) which we recall here to obtain an explicit algebraic descriptionof NL d,g . Lemma 7.
For any f P U , d ą and g ě we have: f P NL d,g if and onlyif X f contains an effective divisor with Hilbert polynomial t ÞÑ dt ` ´ g .Proof. Assume that X f contains an effective divisor C with Hilbert polyno-mial t ÞÑ td ` ´ g . Since X f is smooth, C is a locally principal divisor and givesan element γ of Pic X f . The integer d is the degree of C , so it is the number ofpoints in the intersection with a generic hyperplane, that is d “ γ ¨ h . Moreover, g is the arithmetic genus of C , which is determined by g ´ “ γ ¨ γ (Hartshorne1977, Ex. III.5.3(b) and V.1.3(a)). So f P NL d,g .Conversely, let f P NL d,g . By definition, there is a divisor C on X f such thatits class γ in Pic X f satisfies γ ¨ h “ d and γ ¨ γ “ g ´ . From the Riemann–Rochtheorem for surfaces (ibid., p. V.1.6) we get: dim H p X, O X p C qq ` dim H p X, O X p´ C qq ě γ ¨ γ ` “ g ` ą EPARATION OF PERIODS OF QUARTIC SURFACES 11 so that either C or ´ C must be linearly equivalent to an effective divisor.Since γ ¨ h ą , ´ C can not be effective and therefore C must be. As above, theHilbert polynomial of C is given by t ÞÑ dt ` ´ g . (cid:3) In light of Lemma 7, the algebraicity of NL d,g is proved by using the Hilbertscheme H d,g . The Hilbert scheme H d,g of degree d and genus g curves in P isa projective scheme that parametrizes all the subschemes of P whose Hilbertpolynomial is t ÞÑ dt ` ´ g .The Hilbert scheme H d,g may contain components that are not desirable forour purposes. For example H , , which contains twisted cubics in P , containstwo irreducible components (Piene and Schlessinger 1985): a -dimensionalcomponent that is the closure of the space of all smooth cubic rational curvesin P ; and a -dimensional component parametrizing the union of a planecubic curve with a point in P . We would be only interested in the first, notin the second component. So we introduce H d,g , the union of componentsof H d,g obtained by removing the components that does not correspond tolocally-complete-intersection pure-dimensional subschemes of P .When d ą g ´ , Lemma 7 can be rephrased as(54) NL d,g “ proj (cid:32) p f, C q P U ˆ H d,g ˇˇ C Ă X f ( , where proj denotes the projection U ˆ H d,g Ñ U . Since H d,g is a projectivevariety, and the condition C Ă X f is algebraic, this shows that NL d,g is a closedsubvariety of U (for more details about this construction, see Voisin 2003, §3.3).We note furthermore that NL d,g is clearly invariant under the action of theGalois group of algebraic numbers. Therefore, it can be defined over the rationalnumbers.As a consequence, for any nonnegative integers d and g , there is a squarefreeprimitive homogeneous polynomial NL d,g P Z r u , . . . , u s in the 35 coefficientsof the general quartic polynomial that is unique up to sign and whose zero locusis NL d,g in U . Similarly, we define NL ∆ upto sign as the unique squarefreeprimitive polynomial vanishing exactly on NL ∆ .3.2. Height of multiprojective varieties.
The mainstay of our results is abound on the degree and size of the coefficients of the polynomials NL d,g . Thedetermination of these bounds is based on (54) and involves the theory of heightsof multiprojective varieties as developped by D’Andrea et al. (2013), and, beforethem, Bost et al. (1991), Krick et al. (2001), Philippon (1995), and Rémond(2001a,b), among others. We recall here the results that we need, followingD’Andrea et al. (2013).3.2.1. Heights of polynomials.
Let f “ ř α c α x α P C r x , . . . , x n s . We recall thefollowing different measures of height of f : } f } . “ ÿ α | c α | , (55) } f } sup . “ sup | x |“¨¨¨“| x n |“ | f p x q| , (56) m p f q . “ ż r , s n log ˇˇ f ` e πit , . . . , e πit n ˘ˇˇ d t ¨ ¨ ¨ d t n . (57) Lemma 8 (D’Andrea et al. 2013, Lemma 2.30) . For any homogeneous polynomial f P C r x , . . . , x n s , exp p m p f qq ď } f } sup ď } f } ď exp p m p f qqp n ` q deg f . The extended Chow ring.
The extended Chow ring (ibid., Definition 2.50)is a tool to track a measure of height of multiprojective varieties when performingintersections and projections. We present here a very brief summary. Bold lettersrefer to multi-indices and all varieties are considered over Q . Let n P N r andlet P n be the multiprojective space P n “ P n ˆ ¨ ¨ ¨ ˆ P n m .An algebraic cycle is a finite Z -linear combination ř V n V V of irreduciblesubvarieties of P n . The irreducible components of an algebraic cycle as above arethe irreducible varieties V such that n V ‰ . An algebraic cycle is equidimensional if all its irreducible components have the same dimension. An algebraic cycle is effective if n V ě for all V . The support of X , denoted by supp X , is the unionof the irreducible components of X .Let A ˚ p P n ; Z q be the extended Chow ring, namely(58) A ˚ p P n ; Z q . “ R r η, θ , . . . , θ m s{p η , θ n ` , . . . , θ n m ` m q , where θ i is the class of the pullback of a hyperplane from P n i and η is usedto keep track of heights of varieties. For two elements a and b of this ring, wewrite a ď b when the coefficients of b ´ a in the monomial basis are nonnegative.To an algebraic cycle X of P n we associate an element r X s Z of A ˚ p P n ; Z q (ibid., Definition 2.50). If X is effective, then r X s Z ě . The coefficients of theterms in r X s Z for monomials not involving η record the usual multi-degrees of X . The terms involving η record mixed canonical heights of X . The definitionof these heights is based on the heights of various Chow forms associated to X (ibid., §2.3). For the computations in this paper, we only need the followingresults.Let f P Z r x , . . . , x r s be a nonzero multihomogeneous polynomial with respectto the group of variables x , . . . , x n . We assume that f is primitive , that is, theg.c.d. of the coefficients of f is 1. The element associated in A ˚ p P n ; Z q to thehypersurface V p f q Ă P n is (ibid., Proposition 2.53(2))(59) r V p f qs Z “ m p f q η ` deg x p f q θ ` ¨ ¨ ¨ ` deg x r p f q θ r . To such a polynomial f , we also associate (ibid., Eq. (2.57))(60) r f s sup . “ log p} f } sup q η ` deg x p f q θ ` ¨ ¨ ¨ ` deg x r p f q θ r . Arithmetic Bézout theorem.
Let X be an effective cycle and H a hyper-surface in P n . They intersect properly if no irreducible component of X is in H .When X and H intersect properly, ones defines an intersection product X ¨ H , that EPARATION OF PERIODS OF QUARTIC SURFACES 13 is an effective cycle supported on X X H . If X is equidimensional of dimension r ,then X ¨ H is equidimensional of dimension r ´ .The following statement is an arithmetic Bézout bound that not only boundsthe degree, as with the classical Bézout bound, but also the height of an intersec-tion. Theorem 9 (ibid., Theorem 2.58) . Let X be an effective equidimensional cycle on P n and f P Z r x , . . . , x m s . If X and V p f q intersect properly, then r X ¨ V p f qs Z ďr X s Z ¨ r f s sup . This theorem can be applied (as in ibid., Corollary 2.61) to bound the heightof the irreducible components of a variety in terms of its defining equations.
Proposition 10.
Let Z Ă P n be an equidimensional variety and let X be V p f , . . . , f s q X Z , where f i is a multihomogeneous polynomial of multidegreeat most d and sup-norm at most L . Let X r be the union of all the irreduciblecomponents of X of codimension r in Z . Then r X r s Z ď r Z s Z ˜ log p sL q η ` m ÿ i “ d i θ i ¸ r . Proof.
Let p y ij q be a new group of variables, with ď i ď r and ď j ď s .Let g i . “ ř sj “ y ij f j and X . “ V p g , . . . , g r q in P k ˆ Z , with k “ rs ´ Wefirst claim that P k ˆ X r is a union of components of X . Indeed, let ξ be thegeneric point of P k and ξ be the generic point of a component Y of X r , sothat ξ “ p ξ , ξ q is the generic point of the component P k ˆ Y of P k ˆ X r . Since X has codimension r at ξ , the generic linear combinations g , . . . , g r form a regularsequence at ξ (in other words, they form a regular sequence at ξ for genericvalues of the v ij ). Therefore, X has codimension r at ξ . Since P k ˆ Y Ď X , itfollows that P k ˆ Y is a component of X .Let X r be the union of the components of codimension r of X . The argumentabove shows that r P k ˆ X r s Z ď r X r s Z . Besides, by repeated application of (ibid.,Corollary 2.61),(61) r X r s Z ď r P k ˆ Z s Z r ź i “ r g i s sup . We compute, using (59) that(62) r g i s sup ď log p sL q η ` θ ` s ÿ i “ d i θ i . Finally, we note that r P k ˆ X r s Z “ r X r s Z and r P k ˆ Z s Z “ r Z s Z (ibid., Proposi-tion 2.51.3 and 2.66). (cid:3) Proposition 11.
Let X be an equidimensional closed subvariety of P k ˆ P n andlet Y Ă P n be the projection of X . If Y is equidimensional, then θ k r Y s Z ď θ dim X ´ dim Y r X s Z P A ˚ p P k ˆ P n ; Z q , where θ is the variable attached to P k in the extended Chow ring of P k ˆ P n .Proof. We will argue by induction on r . “ dim X ´ dim Y . When r “ , this is(D’Andrea et al. 2013, Proposition 2.64).Suppose now that r ą and X is irreducible. Let Q r y , x , . . . , x m s denotethe multihomogeneous coordinate ring of P k ˆ P n . There is an i , ď i ď k ,such that H . “ V p y i q Ă P k ˆ P n intersects X properly (otherwise X would beincluded in all V p y i q and would be empty). Since the fibers of X Ñ Y are positivedimensional, H intersects each fiber. In particular, the set-theoretical projectionsof X and X X H coincide. As X is irreducible, so is Y . In particular, there is anirreducible component X Ă X X H that maps to Y . By induction hypothesisapplied to X , θ k r Y s Z ď θ dim X ´ dim Y r X s Z . Moreover, r X s Z ď r X s Z r y i s sup , and,in view of (60), r y i s sup “ θ . The claim follows.If X is reducible, then we apply the inequality above to each of the irreduciblecomponents of Y together with an irreducible component of X mapping ontothat component. (cid:3) Explicit equations for the Noether–Lefschetz loci.
Following Gotz-mann (1978), Bayer (1982), and the exposition of Lella (2012), we describe theequations defining the Hilbert schemes of curves in P . An explicit description ofthe Noether–Lefschetz loci NL d,g follows.3.3.1. The Hilbert schemes of curves.
For d ą and g ě let H d,g be the Hilbertscheme of curves of degree d and genus g in P . It parametrizes subschemes of P with Hilbert polynomial p p m q . “ dm ` ´ g . Smooth curves in P of degree d and genus g , in particular, have Hilbert polynomial p p m q . Let R “ C r w, x, y, z s be the homogeneous coordinate ring of P . For m ě , let R m denote the m thhomogeneous part of R and let q p m q “ dim R m ´ p p m q .The Hilbert scheme H d,g can be realized in a Grassmannian variety as follows.A subscheme X of P is uniquely defined by a saturated homogeneous ideal I of R . If the Hilbert polynomial of X is p , then I is the saturation of the idealgenerated by the degree r slice I r . “ I X R r (Gotzmann 1978; Bayer 1982, §II.10),where(63) r “ ˆ d ˙ ` ´ g, is the Gotzmann number of p (Bayer 1982, §II.1.17). For practical reasons, weneed r ě , so we define instead(64) r “ max ˆˆ d ˙ ` ´ g, ˙ . So X is entirely determined by I r , which is a q p r q -dimensional subspace of R r .Let G be the Grassmannian variety of q p r q -dimensional subspaces of R r . Asa set, one can construct H d,g as the subset of all Ξ P G such that the idealgenerated by Ξ in R defines a subscheme of P with Hilbert polynomial p . Infact, H d,g is a subvariety that is defined by the following condition (ibid., §VI.1): EPARATION OF PERIODS OF QUARTIC SURFACES 15 (65) H d,g “ t Ξ P G | dim p R Ξ q ď q p r ` qu , where R is the space of linear forms in w, x, y, z , so that R Ξ is a subspaceof R r ` .Several authors gave explicit equations for H d,g in the Plücker coordinates(Bayer 1982; Brachat et al. 2016; Gotzmann 1978; Grothendieck 1961). We willprefer here a more direct path that avoids the Plücker embedding.3.4. Equations for the relative Hilbert scheme.
Define the relative Hilbertscheme of curves inside quartic surfaces(66) H d,g p q . “ tp f, C q P P p R q ˆ H d,g | C Ă V p f qu , for each d ą , g ě .We define the following auxiliary spaces to better describe (66). First, definethe following ambient space(67) A . “ P p R q ˆ P ´ End p C q p r q´ N r ´ , R r q ¯ ˆ P ´ End p R r ` , C p p r ` q q ¯ . Second, let B “ tp f, φ, ψ q P A u be the set of all triples satisfying the conditions(i) R r ´ f Ď ker ψ ,(ii) R im p φ q Ď ker ψ ,(iii) im φ X R r ´ f “ ,(iv) φ and ψ are full rank.Finally, we denote by B the Zariski closure of B . Lemma 12.
The map B Ñ H d,g p q defined by p f, φ, ψ q ÞÑ p f, R r ´ f ` im φ q iswell defined and surjective.Proof. Let p f, φ, ψ q P B and let Ξ “ R r ´ f ` im φ . Constraint (iv) impliesthat im φ has dimension q p r q ´ N r ´ . Together with Constraint (iii), wehave dim Ξ “ q p r q . Moreover, Constraint (iv) implies that ker ψ has dimen-sion q p r ` q . In particular Since R Ξ “ R r ´ f ` R im φ , Constraints (i)and (ii) implies that R Ξ has dimension at most q p r ` q . So, Ξ P H d,g p q .Since R r ´ f Ď Ξ , the polynomial f is in the saturation of the ideal generatedby Ξ . Hence, p f, Ξ q P H d,g p q .Conversely, let p f, Ξ q P H d,g p q , then R r ´ f Ă Ξ and there is a full rankmap φ : C q p r q´ N r ´ Ñ R r such that im φ complements R r ´ f in Ξ . Furthermore, dim R Ξ ď q p r ` q , because Ξ P H d,g , so there is a full rank map ψ : R r ` Ñ C p p r ` q such that R Ξ Ď ker ψ . So p f, Ξ q is the image of p f, φ, ψ q P B . (cid:3) Lemma 13.
For any a ě , let B a be the union of the codimension a componentsof B . Then “ B a ‰ Z ď p
15 log p d ` q η ` θ ` θ ` θ q a Proof.
Let B be the closed set defined by the constraints (i) and (ii). Theconstraints (iii) and (iv) are open, so any component of B is a component of B .In particular r B a s Z ď r B a s Z .Constraint (i) is expressed with p p r ` q N r ´ polynomial equations of mul-tidegree p , , q (w.r.t. f , φ and ψ respectively). Namely, ψ p mf q “ for everymonomial m in R r ´ . Each p p r ` q components of the equation ψ p mf q “ involves a sum of 35 terms (since f , as a quartic polynomial, contains only35 terms) with coefficients 1. So the -norm of these constraints is at most (which is also at most N r , since r ě ).Constraint (ii) is expressed with p p r ` qp q p r q ´ N r ´ q polynomial equationsof multidegree p , , q . Namely, ψ p vφ p e qq “ for any basis vector e and anyvariable v P t w, x, y, z u . Each p p r ` q component of the equation ψ p vφ p e qq “ involves a sum of N r terms with coefficients 1. So the -norm of these constraintsis at most N r .The claim is then a consequence of Proposition 10, with s “ p p r ` q N r ´ ` p p r ` qp q p r q ´ N r ´ q and L “ N r . We check routinely, with Mathematica,that sL ď p d ` q . (cid:3) Theorem 14.
There is an absolute constant A ą such that for any d ą and g ě we have deg p NL d,g q ď A d and } NL d,g } ď A d . Proof.
We assume NL d,g is non-emtpy, since these inequalities are triviallysatisfied if NL d,g “ H with NL d,g “ . Let P . “ P ` End p C q p r q´ N r ´ , R r q ˘ and P . “ P ` End p R r ` , C p p r ` q q ˘ denote the second and third factors of A .Let α . “ p q p r q ´ N r ´ q N r ´ and β . “ p p r ` q N r ` ´ denote the dimensionsof P and P respectively. Let E be the projection of B on P p R q ˆ P . The fibersof the map B Ñ E are projective subspaces of P since Constraints (i) and (ii)are linear in ψ . The dimension of these fibers are β . “ p p r ` q ´ . So, byProposition 11,(68) θ β r E s Z ď θ β “ B ‰ Z . Next, the map B Ñ H d,g p q factors through E and the fibers of the correspondingmap E Ñ H d,g p q have dimension α . “ p q p r q ´ N r ´ q q p r q ´ . Finally, let e bethe dimension of the fibers of the map H d,g p q Ñ NL d,g . (If this dimension isnot generically constant, we work one component at a time.) Once again, byProposition 11, we obtain(69) θ α r NL d,g s Z ď θ α ` e r E s Z . Since r NL d,g s Z “ m p NL d,g q η ` deg p NL d,g q θ , taking L “
15 log p d ` q , we get deg NL d,g ď coeff of θ θ α ´ α ´ e θ β ´ β in p Lη ` θ ` θ ` θ q α ` β ´ α ´ β ´ e ` (70) ď α ` β ´ α ´ β ´ e ` . (71) EPARATION OF PERIODS OF QUARTIC SURFACES 17
The exponent is a polynomial in d and g . Unless d ě g ´ , NL d,g is empty.So, we may bound the exponent with a polynomial only in d , which turns out tobe of degree . Therefore, deg NL d,g ď A d for some constant A ą .Similarly, m p NL d,g q ď coeff of ηθ α ´ α ´ e θ β ´ β in p Lη ` θ ` θ ` θ q α ` β ´ α ´ β ´ e ` (72) ď p α ` β ´ α ´ β ´ e ` q L α ` β ´ α ´ β ´ e (73) ď O p d q . (74)By D’Andrea et al. (2013, Lemma 2.30.3),(75) } NL d,g } ď exp p m p NL d,g qq deg NL ∆ , and this implies the claim, for some other constant A ą . (cid:3) For the following, we write a Ò b for a b . This is a right-associative operation. Corollary 15.
There is an absolute constant A ą such that for any ∆ ą , deg p NL ∆ q ď A Ò ∆ Ò and } NL ∆ } ď Ò A Ò ∆ Ò . In fact, one can obtain the following explicit bounds deg p NL ∆ q ď p ∆ ` q { and log } NL ∆ } ď p ∆ ` q p ∆ ` q { . Proof.
The first statement follows directly from (53) and Theorem 14 using adifferent A . The second statement is found by carrying out the arguments in theproof of Theorem 14 with the help of a computer algebra system. (cid:3) How good are these bounds?
We can compare our degree boundsfor NL ∆ to the exact degrees computed by Maulik and Pandharipande (2013),from which it actually follows that(76) deg NL ∆ “ O p ∆ q . This sharper bound does not directly imply a sharper bound on the heightof NL ∆ but it suggests the following conjecture. This would improve subsequentlyTheorems 17 and 20. In particular, Equation (2) would be exponential in thesize of the coefficients, as opposed to being doubly exponential. Conjecture 16.
There is a constant c ą such that for any ∆ ą , } NL ∆ } ď c Ò ∆ Ò . Now we turn to the details of (76). Following Maulik and Pandharipande(2013) (but replacing q by q ), consider the following power series(77) A . “ ÿ n P Z q n , B . “ ÿ n P Z p´ q n q n , Ψ “ ÿ n ą q n , and Θ defined by(78) Θ . “ A ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ´ A B ` B . From (Maulik and Pandharipande 2013, Corollary 2), we have, for any ∆ ą ,(79) deg NL ∆ ď coefficient of q ∆ in Θ ´ Ψ . In fact, this is an equality when the components of NL ∆ are given appropriatemultiplicities. Let Θ r k s denote the coefficient of q k in Θ . By (79), we only needto bound Θ r ∆ s in order to bound deg NL ∆ . To do so, replace every negativesign in the definition of Θ by a positive sign, including those in B , to obtain the coefficientwise inequality(80) Θ ď ˆ ÿ n P Z q n ˙ . The coefficient of q k in ´ř n P Z q n ¯ is(81) r p k q . “ p a , . . . , a q P Z ˇˇˇˇˇ ÿ i a i “ k + . The asymptotic bound r d p k q “ O p x d ´ q , for d ą , is well known (e.g. Krätzel2000, Satz 5.8). 4. Separation bound
We now state and prove the main results. Recall that a Ò b “ a b is rightassociative and for γ P H Z we defined the discriminant ∆ p γ q as p γ ¨ h q ´ γ ¨ γ . Theorem 17.
For any f P Z r w, x, y, z s X U there is a computable constant c ą such that for any γ P H p X f , Z q , if γ ¨ ω f ‰ , then | γ ¨ ω f | ą ` Ò c Ò ∆ p γ q Ò ˘ ´ . Multiplicity of Noether–Lefschetz loci.
The multiplicity of some nonzeropolynomial F P C r x , . . . , x s s at a point p P C s is the unique integer k such thatall partial derivatives of F of order ă k vanish at p and some partial derivativeof order k does not. It is denoted by mult p F .The multiplicity of NL ∆ at some f P U is related to the elements of Pic p X f q with discriminant ∆ . For ∆ ą , let E ∆ be a set of representatives of theequivalence classes of the relation „ on H Z defined by(82) γ „ γ if D a P Q ˚ , b P Q : γ “ aγ ` bh. EPARATION OF PERIODS OF QUARTIC SURFACES 19
Lemma 18.
For any f P U and any ∆ ą , mult f NL ∆ “ p Pic X f X E ∆ q . Proof.
Let Ą NL ∆ be the lift of NL ∆ in r U . Arguing as in §3.1.1, Ą NL ∆ is theunion of smooth analytic hypersurfaces:(83) Ą NL ∆ “ ď η P E ∆ P ´ t w P D | w ¨ η “ u . Then the same holds locally for NL ∆ .For any f P U it follows from the smoothness of branches of NL ∆ that mult f NL ∆ is exactly the number of branches meeting at f . The branchesmeeting at f are described by the elements of Pic X f with discriminant ∆ . Twoelements γ and γ describe the same branch (that is the same hyperplane sectionof D ) if and only if γ „ γ . So mult f NL ∆ is exactly the number of equivalenceclasses in t γ P Pic X f | ∆ p γ q “ ∆ u for this relation. (cid:3) Proof of Theorem 17.
We first apply Corollary 3. Let ε “ | γ ¨ ω f | . Thecorollary gives constants C f ą and ε f ą (depending only on f ) such thatif ε ă ε f then there exists a monomial m P R and t P C such that(84) | t | ď C f ε and(85) γ P Pic X f ` tm . Assume that ε ă ε f and let t and m be as above. As u varies, the num-ber p Pic p X f ` um q X E ∆ q has a strict local maximum at u “ t . By Lemma 18,so does mult f ` um NL ∆ p γ q . In particular, there is some higher-order partial deriv-ative of NL ∆ which vanishes at f ` tm but not at f ` um , for u close to but notequal to t . Let α P N be the multi-index for which(86) P . “ α ! ¨ ¨ ¨ α ! B | α | NL ∆ B u α P Z r u , . . . , u s is this derivative. For a monomial u β . “ u β ¨ ¨ ¨ u β we have(87) α ! ¨ ¨ ¨ α ! B | α | u β B u α “ ź i “ ˆ β i α i ˙ u β ´ α . Since ` β i α i ˘ ď β i , it follows that(88) ›››› α ! ¨ ¨ ¨ α ! B | α | NL ∆ B u α ›››› ď deg NL ∆ } NL ∆ } . Let Q P Z r x s be the integer polynomial Q p x q . “ P p f ` xm q . By construc-tion Q ‰ and Q p t q “ . Clearly deg Q ď deg NL ∆ , and we check that(89) } Q } ď } P } p} f } ` q deg P . and then(90) } Q } ď deg NL ∆ } NL ∆ } p} f } ` q deg NL ∆ . From Corollary 15, we find a constant c depending only on f such that(91) deg Q ď c Ò ∆ Ò and } Q } ď Ò c Ò ∆ Ò . We write Q “ ř deg Qi “ q i x i . Let k be the smallest integer such that q k ‰ .Since Q p t q “ , it follows that(92) ˇˇ q k t k ˇˇ ď deg Q ÿ i “ k ` ˇˇ q i t i ˇˇ If ε ă C ´ f , we have | t | ă , by (84), and it follows that ˇˇ q k t k ˇˇ ď ˇˇ t k ` ˇˇ } Q } .Since q k P Z and nonzero, it follows that(93) | t | ě } Q } . By (84), this leads to(94) ε ě ` Ò c Ò ∆ Ò ˘ ´ , for some other constant c which depends only on f . Recall that (94) holds withthe assumption that ε ď ε f and ε ă C ´ f . However, we can choose c large enoughso that the right-hand side of (94) is smaller that ε f and C ´ f . Then (94) holdsunconditionally, concluding the proof of Theorem 17. (cid:3) Quartics with algebraic coefficients.
Let K Ă C be a number field ofdegree D “ r K : Q s . Suppose that f P K r w, x, y, z s X U has coefficients thatare algebraic integers in K . Let H ą be an upper bound for the absolutelogarithmic Weil height for the coefficient vector of f (Waldschmidt 2000, p.77). Theorem 19.
Let f and H, D ą be as above. Then there is a computableconstant c ą such that for any γ P H p X f , Z q , if γ ¨ ω f ‰ , we have | γ ¨ ω f | ą ` Ò c Ò ∆ p γ q Ò ˘ ´ D p ` H q . Proof.
The proof of Theorem 17 carries through, with the sole exception that Q p x q no longer has integer coefficients. If q k is the first non-zero coefficient of Q p x q , then q k is an algebraic integer defined by a polynomial expression r q k p f q incoefficients of f with r q k having integer coefficients. Therefore, the number “ ”in (93) must be replaced by a suitable lower bound on the norm of q k . For this,we use Liouville’s inequality (ibid., Proposition 3.14):(95) | q k | ě } r q k } ´ D ` e ´ DH deg r q k . It is easy to see that deg r q k ď deg NL ∆ and } r q k } ď deg NL ∆ } NL ∆ } , the lattercan be bounded by } Q } . The result follows for some other c . (cid:3) Numbers à la
Liouville.
Let p θ i q i ě be a sequence of positive integerssuch that θ i is a strict divisor of θ i ` for all i ě (in particular θ i ě i .) Consider EPARATION OF PERIODS OF QUARTIC SURFACES 21 the number L θ . “ ÿ i “ θ ´ i . As a corollary to the separation bound obtained in Theorem 17, the followingresult states that L θ is not a ratio of periods of quartic surfaces when θ growstoo fast. Theorem 20. If | θ i ` | ě Ò Ò θ i Ò , for all i large enough, then L θ is notequal to γ ¨ ω f γ ¨ ω f for any γ , γ P H Z and any f P U with rational coefficients.Proof. Let l k “ ř ki “ θ ´ i . Since θ i divides θ i ` , we can write l k “ u k θ k for someinteger u k . And since the divisibility is strict, θ i ě i and u k ď θ k . Moreover(96) ă L θ ´ l k ď θ ´ k ` , using θ k ` i ` ě i θ k ` , for any i ě . Assume now that L θ “ γ ¨ ω f γ ¨ ω f forsome γ , γ P H Z and some f P U with rational coefficients. Then, with(97) γ k . “ θ k γ ´ u k γ , we check that ∆ p γ k q “ O p θ k q and that(98) ă | γ k ¨ ω f | “ | θ k | | γ ¨ ω f | | L θ ´ l k | ď C θ k θ k ` , for some constant C . By Theorem 17, we obtain therefore(99) p Ò c Ò θ k Ò q ´ ď C θ k θ k ` , for some constant c ą which depends only on f . This contradicts the assumptionon the growth of θ . (cid:3) Computational complexity.
Given a polynomial f P Z r w, x, y, z s X U and a cohomology class γ P H p X f , Z q , we can decide if γ P Pic p X f q (thatis γ ¨ ω f “ ) as follows:(a) Compute the constant c in Theorem 17;(b) Let ε “ ` Ò c Ò ∆ p γ q Ò ˘ ´ and compute an approximation s P C ofthe period γ ¨ ω f such that | s ´ γ ¨ ω f | ă ε .Then γ is in Pic p X f q if and only if | s | ă ε .Computing the Picard group itself is an interesting application of this procedure.Algorithms for computing the Picard group of X f , or even just the rank of it,break the problem into two: a part gives larger and larger lattices inside Pic p X f q while the other part gets finer and finer upper bounds on the rank of Pic p X f q (Charles 2014; Hassett et al. 2013; Poonen et al. 2015). The computation stopswhen the two parts meet. Approximations from the inside are based on findingsufficiently many elements of Pic p X f q . So while deciding the membership of γ in Pic p X f q can be solved by computing Pic p X f q first, it makes sense not to assume prior knowledge of the Picard group and to study the complexity ofdeciding membership as ∆ p γ q Ñ 8 , with f fixed.Step (a) does not depend on γ , so only the complexity of Step (b) matters,that is the numerical approximation of γ ¨ ω f . This approximation amountsto numerically solving a Picard–Fuchs differential equation (Sertöz 2019) andthe complexity is p log ε q ` o p q (Beeler et al. 1972; Hoeven 2001; Mezzarobba2010, 2016). With the value of ε in Step (b), we have a complexity bound of exp p ∆ p γ q O p q q for deciding membership.For the sake of comparison, we may speculate about an approach that woulddecide the membership of γ in Pic p X f q by trying to construct an explicit algebraicdivisor on X f whose cohomology class is equal to γ . It would certainly need todecide the existence of a point satisfying some algebraic conditions in some Hilbertscheme H d,g , with d “ O p ∆ q and g “ O p ∆ q (see §3.1.1). Embedding H d,g (orsome fibration over it, as we did in §3.4) in some affine chart of a projectivespace of dimension d O p q will lead to a complexity of exp p ∆ p γ q O p q q for decidingmembership in this way.However, if Conjecture 16 holds true, then the complexity of the numericalapproach for deciding membership would reduce to ∆ p γ q O p q . References
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Email address : [email protected] URL : pierre.lairez.fr Emre Can Sertöz, Max Planck Institute for Mathematics, Bonn, Germany
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