WWEAK APPROXIMATION FOR CUBIC HYPERSURFACES
ZHIYU TIAN
Abstract.
We prove weak approximation for smooth cubic hypersur-faces of dimension at least 2 defined over the function field of a complexcurve. introduction Given an algebraic variety X over a number field or function field F , anatural question is whether the set of rational points X ( F ) is non-empty.If it is non-empty, then how many rational points are there? In particular,are they Zariski dense? Do they satisfy weak approximation ?In this article, we address the weak approximation question for cubichypersurfaces defined over the function field of a complex curve.Smooth cubic hypersurfaces of dimension at least 2 belong to the classof “rationally connected varieties”. Roughly speaking, a smooth projectivevariety X is rationally connected if for any two general points x and y ,there is a rational curve connecting them. For more precise definition andproperties of rationally connected varieties, see [Kol96], Chap. IV.After the pioneering work of Graber-Harris-Starr [GHS03], which estab-lished the existence of rational points for rationally connected varieties de-fined over the function field of a curve, there has been much research centeredaround the arithmetic of such varieties, see, e.g. [HT06], [HT09], [HT08],[TZ13], [Kne13], [Xu12b].In particular, Hassett and Tschinkel [HT06] conjectured that weak ap-proximation holds for all smooth projective rationally connected varietiesdefined over the function field of a complex curve.We first state the conjecture in the arithmetic form. Let B be a smoothprojective connected complex curve. For any closed point b ∈ B , denoteby (cid:98) O B,b the completion of the local ring at the point b and Frac (cid:98) O B,b thecorresponding fraction field. Let A = ◦ (cid:89) b ∈ B Frac (cid:98) O B,b be the ad´eles over the function field C ( B ), where all but finitely many ofthe factors in the product are in (cid:98) O B,b . Finally let X η be a smooth rationallyconnected variety defined over the function field C ( B ). Then the weak Date : October 17, 2018. a r X i v : . [ m a t h . AG ] J u l TIAN approximation conjecture can be formulated as saying that the set of rationalpoints X η ( C ( B )) is dense in X η ( A ).For our purpose, it is more useful to formulate the conjecture in thefollowing geometric form. Conjecture 1.1 ([HT06]) . Let π : X → B be a flat surjective morphismfrom a projective variety to a smooth projective curve such that a generalfiber is smooth and rationally connected (such a map X → B is called a model of the generic fiber). Then the morphism π satisfies weak approxima-tion . That is, for every finite sequence ( b , . . . , b m ) of distinct closed pointsof B , for every sequence ( (cid:98) s , . . . , (cid:98) s m ) of formal power series sections of π over b i , and for every positive integer N , there exists a regular section s of π which is congruent to (cid:98) s i modulo m NB,b i for every i = 1 , . . . , m .For the equivalence of the two formulations, see section 1 of the surveyarticle [Has10], which provides a nice introduction and summary of knownresults of weak approximation in the function field case (as of 2008).Some special cases of the conjecture are known, e.g. • P n , conic bundles over P , del Pezzo surfaces of degree at least 4,[CTG04], • low degree complete intersections of degree ( d , . . . , d c ) such that (cid:80) d i ≤ n + 1, [dJHS11], [Has10], • smooth cubic hypersurfaces in P n , n ≥ • isotrivial families [TZ13], • at places of good reduction (for any family) [HT06], • a general family of del Pezzo surfaces of degree at most 3, [HT08],[Kne13],[Xu12b], and • a smooth hypersurface with square-free discriminant [HT09].We notice an interesting difference between the cases completely under-stood and the other cases. Namely, the former cases are proved by studyingthe global geometry over the function field while the latter cases are provedby studying the local singular fibers and the singularities of the total space.In some sense, our paper is a combination of the two approaches. Themain theorem is the following. Theorem 1.2.
Let X be a smooth cubic hypersurface of dimension at least defined over the function field of a complex curve. Then weak approximationholds at all places. We conclude this introduction by explaining the idea of the proof. Firstof all, by choosing a Lefschetz pencil and using standard facts about weakapproximation, one reduces to prove weak approximation for cubic surfaces.There are two new ingredients in the proof. The first one is local. Theobservation is that when the central fiber is a cone over an irreducible planecubic curve or non-normal (i.e. the worst degenerate case for families ofcubic surfaces, see 3.1), one can make a ramified base change and a birationalmodification so that the new central fiber has at worst du Val singularities.
EAK APPROXIMATION FOR CUBIC HYPERSURFACES 3
One just needs to keep track of the Galois action to get back to the originalfamily. The singularities have been greatly improved during this process.The second new idea is global and geometric. Given a local formal sectionthat we want to approximate, say (cid:98) s , we choose a section s , whose restrictionto the formal neighborhood gives a formal section (cid:98) s . Then the two formalsections determine a unique line (cid:98) L , which intersects the family at a thirdlocal formal section (cid:98) s (cid:48) . One can find a line L defined over the functionfield, which contains the rational point corresponding to the section s , andapproximate (cid:98) L to order N (i.e. weak approximation for the space of linescontaining s ). The line L (generally speaking) intersects the family at s anda degree 2 multisection σ .The next step is to deform σ so that it approximates the formal sections (cid:98) s and (cid:98) s (cid:48) . This is equivalent to a special case of weak approximation after adegree 2 base change. However, the multisection σ already approximates (cid:98) s (cid:48) (even though we have no control on this formal section) to order N . Thusone only needs to approximate (cid:98) s , which comes from the section and can becarefully chosen to lie in the smooth locus of the fibration (Lemma 5.1) sothat weak approximation is possible (by the local approach, i.e. the studyof singular fibers).Once we approximate the formal sections (cid:98) s (cid:48) and (cid:98) s , we take the linespanned by the degree 2 multisection and the third intersection point withthe cubic surface is what we need.Both Swinnerton-Dyer [SD01] and Madore [Mad06] have used the com-position law of the cubic surface to study weak approximation on cubicsurfaces. However, the basic strategy seems quite different. Their idea is touse a unirational parameterization to approximate v -adic points, which onlyworks for places of good reduction. Our approach is to use the compositionlaw to reduce the problem to a special and easier case, which can be provedvia the deformation technique of Koll´ar-Miyaoka-Mori [KMM92]. Acknowledgments:
I would like to thank Tom Graber for many helpfuldiscussions and for saving me from making many false statements, ChenyangXu and Runpu Zong for their interest in the project, Letao Zhang for herhelp with the preparation of the manuscript, and the referees for so manyhelpful suggestions that have greatly improved the paper, both in its contentand in its way of presentation.2.
Preliminaries
Everything with a cyclic group action.
In this subsection we col-lect some useful results from [TZ13]. Let k be any field and G a cyclic groupof order l such that l is invertible in k .First, we are concerned with the following infinitesimal lifting problem.Let S and R be k -algebras with a G -action and f : S → R be an algebrahomomorphism compatible with the action. Let A be an Artinian k -algebrawith a G -action, I ⊂ A an invariant ideal such that I = 0. Consider TIAN the following commutative diagram, where p is a G -equivariant k -algebrahomomorphism. S f −−−−→ R (cid:121) (cid:121) p A π −−−−→ A/I −−−−→ G -equivariant lifting h : R → A .The following lemma completely answers this question. Lemma 2.1.
If we can lift the map p to a k -algebra homomorphism h : R → A such that π ◦ h = p , then we can find an equivariant lifting ˜ h : R → A with the same property.Proof. For every element g in G , define a map h g : R → A by h g ( r ) = g · h ( g − · r ). This is an S -algebra homomorphism and also a lifting of themap p : R → A/I . The map h is G -equivariant if and only if h g ( r ) = h ( r ) for every g ∈ G and every r ∈ R . The difference of any two suchliftings is an element in Hom (Ω R/S , I ), where Ω
R/S is the module of relativedifferentials. Therefore one has θ ( g )( r ) = h g ( r ) − h ( r ) in Hom (Ω R/S , I ).Notice that
Hom (Ω R/S , I ) is naturally a G -module with the action of G on Hom (Ω R/S , I ) given by G × Hom (Ω R/S , I ) → Hom (Ω R/S , I )( g, η ) (cid:55)→ g · η = ( ω (cid:55)→ g · η ( g − · ω )) . It is easy to check that θ ( gh ) = g · θ ( h ) + θ ( g )Thus θ defines an element [ θ ] in H ( G, Hom (Ω R/S , I )). The existence ofan equivariant lifting is equivalent to the existence of an element Θ ∈ Hom (Ω R/S , I ) such that g Θ − Θ = θ , i.e, the class defined by θ is zero in H ( G, Hom (Ω R/S , I )). Since the characteristic of the field is relatively primeto the order of G , all the higher cohomology groups H i ( G, Hom (Ω R/S , I )) , i ≥ G vanish ([Wei94], Proposition 6.1.10, Corollary 6.5.9). The vanishingcan be proved by the usual averaging argument. (cid:3) Corollary 2.2.
Let X and Y be two k -schemes with a G -action and f : X → Y be a finite type G -equivariant morphism. Let x ∈ X be a fixed point,and y = f ( x ) (hence also a fixed point). Assume that f is smooth at x .Then there exists a G -equivariant section s : Spec (cid:98) O y,Y → X . In particular,assume that Y is irreducible and the G action on Y is trivial. If there is afixed point in X , then the set the fixed points of X dominates Y .Proof. Let S be the local ring at y, and R be the local ring at x. There isan obvious G action on both of these k -algebras. We start with the section s : Spec k ( y ) → f − ( y ) , Spec k ( y ) (cid:55)→ x , which is clearly G -equivariant. Bythe smoothness assumption, a section from Spec ( (cid:98) O y,Y / m ny ) always lifts to EAK APPROXIMATION FOR CUBIC HYPERSURFACES 5 a section from Spec ( (cid:98) O y,Y / m yn +1 ). Now apply Lemma 2.1 inductively tofinish the proof. (cid:3) We also need the following G -equivariant smoothing result, which is aslight generalization of the corresponding results in [TZ13]. Lemma 2.3.
Let X be a smooth quasi-projective rationally connected varietyover C and G be a cyclic group of order l acting on X . Fix an action of G on P by z (cid:55)→ ζz , where ζ is a primitive l -th root of unity. Assume thatthere is a very free rational curve through every point of X . (1) Let f : P → X be a G -equivariant map. Then there exists a G -equivariant map ˜ f : P → X such that ˜ f (0) = f (0) , ˜ f ( ∞ ) = f ( ∞ ) ,and ˜ f is very free. (2) Let f i : C i → X, ≤ i ≤ n be a chain of equivariant maps, i.e. foreach i , C i ∼ = P , and f i is a G -equivariant map such that f i ( ∞ ) = f i +1 (0) for ≤ i ≤ n − . Then there is a G -equivariant map ˜ f : P → X such that ˜ f (0) = f (0) and ˜ f ( ∞ ) = f n ( ∞ ) . Recall that a quasi-projective complex variety is rationally connected ifthere is a rational curve through a general pair of points. The assumptionthat there is a very free rational curve through every point of X is sayingthat X is strongly rationally connected in the sense of Hassett-Tschinkel[HT08]. Proof.
For part (1), we may assume that the equivariant map f is an em-bedding and dim X ≥ X with X × P M for some large M andprojecting deformations to the first factor X . Let C be the image of themorphism f .We first attach very free curves C i at general points p i ∈ C along generaltangent directions at p i . Let D be the nodal curve assembled in this way. ByLemma 2.5 [GHS03], after attaching enough such curves, the twisted normalsheaf N D /X ( − − ∞ ) is globally generated and H ( D , N D /X ( − − ∞ ) ⊗ L ) = 0, where L is any line bundle on D which has degree − l on C and0 on all the other irreducible components C i .We then attach all the curves that are G -conjugate to C i ’s (we may choose C i ’s such that the G -orbits do not intersect each other). The new nodal curveis denoted by D . The map D → X is G -equivariant. As in the previousparagraph, D has the property that H ( D, N D/X ( − − ∞ ) ⊗ L ) = 0, where L is any extension of the line bundle L on D which has degree − l on C and 0 on all the attached rational tails. Denote by R j the rational curveattached to C at the point p j .We have the following two exact sequences:0 → ⊕ j N D/X ( − −∞ ) | R j ( − n j ) → N D/X ( − −∞ ) → N D/X ( − −∞ ) | C → , → N C/X ( − − ∞ ) → N D/X ( − − ∞ ) | C → ⊕ j Q j → , TIAN where n j ’s are the nodal points on R j , and Q j ’s are torsion sheaves supportedon the points p j ∈ C . Every sheaf has a natural G -action and the G -equivariant deformations are given by G -invariant sections of N D/X . To finda G -equivariant deformation smoothing all the nodes of D and fixing 0 and ∞ , one just needs to find a G -invariant section in H ( D, N D/X ( − − ∞ )) G which, for any j , is not mapped to 0 under the composition of maps H ( D, N D/X ( − − ∞ )) → H ( C, N D/X ( − − ∞ ) | C ) → Q j for all j .Since H ( D, N D/X ( − − ∞ ) ⊗ L ) = 0, we also have H ( C, N D/X ( − − ∞ ) ⊗ L ⊗ O C ) = 0 . Let c , . . . , c l be an orbit of the G action on C . By the vanishing of H ( C, N D/X ( − − ∞ ) ⊗ O C ( − x − . . . − x l )), the map H ( C, N D/X ( − − ∞ ) ⊗ O C ( − c − . . . − c l − )) → N D/X ( − − ∞ ) | c l is surjective. Thus there is a section of N D/X ( − − ∞ ) | C which vanisheson c , . . . , c l − but not on c l . Then taking the average over G gives a G -invariant section of N D/X ( − − ∞ ) | C which does not vanish on any of thepoints c , . . . , c l . In particular, for any l nodes on C which lie in a G -orbit,we can find a G -invariant section of N D/X ( − − ∞ ) which does not vanishon them. Then a general G -invariant section of N D/X ( − − ∞ ) | C does notvanish on any of the nodes p i .We have a surjection map H ( D, N D/X ( − − ∞ )) → H ( C, N D/X ( − − ∞ ) | C )and (consequently) H ( D, N D/X ( − − ∞ )) G → H ( C, N D/X ( − − ∞ ) | C ) G is also surjective. So a general G -invariant section in H ( D, N D/X ( − −∞ )) G does not vanish on the nodes. We take the G -equivariant deformationgiven by this section, which necessarily smooths all the nodes of D with 0and ∞ fixed. A general smoothing is very free since the normal bundle isample by upper-semicontinuity.For the second part, we may assume that all the f i ’s are very free by thefirst part. Let f be the G -equivariant map obtained by gluing the f i ’s.Let ( T, o ) be a pointed smooth curve with trivial G -action, and let ˜Σbe P × T with the natural diagonal action. There are two G -equivariantsections, s = 0 × T, s ∞ = ∞ × T . Now blow up the point s ∞ ( o ) and stilldenote the strict transforms of the two sections by s and s ∞ . The G -actionextends to the blow-up. We can make the fiber over o ∈ T a chain of rationalcurves with n irreducible components by repeating this operation. Then weget a smooth surface Σ with a G -action such that the projection to T is G -equivariant. EAK APPROXIMATION FOR CUBIC HYPERSURFACES 7
Let h : s → X × T and h ∞ : s ∞ → X × T be T -morphisms suchthat h ( s ) = f (0) × T and h ∞ ( s ∞ ) = f n ( ∞ ) × T . Consider the relativeHom-scheme Hom T (Σ , X × T, h , h ∞ ) parameterizing T -morphisms from Σto X × T fixing h and h ∞ . It has a natural G action and the map µ :Hom T (Σ , X × T, h , h ∞ ) → T is G -equivariant. Now µ is smooth at f .By Corollary 2.2, there is a G -equivariant formal section. So there are G -equivariant smoothings of the morphism f . (cid:3) Finally we quote the following theorem from [TZ13]. For our purpose, weonly need to find equivariant rational curves in a few cubic surfaces, which,however, might be singular. Furthermore we want the curve to lie in thesmooth locus, so the proof in [TZ13] does not directly carry over. Howeverit is good to know that such curve exists at least in a desingularization. Wewill discuss this problem in more detail later in 3.4.
Theorem 2.4.
Let X be a smooth projective rationally connected varietyand let G be a cyclic group of order l with an action on X . Choose aprimitive l -th root of unity ζ and let G act on P by [ X , X ] (cid:55)→ [ X , ζX ] .Then for each pair ( x, y ) of fixed points in X , there is a G -equivariant map f : P → X such that f (0) = x and f ( ∞ ) = y . Iterated blow-up.
Let π : X → C be a flat proper family over asmooth projective connected curve C . Let c ∈ C be a closed point and (cid:98) s : Spec (cid:98) O c,C → X be a formal section. Assume that (cid:98) s lies in the smoothlocus of X → C . The N -th iterated blow-up associated to (cid:98) s is definedinductively as follows.The 0-th iterated blow-up X is X itself. Assume the i -th iterated blow-up X i has been defined. Let (cid:98) s i be the strict transform of (cid:98) s in X i . Then X i +1 is defined as the blow-up of X i at the point (cid:98) s i ( c ).We remark that if both X and C have a G -action such that • the map π : X → C is G -equivariant. • The point c is the fixed point of G and (cid:98) s is G -equivariant,then each X i has a G -action such that the natural morphisms X i +1 → X i and the formal sections (cid:98) s i are G -equivariant. In particular, the intersectionof (cid:98) s i with the central fiber is a fixed point of G .One can also do this at fibers over a G -orbit in C , provided the formal sec-tions over these points are conjugate to each other under the G -action. Thenthe iterated blow-up still has a G -action and every morphism is compatiblewith the action.On X N , the fiber over the point c consists of the strict transform of X | c and exceptional divisors E , . . . , E N , and • E i , i = 1 , . . . , N −
1, is the blowup of P d at r i (= (cid:98) s i ( c )), the pointwhere the proper transform of (cid:98) s (i.e. (cid:98) s i ) meets the fiber over c ofthe ( i − • E N ∼ = P d , TIAN where d is the dimension of the fiber.The intersection E i ∩ E i +1 is the exceptional divisor P d − ⊂ E i , and aproper transform of a hyperplane in E i +1 , for i = 0 , . . . , N − (cid:98) s to the N -th order is thesame as finding a section in X N +1 intersecting the fiber over c at E N +1 , orequivalently, a section in X N which intersects the exceptional divisor E N atthe point r N = (cid:98) s N ( c ) (Proposition 11, [HT06]).3. Standard models of cubic surfaces over a Dedekind domain
Standard models.
Corti [Cor96] developed a theory of standard mod-els of cubic surfaces over Dedekind domains. Let C be a smooth projectiveconnected curve and p a point in C . Denote by O the spectrum of O C,p or (cid:98) O C,p and K the quotient field of O C,p or (cid:98) O C,p . Let X K be a cubic surfacedefined over K . A model of X K over O is a flat projective family X O suchthat the generic fiber is X K . Definition 3.1.
A standard model of X K over O is a model X O over O such that(1) X O has terminal singularities of index 1.(2) The central fiber X is reduced and irreducible.(3) The anticanonical system − K X O is very ample and defines an em-bedding X O ⊂ P O .The main theorem of [Cor96] is the following. Theorem 3.2. [Cor96]
A standard model exists over O . Gluing local models together one gets a standard model over the curve C .Note that in dimension 3 terminal singularities of index 1 are isolated andhave multiplicity 2.For the singularities of the central fiber, we have the following, proved in[BW79]. Lemma 3.3 ( [BW79]) . An integral cubic surface is either a cubic surfacewith du Val (=ADE) singularities, or a cone over an irreducible, possiblysingular, plane cubic curve or a non-normal surface with only multiplicity singularities (along a line). To prove this lemma, one can look at the cases of singular cubic surfaceslisted in [BW79]. Note that an integral cubic surface is normal if and onlyif it has isolated singularities. In the list of [BW79], classes (A)-(C) corre-sponds to cubic surfaces with du Val singularities, class (D) the cone over asmooth plance cubic, class (E) the case of a non-normal cubic surface whichis not a cone, and class (F) the cone over a nodal or cuspidal plane cubic.In the following, we will analyze the local structure of the standard modelover the formal neighborhood
X →
Spec C [[ t ]] in the last 2 cases.The main result is the following. EAK APPROXIMATION FOR CUBIC HYPERSURFACES 9
Proposition 3.4.
Let X be a standard model over Spec C [[ t ]] whose centralfiber does not have du Val singularities. Then after a ramified base change t = r l and a birational modification, we get a new family X (cid:48) → Spec C [[ r ]] whose central fiber has du Val singularities. Furthermore, the Galois group G ∼ = Z /l Z acts on the new total space and the projection X (cid:48) → Spec C [[ r ]] is G -equivariant.Moreover, formal sections of the family X contained in the smooth locusinduce G -equivariant formal sections of X (cid:48) → Spec C [[ r ]] contained in thesmooth locus.Finally, given two formal sections of the family X intersecting the centralfiber in the smooth locus, let x, y be the intersection points of the correspond-ing new G -equivariant sections with the new central fiber. Then x, y are fixedpoints (under the G action) in the smooth locus of the central fiber X (cid:48) of thenew family. Moreover, there is a G -equivariant map f : P → X (cid:48) such that f (0) = x, f ( ∞ ) = y and the image of f lies in the smooth locus of X (cid:48) . Wemay also assume that f is very free. The proof of this proposition will be given in the following subsections.The first two consist of explicit computations of the base change and thecorrespondences between the sections. The last one proves the existence ofequivariant very free curves in the smooth locus.3.2.
Base change computation I: Cone over a plane cubic.
In thefollowing two subsections, we will always denote the defining polynomial ofthe family as H ( t, X , X , X , X ), which is a formal power series in t withcoefficients in X , . . . , X . Convention 3.5.
We say that a monomial M ( t, X , . . . , X ) is in the defin-ing polynomial H if after taking the power series expansion of H , it appearsas a monomial in H .Assume the central fiber is a cone over an irreducible plane cubic curve,defined by equation F ( X , X , X ) = 0. Then the total space has multiplic-ity 3 at vertex [1 , , ,
0] unless we have tX , t X , or tX X i , i = 1 , , t = r n for some n and a birational modification so that the central fiber has at worst du Valsingularities.Case (1): If tX is contained in the defining polynomial H , we assignweight (0 , , , ,
1) to ( X , X , X , X , r ). Then the homogeneous polyno-mial F has weight 3. We make a degree 3 base change t = r . Thereis exactly one more monomial with weight 3 in H ( r , X , . . . , X ), namely r X and the other monomials all have weight strictly greater than 3. Soafter a degree 3 base change t = r and change of variables Y = X , rY = X , rY = X , rY = X , the new defining equation has the form r ( Y + F ( Y , Y , Y )) + r ≥ R ( Y , Y , Y , Y , r ) = 0 , or equivalently, Y + F ( Y , Y , Y ) + r ≥ R ( Y , Y , Y , Y , r ) = 0 . The new family X (cid:48) → Spec C [[ r ]] has a Z / Z action on the total spacecompatible with Galois group action on Spec C [[ r ]]. The central fiber of thenew family is Y + F ( Y , Y , Y ) = 0 . Taking partial derivative with respect to Y shows that the only possiblesingularities of the new central fiber lie in the plane Y = 0 and comefrom singularities of the elliptic curve. Such singularities are isolated andhave multiplicity 2, and thus are du Val singularities. Furthermore, thesingularities of X (cid:48) lie in the curve Y = F ( Y , Y , Y ) = 0.Case (2): If tX is not contained in the defining polynomial H , and tX X i (for some i = 1 , ,
3) is contained in H , we again assign weight (0 , , , , X , X , X , X , r ). Then the homogeneous polynomial F has weight 3.One can make a degree 2 base change t = r , and the change of variables Y = X , rY = X , rY = X , rY = X . After a linear change of coordinates, one can write the equation for the newcentral fiber as Y Y + F (cid:48) ( Y , Y , Y ) = 0 , where Y = F (cid:48) ( Y , Y , Y ) = 0 defines an irreducible plane cubic C .The singularities (if they exist) of the new central fiber are defined byequations Y Y = Y + ∂F (cid:48) ∂Y = ∂F (cid:48) ∂Y = ∂F (cid:48) ∂Y = 0 . They are of two kinds. One possible singularity lies in the plane Y = 0 andcomes from singularities of the plane cubic C . The other singularities lie inthe plane Y = 0. If ∂F (cid:48) ∂Y vanishes at the singularities, then so does Y . Thusthese singularities belong to the previous kind. If ∂F (cid:48) ∂Y does not vanish at thesingularities, then Y is non-zero and Y = 0 is tangent to the curve C at asmooth point. There are two singularities of the second kind coming fromtwo solutions of the equation Y + ∂F (cid:48) ∂Y = 0, which satisfies the property Y (cid:54) = 0, and the two singular points are conjugate to each other under the Z / Z action. Again the new central fiber has du Val singularities only.Case (3): If neither tX nor tX X i , i = 1 , , H , then t X has to be contained in H . So one can make a degree 3 base change t = r and change of variables Y = X , r Y = X , r Y = X , r Y = X . Then the central fiber is Y + F ( Y , Y , Y ) = 0 , EAK APPROXIMATION FOR CUBIC HYPERSURFACES 11 which has du Val singularities only.Finally, a formal section (cid:98) t : Spec C [[ t ]] → X induces a G -equivariantformal section (cid:98) r : Spec C [[ r ]] → X (cid:48) . Let [ a, b, c, d ] be the intersection of (cid:98) t with the central fiber. Assume it is a smooth point. So in particular, one of b, c, d is non-zero. Then the induced section (cid:98) r intersects the new central fiberat [0 , b, c, d ], which lies in the plane elliptic curve C = { Y = F ( Y , Y , Y ) =0 } or { Y = F (cid:48) ( Y , Y , Y ) = 0 } . Moreover, the point [0 , b, c, d ] is a smoothpoint of the elliptic curve, otherwise the point [ a, b, c, d ] is a singular pointof the surface F ( X , X , X ) = 0 since it lies in the line spanned by thesingular point in the curve C and the vertex [1 , , , Base change computation II: Non-normal and not a cone.
When the central fiber is non-normal but not a cone, by [BW79], p. 252,case E, the equation of the surface can be uniquely written as X X + X X = 0 , or X X + X X X + X = 0 . In both cases the singular locus is the line X = X = 0. Since thetotal space is smooth along the generic point of the line, we have a term tF ( X , X ) in the defining polynomial H .In the first case, make a degree 2 base change t = r and change ofvariables Y = X , Y = X , rY = X , rY = X , one can get a new family X (cid:48) → Spec C [[ r ]], together with a Z / Z actionon the total space compatible with action r (cid:55)→ − r . The central fiber of thefamily is defined by F ( Y , Y ) + Y Y + Y Y = 0 , which has only du Val singularities. One has a blow-up/blow-down de-scription of this change of variables similar to the previous case. As in thethe previous case, a formal section (cid:98) t induces a G -equivariant formal section (cid:98) r : Spec C [[ r ]] → X (cid:48) . If the original section intersects the central fiber in thesmooth locus (i.e. one of the coordinates X or X is non-zero), then thenew formal section (cid:98) r intersects the new central fiber in the line Y = Y = 0,which lies in the smooth locus.In the second case, we need to make different base changes and list themas follows.(1) If tX is contained in the defining polynomial H , then make a degree6 base change t = r .(2) If tX is not in H and tX X is in H , then make a degree 5 basechange t = r .(3) If neither tX nor tX X is in H but tX X is in H , then make adegree 4 base change t = r . (4) If none of tX , tX X , tX X are in H , and tX X is in H , thenmake a degree 3 base change t = r .(5) If none of tX , tX X , tX X are in H , but tX X is in H , thenmake a degree 4 base change t = r .After the base change, make the following change of variables Y = X , rY = X , r Y = X , r Y = X . After the base change and change of variables, the central fiber has the form Y Y + Y Y Y + Y + G = 0 , where G is one of the polynomials Y , Y Y , aY Y + bY Y , Y + cY Y .This defines a cubic surface with at worst du Val singularities.A formal section (cid:98) t induces a new formal section (cid:98) r : Spec C [[ r ]] → X (cid:48) . Ifthe original section (cid:98) t intersects the central fiber in the smooth locus (i.e.the coordinate X is non-zero), then the new formal section (cid:98) r intersects thenew central fiber at the point [0 , , , Equivariant curves.
We first show the following.
Lemma 3.6.
Let X be a smooth quasi-projective variety with an action ofa finite cyclic group G of order l , and let T be an irreducible component ofthe fixed point loci of G . Assume that there is a very free curve thoroughevery point of X . Fix a G -action on P by [ X , X ] (cid:55)→ [ X , ζX ] , where ζ is a primitive l -th root of unity. Then given any two fixed points x, y in T ,there is a G -equivariant very free curve f : P → X connecting x and y .Proof. Given a point x in T , the constant map P → x is G -equivariant.By assumption, there is a very free rational curve in the smooth locus andpassing through that point x . Then Lemma 2.3, (1), applied to X , showsthat there is a very free G -equivariant rational curve mapping 0 and ∞ to x . We can deform the curve with 0 mapped to x in a G -equivariant way toget a very free G -equivariant map connecting x and a general point in T .To do this, first define two morphisms s : 0 × T ⊂ P × T → X × T (0 , t ) (cid:55)→ ( x, t )and s ∞ : ∞ × T ⊂ P × T → X × T ( ∞ , t ) (cid:55)→ ( t, t ) . Consider the relative Hom-scheme over T fixing the two morphisms s and s ∞ Hom T ( P × T, X × T, s , s ∞ ) . There is a G -action on the relative Hom-scheme. The projection to T is G -equivariant and smooth at the point represented by the very free curve EAK APPROXIMATION FOR CUBIC HYPERSURFACES 13 mapping 0 and ∞ to x ∈ T . So by Corollary 2.2, the map from the Hom-scheme to T is dominant and one can find such a deformation.If y is another point in T , the same construction gives a G -equivariantvery free curve in the smooth locus connecting y and a general point in T .To connect x and y , take a common general point z in T and two very free G -equivariant rational curves connecting x (resp. y ) to z . Then part twoof Lemma 2.3 shows that there is a very free G -equivariant rational curveconnecting x and y . (cid:3) By the description of the base change, and how the sections correspond toeach other, the fixed points we need to connect in Proposition 3.4 are con-tained in the smooth locus of the central fiber and lie in a single irreduciblecomponent of the fixed point loci. Furthermore by [Xu12a], or by Theorem21, [HT08], all cubic surfaces with at worst du Val singularities satisfy thecondition that for any point in the smooth locus of the cubic surface, thereis a very free rational curve in the smooth locus and passing through thatpoint. Thus the last statement in Proposition 3.4 follows from the abovelemma. 4.
Approximation in the smooth locus
This section is devoted to a special case of weak approximation.4.1.
Finding G -equivariant sections. We first develop the techniques in[HT06] in a G -equivariant setting. Theorem 4.1.
Let G be a cyclic group of order l and let X (resp. C ) bea smooth proper variety (resp. a smooth projective curve) with a G -action.Let π : X → C be a flat family of rationally connected varieties. Assumethe following: (1) The morphism π is G -equivariant. (2) There is a G -equivariant section s : C → X . (3) The G -action on C has a fixed point p and the action of G near p is given by t (cid:55)→ ζt , where t is a local parameter and ζ is a primitive l -th root of unity. (4) The fiber of π : X → C over the point p is smooth.Then for any positive integer N , and any G -equivariant formal section (cid:98) s : Spec (cid:98) O p,C → X , there is a G -equivariant section s (cid:48) which agrees with theformal section (cid:98) s to order N . The idea of the proof goes back to [HT06]. Namely, we would like toadd suitable rational curves to the given section and make G -equivariantdeformations to produce a new section with prescribed jet data. The onlysubtlety in the proof is that in general we cannot choose the rational curvesto be immersed. So instead of working with the normal sheaf as is done in[HT06], we work with the complex Ω f defined as − f ∗ Ω X df † −−−−→ Ω C . and its derived dual in the derived category. All the tensor products, du-als, pull-backs, and push-forwards in the proof should also be taken as thederived functors in the derived category.The following is a general form of the commonly used short exact se-quences (of normal sheaves) which govern the deformation of a stable mapfrom a nodal domain. Lemma 4.2.
Let f : C ∪ D → X be a morphism from a nodal curve C ∪ D with a single node to a smooth variety X and f (resp. f ) the restrictionof f to C (resp. D ). Then (1) We have the following distinguished triangles: Ω ∨ f ⊗ O D ( − n ) → Ω ∨ f → Ω ∨ f ⊗ O C → Ω ∨ f ⊗ O D ( − n )[1]Ω ∨ f → Ω ∨ f ⊗ O C → (cid:15) [ − → Ω ∨ f [1] where n is the preimage of the node in D , and (cid:15) is a skyscraper sheafsupported at the preimage of the node in C . (2) Let G be a cyclic group of order l . Assume that there is a G -actionon C ∪ D fixing each irreducible component. Then the node is afixed point of the action and there is a natural G -action on all thecomplexes above. If locally around the node, the action is given by C [ x, y ] /xy −−−−→ C [ x, y ] /xy ( x, y ) −−−−→ ( ζx, ζ − y ) , where ζ is a primitive l -th roots of unity, then the G -action on (cid:15) istrivial.Proof. The first distinguished triangle comes from restriction to the compo-nent C .For the second distinguished triangle, consider the following distinguishedtriangles and the map between them:Ω C ∪ D ⊗ O C −−−−→ Ω f ⊗ O C −−−−→ f ∗ Ω X ⊗ O C [1] −−−−→ Ω C ∪ D ⊗ O C [1] (cid:121) (cid:121) (cid:13)(cid:13)(cid:13) (cid:121) Ω C −−−−→ Ω f −−−−→ f ∗ Ω X [1] −−−−→ Ω C [1]Therefore we have distinguished trianglesΩ C ∪ D ⊗ O C → Ω C → Q [1] → Ω C ∪ D ⊗ O C [1](1) Ω f ⊗ O C → Ω f → Q (cid:48) [1] → Ω f ⊗ O C [1] ,Q [1] → Q (cid:48) [1] → → Q [2] . EAK APPROXIMATION FOR CUBIC HYPERSURFACES 15 where Q is a skyscraper sheaf supported at the node. The last distinguishedtriangle shows that Q ∼ = Q (cid:48) . Taking dual of the distinguished triangle (1)gives the second triangle in the lemma.Part 2 of the lemma can be proved by a local computation. Or we canargue that the sheaf (cid:15) corresponds to a G -equivariant smoothing of the node.Therefore it has to be G -invariant. (cid:3) Now we begin the proof.
Proof of Theorem 4.1.
The proof is divided into two steps.
Step 1: Approximation at -th order. We may assume that H ( C, N C/ X ( − p )) = 0by the same argument as in Lemma 2.3.The section s and the formal section (cid:98) s intersect the fiber X p at two fixedpoints of the G -action. Take a rational curve D ∼ = P with a G -action as[ X , X ] (cid:55)→ [ ζX , X ], where ζ is the primitive l -th root of unity in theassumptions. By Theorem 2.4, there is a G -equivariant very free curve D ∼ = P → X p → X which maps 0 = [1 ,
0] to s ( p ) and ∞ = [0 ,
1] to (cid:98) s ( p ).Let f : C ∪ D → X be the nodal curve by combining the section and thecurve D and f (resp. f ) the restriction of f to C (resp. D ).By Lemma 4.2, we have the following distinguished triangles:Ω ∨ f ( −∞ ) ⊗ O C ( − p ) → Ω f ∨ ( −∞ ) → Ω f ∨ ( −∞ ) ⊗ O D → Ω ∨ f ⊗ O C ( − p )[1]Ω ∨ f ( − p ) → Ω ∨ f ⊗ O C ( − p ) → (cid:15) [ − → Ω ∨ f ( − p )[1]Ω ∨ f ⊗ O D ( −∞ ) → Ω ∨ f ⊗ O D ( −∞ ) → (cid:15) (cid:48) [ − → Ω ∨ f ⊗ O D ( −∞ )[1]where (cid:15) and (cid:15) (cid:48) are torsion sheaves supported at the node of C and D . Everycomplex has a natural G -action, and the G -actions on (cid:15) and (cid:15) (cid:48) are trivial.Also note that Ω f ( −∞ ) ⊗ O C ∼ = Ω f ⊗ O C . Taking hypercohomology gives long exact sequences0 → H (Ω ∨ f ⊗ O C ( − p )) → H (Ω ∨ f ( −∞ )) → H (Ω ∨ f ( −∞ ) ⊗ O D )(2) → H (Ω ∨ f ⊗ O C ( − p )) → H (Ω ∨ f ( −∞ )) → H (Ω ∨ f ( −∞ ) ⊗ O D ) → . . . , → H (Ω ∨ f ⊗ O C ( − p )) → H (Ω ∨ f ⊗ O C ( − p )) → (cid:15) (3) → H (Ω ∨ f ⊗ O C ( − p )) → H (Ω ∨ f ⊗ O C ( − p )) → , and 0 → H (Ω ∨ f ⊗ O D ( −∞ )) → H (Ω ∨ f ⊗ O D ( −∞ )) → (cid:15) (cid:48) (4) → H (Ω ∨ f ⊗ O D ( −∞ )) → H (Ω ∨ f ⊗ O D ( −∞ )) → . Note that Ω ∨ f is quasi-isomorphic to N C/ X [ − H (Ω ∨ f ⊗ O C ( − p )) = H (Ω ∨ f ⊗ O C ( − p )) = 0 . Note that Ω ∨ f is quasi-isomorphic to a shifted sheaf N [ − N isdefined as the quotient in0 → T D → f ∗ T X → N ∼ = f ∗ T X /T D → . Since f ∗ T X is globally generated, H (Ω ∨ f ⊗ O D ( −∞ )) = 0. Then by thethird long exact sequence, H (Ω ∨ f ⊗ O D ( −∞ )) = 0 . Therefore by the long exact sequence (2), H (Ω ∨ f ( −∞ )) = 0 , and thus the G -equivariant deformation of the nodal curve C ∪ D with thepoint ∞ fixed is unobstructed.Then by the long exact sequences (2), (4) and the vanishing, the compo-sition of maps H (Ω ∨ f ( −∞ )) G → H (Ω ∨ f ( −∞ ) ⊗ O D ) G → (cid:15) (cid:48) is surjective. Thus there is a G -equivariant deformation with ∞ fixed whichsmooths the node between C and D . Step 2: Approximation at higher order.
Assume that we have a section, still denoted by s , which agrees with (cid:98) s to the k ( ≥ (cid:98) s to order k + 1.Now let X k +1 be the ( k + 1)-th iterated blow-up of X associated to theformal section (cid:98) s . Then G also acts on X k +1 and the projective to C is G -equivariant. By abuse of notations, still denote the strict transforms of s and (cid:98) s by s and (cid:98) s . Then they both intersect the exceptional divisor E k +1 ∼ = P d at fixed points of G . Assume the intersection points are different, otherwisethere is nothing to prove.Again we assume that H ( C, N C/ X k +1 ( − p )) = 0.The key lemma is the following. Lemma 4.3.
There is a comb f : C ∪ D → X k +1 from a nodal domainconsisting of the given section s ( C ) and suitable rational curves in the fibersuch that • D = D k +1 ∪ ∪ lj =1 R j , where D k +1 ∼ = P and R j = ∪ ki =1 D ij is a chainof rational curves. Denote by x j the node that connects D k +1 to R j . • There is a G -action on D in the following way. The G -action on D k +1 is given by [ X , X ] (cid:55)→ [ X , ζ − X ] . EAK APPROXIMATION FOR CUBIC HYPERSURFACES 17
The group G acts on R j , j = 1 , . . . , l via a cyclic permutation amongthem. In particular, the points x j ∈ D k +1 are conjugate to eachother under the G -action. • The morphism f : C ∪ D → X is G -equivariant. • The G -fixed point ∞ = [0 , on D k +1 is mapped to (cid:98) s ( p ) , and , on D k +1 connects C . • The morphism f : C ∪ D is an immersion except at and ∞ in D k +1 . • The complex Ω ∨ f satisfies the following vanishing conditions. (5) H (Ω ∨ f ⊗ O C ( − p )) = H (Ω ∨ f ⊗ O D k +1 ( − − ∞ )) = H (Ω ∨ f ⊗ O D ij ( − , (6) H (Ω ∨ f ( −∞ )) = 0 , (7) H (Ω ∨ f ⊗ O D k +1 ( −∞ − x − . . . − x l )) G = 0 . The construction is essentially the same as the one in [HT06], with theonly difference coming from the consideration of the G -action. For an illus-tration of the comb C ∪ D , see Figure. 1 below and for the configuration ofthe comb with respect to the iterated blow-up X k +1 , see Figure. 2. Proof of Lemma 4.3.
The line L in E k +1 ∼ = P d joining s ( p ) and (cid:98) s ( p ) is in-variant and intersects the exceptional divisor E k of X k +1 at a unique point y k , which is necessarily a fixed point of G . Then there are 3 fixed points inthe line L and thus all points are fixed points of G . Take a curve D k +1 ∼ = P .We impose a G -action on it by[ X , X ] (cid:55)→ [ X , ζ − X ] . Take an l -to-1 G -equivariant map from D k +1 to the line L such that 0 = [1 , s ( p ) and ∞ = [0 ,
1] is mapped to (cid:98) s ( p ). There are l points x , . . . , x l , which lie in the same orbit of G , being mapped to the point y k ∈ E k ∩ E k +1 , where E k and E k +1 are exceptional divisors of the ( k + 1)-th iterated blow-up.The exceptional divisor E k is isomorphic to the blow-up of P d at a point,thus is a P -bundle over P d − . Let D k, , . . . , D k,l be l copies of P eachmapped isomorphically to the fiber curve P containing the point y k .Inductively, let y i be the intersection point of D i +1 , with E i and D i, , . . . , D i,l be l copies of P each mapped isomorphically to the fiber P containing thepoint y i for all i = k − , . . . , y be the point of the intersection of D , with the stricttransform of X | p and let D , . . . , D ,l be l copies of P mapped to a veryfree curve in the strict transform of X p intersecting E at the point y . Wemay also assume that the maps are immersions. D k +1 R x . . . . . .R l D k x l D k − , D kl D k − ,l . . . . . .. . . . . .C Figure 1.
The comb C ∪ D Let R j be the chain of rational curves ∪ ki =1 D i,j connected to D k +1 at thepoint x j for j = 1 , . . . , l , and let D be the curve D k +1 ∪ ∪ lj =1 R j . There isa natural G -action on D , which permutes the l -chains of rational curves R j and acts on the irreducible component D k +1 as specified above.The restriction of the complex Ω ∨ f to each curve D i,j is quasi-isomorphicto the normal sheaf with a shift N f [ −
1] (since the comb is an immersionalong such curves). One can compute the restriction of N f to each curve D i,j as follows (see the proof of Sublemma 27, [HT06]).(8) N f | D i,j = (cid:40) O ⊕ d , ≤ i ≤ k, ⊕ d − n =1 O ( a n ) ⊕ O , a n ≥ i = 0 . We now compute Ω ∨ f k +1 on D k +1 , where f k +1 is the restriction of the mapto D k +1 (i.e. the degree l multiple cover of the line in P d − ). This complexis quasi-isomorphic to the complex0 1 T D k +1 ∼ = O (2) −−−−→ f ∗ k +1 T X k +1 ∼ = O (2 l ) ⊕ ⊕ d − i =1 O ( l ) ⊕ O ( − l ) , Also note that the sheaf map T D k +1 → f ∗ k +1 T X k +1 is injective and is thecomposition of maps O (2) → O (2 l ) → f ∗ k +1 T X k +1 . EAK APPROXIMATION FOR CUBIC HYPERSURFACES 19 E k +1 ∼ = P d D k +1 l :1 −→ L (cid:98) s ( p ) S ( p ) y k y k − D k . . . D kl are mappedto this fiber E k · · · y D . . . D l are mappedto this fiber E y D · · · D l E P S ( C ) C Figure 2.
Construction of the comb C ∪ D We have a distinguished triangle(9) Ω ∨ f k +1 → Ω ∨ f ⊗ O D k +1 → (cid:15) [ − ⊕ ⊕ lj =1 (cid:15) j [ − → Ω ∨ f k +1 [1] , where (cid:15) is a torsion sheaf supported at the node connecting D k +1 and C ,and (cid:15) j is a torsion sheaf supported at the node connecting D k +1 and D k,j .The group G acts on (cid:15) by the trivial action and acts on (cid:15) j by permutation.So the restriction of Ω ∨ f to D k +1 is quasi-isomorphic to the complex0 1 O (2) −−−−→ O (2 l ) ⊕ ⊕ d − i =1 O ( l ) ⊕ O (1) . Since the above map maps the sheaf O (2) injectively into the sheaf O (2 l ),this complex is quasi-isomorphic to the shifted sheaf Q ⊕ ⊕ d − i =1 O ( l ) ⊕ O (1)[ − , where Q is the torsion sheaf defined as the quotient of O (2) → O (2 l ). Notethat the O (1) direction is the normal direction of the fiber.Finally, the restriction of Ω ∨ f to C fits into the distinguished triangle N C/ X k +1 [ − → Ω ∨ f ⊗ O C → (cid:15) → N C/ X k +1 , where (cid:15) is a torsion sheaf supported at the node. Then the vanishing conditions (5) are immediate from the identificationsabove.By the distinguished triangleΩ ∨ f ⊗ O C ( − p ) → Ω ∨ f ( −∞ ) → Ω ∨ f ⊗ O D ( −∞ ) → Ω ∨ f ⊗ O C ( − p )[1]and the three vanishing results in 5, we know that H (Ω ∨ f ( −∞ )) = 0 . This is the vanishing in (6).The vanishing in (7) needs a little bit more work since it is only the G -invariant part of the hypercohomology group that vanishes. First notice thefollowing. Lemma 4.4.
Assume only that the comb C ∪ D satisfies vanishing results(5) and (6). Then a general G -equivariant deformation of C ∪ D with ∞ fixed is unobstructed and smooths the node connecting C and D k +1 .Proof. The vanishing result (6) implies that the G -equivariant deformationof C ∪ D with ∞ fixed is unobstructed.We first consider the following distinguished triangles(10)Ω ∨ f ⊗ O D ( −∞ − → Ω ∨ f ( −∞ ) → Ω ∨ f ( −∞ ) ⊗ O C → Ω ∨ f ⊗ O D ( −∞ − ⊕ lj =1 Ω ∨ f ⊗ O R j ( − x j ) → Ω ∨ f ⊗ O D ( −∞ − → Ω ∨ f ⊗ O D k +1 ( −∞ − → ⊕ lj =1 Ω ∨ f ⊗ O R j ( − x j )[1] . Recall that Ω ∨ f ⊗ O R j is quasi-isomorphic to a shifted normal sheaf N f ⊗O R j [ − N ⊗ O R j are locally free and globally generatedby (8). Therefore H ( ⊕ lj =1 Ω ∨ f ⊗ O R j ( − x j )) = 0 , and thus by the distinguished triangle (11), H (Ω ∨ f ⊗ O D ( −∞ − , which, combined with the long exact sequence of hypercohomology of thedistinguished triangle (10), implies that the map(12) H (Ω ∨ f ( −∞ )) G → H (Ω ∨ f ( −∞ ) ⊗ O C ) G is surjective.Then we look at the distinguished triangleΩ ∨ f → Ω ∨ f ⊗ O C → (cid:15) [ − → Ω ∨ f [1] , where f is the restriction of f to C and (cid:15) is a skyscraper sheaf supportedat the point p .By the vanishing results (5), the map(13) H (Ω ∨ f ( −∞ ) ⊗ O C ) G → ( (cid:15) ) G = (cid:15) is surjective.Note that Ω ∨ f ⊗ O C ∼ = Ω ∨ f ( −∞ ) ⊗ O C . Combining this identification andthe surjectivity of maps in (13) and (12), we have proved that a general G -equivariant deformation with ∞ fixed smooths the node connecting C and D k +1 . (cid:3) We have a distinguished triangleΩ ∨ f k +1 ( −∞ ) → Ω ∨ f ⊗ O D k +1 ( −∞ ) → (cid:15) [ − ⊕ ⊕ lj =1 (cid:15) j [ − → Ω ∨ f k +1 ( −∞ )[1] , where (cid:15) is a torsion sheaf supported at 0 ∈ D k +1 . This induces a map(14) H (Ω ∨ f ⊗ O D k +1 ( −∞ )) G → (cid:15) By Lemma 4.4, a general deformation of C ∪ D with ∞ fixed is unobstructedand smooths the node connecting C and D k +1 (note that the proof of thisresult is independent of the vanishing (7)). Thus the composition H (Ω ∨ f ( −∞ ) → H (Ω ∨ f ⊗ O D k +1 ( −∞ )) G → (cid:15) is surjective. So the map in (14) is also surjective.Recall that Ω ∨ f ⊗ O D k +1 ( −∞ ) is quasi-isomorphic to the shifted sheaf( Q ⊕ ⊕ d − i =1 O ( l ) ⊕ O (1)) ⊗ O D k +1 ( −∞ )[ − , and the O (1) direction is the normal direction of the fiber.Moreover the map in (9) is can be written as Q ⊕ ⊕ d − i =1 O ( l ) ⊕ O ( − l )[ − → Q ⊕ ⊕ d − i =1 O ( l ) ⊕ O (1)[ − → (cid:15) [ − ⊕ ⊕ lj =1 (cid:15) j [ − → Q ⊕ ⊕ d − i =1 O ( l ) ⊕ O ( − l )Thus only the O (1) ⊗ O D k +1 ( −∞ ) summand may have a non-zero map to (cid:15) in the above evaluation map in (14). Thus the unique section in this sum-mand (i.e. the section of H ( O (1) ⊗ O D k +1 ( −∞ )) = H ( O D k +1 )) is mappedto a non-zero element in (cid:15) . Furthermore, this unique section, thought of asa section in H (Ω ∨ f ⊗ O D k +1 ) G via the inclusion H (Ω ∨ f ⊗ O D k +1 ( −∞ )) G → H (Ω ∨ f ⊗ O D k +1 ) G only vanishes at ∞ ∈ D k +1 . Therefore the map(15) H ( O (1) ⊗ O D k +1 ( −∞ )) G → ( ⊕ lj =1 (cid:15) j ) G is surjective.To prove the vanishing in (7), we only need to consider the O (1) sum-mand since all the other summands have enough positivity to kill the highercohomology H . For the O (1) summand, consider the short exact sequence0 → O (1) ⊗O D k +1 ( −∞− x − . . . − x l ) → O (1) ⊗O D k +1 ( −∞ ) → ⊕ lj =1 (cid:15) j → , which induces a map on the G -invariant part of cohomology H ( O (1) ⊗ O D k +1 ( −∞ )) G → ( ⊕ lj =1 (cid:15) j ) G → H ( O (1) ⊗ O D k +1 ( −∞ − x − . . . − x l )) G → H ( O (1) ⊗ O D k +1 ( −∞ )) G . Since the map (15) is surjective and H ( O (1) ⊗ O D k +1 ( −∞ )) G vanishes, wehave H ( O (1) ⊗ O D k +1 ( −∞ − x − . . . − x l )) G = 0 , and thus H (Ω ∨ f ⊗ O D k +1 ( −∞ − x − . . . − x l )) G = 0 . (cid:3) We now finish the proof of step 2. Consider the distinguished trianglesΩ ∨ f ( −∞ ) ⊗ O C ( − p ) → Ω ∨ f ( −∞ ) → Ω ∨ f ( −∞ ) ⊗ O D → Ω ∨ f ⊗ O C ( − p )[1]Ω ∨ f ⊗ O D k +1 ( −∞ − x − . . . − x l ) → Ω ∨ f ⊗ O D ( −∞ ) → ⊕ lj =1 Ω ∨ f ⊗ O R j → Ω ∨ f ⊗ O D k +1 ( −∞ − x − . . . − x l )[1] . The vanishings in (5), (7) imply that the map(16) H (Ω ∨ f ( −∞ )) G → H (Ω ∨ f ⊗ O D ( −∞ )) G → H ( ⊕ lj =1 Ω ∨ f ⊗ O R j ) G is surjective (note that Ω ∨ f ⊗ O R j ∼ = Ω ∨ f ( −∞ ) ⊗ O R j ).Since the G -action on the chain of rational curves R j is permutation.There is a section of H ( ⊕ lj =1 Ω ∨ f ⊗ O R j ) G which is mapped to a non-zero element in the G -invariant part of the torsionsheaf supported at the nodes on R j , j = 1 , . . . , l if and only if there is asection of H (Ω ∨ f ⊗ O R j )which is mapped to a non-zero element in the torsion sheaf supported at thenodes on R j and for some (and hence for all) j .Since the restriction of Ω ∨ f to R j is quasi-isomorphic to N f ⊗ O R j [ −
1] and N f ⊗ O R j is locally free and globally generated by the vanishing (5) or (8),this follows from the same argument as in [HT06] (in particular, the bottomof P. 187 and P. 188).So combining this observation with the surjectivity of the map in (16)and Lemma 4.4, we have proved that a general G -equivariant deformationwith ∞ fixed smooths all the nodes and produces a new section which agreeswith (cid:98) s to order k + 1. (cid:3) EAK APPROXIMATION FOR CUBIC HYPERSURFACES 23
Weak approximation in the smooth locus.
The following is aspecial case of weak approximation, which turns out to be all one needsto finish the proof. The basic idea is that when the central fiber is verysingular, a base change and a birational modification will greatly improvethe singularities. Then one just need to keep track of the Galois group actionto get back to the original family.
Lemma 4.5.
Let π : X → B be a standard model of families of cubicsurfaces over a smooth projective curve B and s : B → X be a section. Let b , . . . , b k , b k +1 , . . . , b m be finitely many points in B , and (cid:98) s j , k + 1 ≤ j ≤ m be formal sections over the points b j , k + 1 ≤ j ≤ m , which lie in the smoothlocus of π : X → B . Assume that the section s intersects the fibers over b k +1 , . . . , b m in the smooth locus. Then given a positive integer N , there isa section s (cid:48) : B → X such that s (cid:48) is congruent to s modulo m NB,b i for all ≤ i ≤ k , and congruent to (cid:98) s j modulo m NB,b j , for all k + 1 ≤ j ≤ m .Proof. By the iterated blow-up construction, keeping the jet data is thesame as keeping section intersect certain exceptional divisors in the iteratedblow-up, which, in turn, is equivalent to keeping the intersection numbersof the section with exceptional divisors in the iterated blow-up. In thefollowing we will only use deformation/smoothing argument and we willonly use a general deformation (i.e. without specialization) to prove theweak approximation result. Thus the intersection numbers are always kept.Since we start with a section which intersects the fibers over given points inthe smooth locus, the section we produce by adding curves in the smoothlocus of π and smoothing also intersect the fibers over the given points inthe smooth locus. Therefore we can reduce the general case to the case that b k +1 , . . . , b m is just a single point b ∈ B .First of all we may approximate the formal section if the fiber over b hasat worst du Val singularities. Indeed the section s and the formal sectionwe need to approximate lie in the smooth locus over b by assumption. Sothe statement follows from Theorem 1, [HT08] and the fact that the smoothlocus of log del Pezzo surfaces are strongly rationally connected [Xu12a] (orTheorem 21, [HT08] for the case of cubic surfaces).From now on we assume the fiber over the point b is either a cone overa plane cubic curve or non-normal. By Proposition 3.4, at least for theformal neighborhood, we can find a ramified base change and a birationalmodification so that the new central fibers have du Val singularities onlyand the Galois group acts on the total space of the formal neighborhood.The next goal is to show that we can make the base change globally onthe curve B .Given finitely many points x , x , . . . , x n in B , and any positive integer l , there is a cyclic cover of degree l of B which is totally ramified over x , . . . , x n (and other points). To see this, take a general Lefschetz pencilwhich maps x , . . . , x n (and other points) to 0 ∈ P and is unramified overthese points. Take a degree l map B = P → P , [ X , X ] (cid:55)→ [ X l , X l ] and let C = B × P B be the fiber product. Then C is the desired cyclic cover.We may also choose the cover C → B so that the preimages of b , . . . , b k are l distinct points.Let C → B be a cyclic cover of degree l (which is determined in subsec-tions 3.2 and 3.3 according to the type of singularities), totally ramified atthe point b (and other points). There is a new family over the curve C bybase change. The cyclic group G = Z /l Z acts on C and the total space of thenew family over C in such a way that the projection to C is G -equivariant.Let c be the points in C which is mapped to b . One can modify the familylocally around c as in subsections 3.2 and 3.3.Let X (cid:48) → C be the family after the base change and birational modifica-tions. The group G still acts on the total space X (cid:48) and the projection to C is G -equivariant.The section s induces a G -equivariant section of the new family X (cid:48) → C and has the desired jet data at all points mapped to b , . . . , b k . Still denotethe new section by s . Moreover, the new G -equivariant section s intersectsthe fiber over the point c in the smooth locus (Proposition 3.4). So do thenew formal sections we want to approximate.Now the argument in Theorem 4.1 proves weak approximation in thiscase. The theorem needs the assumption that the fiber over c is smooth.But this can be weakened as the following: • the section and the formal sections intersect the fibers over c in thesmooth locus, and • there are G -equivariant very free curves in the smooth locus connect-ing the intersection points of the central fiber with the G -equivariantsection and formal sections over the points c .The second condition is proved in Proposition 3.4. (cid:3) Proof of the main theorem
We first show that there are “nice” sections for a standard model of afamilies of cubic surfaces. The idea goes back to an argument of Keel-M c Kernan [KM99] Sec. 5, in particular, the proof of Corollary 5.6. Hassett-Tschinkel also used the idea of Keel-M c Kernan to study strong rationalconnectedness in [HT09], which is very similar to the argument presentedhere.
Lemma 5.1.
Let π : X → B be a standard model of family of cubic surfacesover a smooth projective curve B and s : B → X be a section. Given finitelymany points b , . . . , b k in B , and a positive integer N , there is a section s (cid:48) : B → X such that s (cid:48) is congruent to s modulo m NB,b i and s (cid:48) ( B − ∪ b i ) liesin the smooth locus of π : X → B .Proof. One first resolves the singularities of X along the fibers over b i insuch a way that the partial resolution is an isomorphism except along thesefibers. Then use the iterated blow-up construction according to the jet data EAK APPROXIMATION FOR CUBIC HYPERSURFACES 25 of s near the points b i . After sufficiently many iterated blow-ups, fixing thejet data is the same as passing through fixed components. Call the newspace X .Then the lemma is reduced to showing that there is a section of thenew family X → B which has desired intersection number with irreduciblecomponents of the fibers over b , . . . , b k in B and lies in the smooth locus of X → B .In the following proof, we will show that the given section s , after addingvery free rational curves in general fibers, deforms away from the singularlocus of the total space. Since the deformation will not change the inter-section numbers with divisors, we get a deformation into the smooth locuswith the jet data fixed.Take a resolution of singularities X → X which is an isomorphism overthe smooth locus such that the exceptional locus in X consists of simplenormal crossing divisors E i , i = 1 , . . . , n . After adding very free curves ingeneral fibers and smoothing, we may assume the strict transform of thesection s , denoted by f : B → X , passes through g + 1 very general points p , . . . , p g +1 in X , where g is the genus of B .First consider the Kontsevich moduli space of stable maps M g,g +1 ( X )parameterizing stable maps from genus g curves with g + 1 marked pointsto X , which pass through g + 1 very general points p , . . . , p g +1 in X . Let V be an irreducible component containing the point represented by the map f : B → X .Next consider the Kontsevich moduli space of stable maps M g,g +1 ( X )parameterizing stable maps from genus g curves with g + 1 marked points to X , which pass through p , . . . , p g +1 . There is a natural forgetful map from M g,g +1 ( X ) to the corresponding moduli space of stable maps to X . Take U to be the inverse image of V and write the restriction of the forgetful mapas F : U → V .Note that U has at most countably many irreducible components. Clearlythe forgetful map F surjects onto an open dense subset of V since we canalways lift a section from X to X , with all the conditions still satisfied.Thus there is an irreducible component U of U which dominates V (herewe are using the fact that C is uncountable). By the following lemma (to beproved later), dim U = − K X · D − g + 1), where f (cid:48) : D → X is generalpoint (hence D is irreducible) in U . Lemma 5.2.
Let X be a smooth -fold. Assume that given g +1 very generalpoints x , . . . , x g +1 in X , there is an embedding f : C → X of a smoothprojective curve of genus g in X which maps g + 1 points c , . . . , c g +1 in C to x , . . . , x g +1 . Let f (cid:48) : ( D, d , . . . , d g +1 ) → ( X, x , . . . , x g +1 ) be a generaldeformation of C . Then H ( D, N D/X ( − d − . . . − d g +1 )) = 0 . In particular,every irreducible component containing the morphism f : ( C, c , . . . c g +1 ) → ( X, x , . . . , x g +1 ) has dimension − K X · C − g + 1) . The standard model has isolated cDu Val singularities, i.e. 3-fold terminaland local complete intersection singularities. So does the new total space X by construction. Therefore every irreducible component containing thepoint f (cid:48) : D → X → X has dimension at least − K X − g + 1) since X has local complete intersection singularities. Furthermore, by definition ofterminal singularities, we have − K X = − K X + n (cid:88) i =1 a i E i , a i > . Thus if the image of D in X intersects the singular locus, − K X · D is strictlylarger than − K X , which is impossible.Therefore we have a section s (cid:48) : B → X which has the desired intersectionnumbers and lies in the smooth locus of the total space X . Finally notethat if a section lies in the smooth locus of the total space X , then thesection lies in the smooth locus of the morphism π : X → B . (cid:3) Proof of the lemma 5.2.
Let M be an irreducible component of the Kontse-vich moduli space of genus g stable maps with g + 1 marked points to X containing f : C → X . Then the evaluation map ev : M → X × . . . × X (cid:124) (cid:123)(cid:122) (cid:125) g +1 is dominant (here we use the fact that C is uncountable). Let f (cid:48) : ( D, d , . . . , d g +1 ) → X be a general point in the moduli space M . Then D is also embedded andone can fix g ( ≥
0) general points in the curve D and deform the curve alongthe normal direction at a general point. This implies that we have an exactsequence of sheaves: H ( D, N D/X ( − d − . . . − d g )) ⊗ O D → N D/X ( − d − . . . − d g ) → Q → , where Q is a torsion sheaf on D and d , . . . , d g are general points in D .It follows from the exact sequence that H ( D, N D/X ) = 0 since a generaldegree g line bundle has no H .We also have short exact sequences0 → N D/X ( − d − . . . − d k +1 ) → N D/X ( − d − . . . − d k ) →N D/X ( − d − . . . − d k ) | d k +1 → , for all k = 0 , . . . , g .Again if the points d i are general, then the maps on global sections H ( D, N D/X ( − d − . . . − d k )) → N D/X ( − d − . . . − d k ) | d k +1 , ≤ k ≤ g EAK APPROXIMATION FOR CUBIC HYPERSURFACES 27 are surjective. Thus we have H ( D, N D/X ) = H ( D, N D/X ( − d )) = . . . = H ( D, N D/X ( − d − . . . − d g +1 ) = 0 . (cid:3) Now we have all the results needed for the proof of Theorem 1.2.
Proof of Theorem 1.2.
Given a smooth cubic hypersurface X over the func-tion field C ( B ) of a smooth projective curve B , one can find a Lefschetzpencil over P C ( B ) whose general fiber is a smooth cubic hypersurface of onedimension lower. If weak approximation holds for general fibers, then weakapproximation holds for the total family (c.f. Theorem 3.1, [Has10]). Thusit suffices to prove weak approximation for all smooth cubic surfaces.Let X → B be a standard model for a smooth cubic surface defined overthe function field C ( B ).By Lemma 5.1, one may choose a section s of π : X → B which lies in thesmooth locus of π . Given a finite number of formal sections (cid:98) s i , ≤ i ≤ m over b i ∈ B, ≤ i ≤ m , let (cid:98) L i be the line in the formal neighborhood X (cid:98) O B,bi which joins the formal sections (cid:98) s i and s . Let (cid:98) s (cid:48) i be the third intersectionpoints of (cid:98) L i with the formal neighborhood. Up to perturbing the formalsections (cid:98) s i , we may assume the three local formal sections are not the same.Choose an integer N large enough. We claim that there is a line L definedover the field C ( B ), which contains the rational point p corresponding to s , intersects the generic fiber X η at a cycle of degree 3, and agrees with (cid:98) L i to order N . This is equivalent to weak approximation for the space oflines through the rational point p . Since the space is isomorphic to P , weakapproximation holds.If the line L intersects the generic fiber at two other rational points, thenwe connect the two sections corresponding to the two rational points withrational curves in general fibers and smooth them with the jet data fixed.So we may assume that there is a degree 2 multisection C → X such thatthe formal sections induced by C agree with (cid:98) s i and (cid:98) s (cid:48) i to order N . See Figure3 for an illustration of the situation (we draw the section s and its restrictionto the formal neighborhood differently so that it is easier to visualize).Make the base change C → B , which is et´ale over the points b i . Denotethe preimages of b i to be c i and c (cid:48) i . The formal sections (cid:98) s i and (cid:98) s (cid:48) i induceformal sections over c i and c (cid:48) i , which will still be denoted by (cid:98) s i and (cid:98) s (cid:48) i . Thesection s also induces a section of the new family, still denoted by s . Thedegree 2 multi-section C induces another section s C of the new family, whichagrees with (cid:98) s (cid:48) i to order N .We may arrange the degree 2 cover C → B to be ´etale over all thepoints whose fibers are singular. So the ´etale neighborhood of singularfibers of the new family over C is isomorphic to the ´etale neighborhood X S | Spec (cid:98) O b ,B (cid:98) S (cid:48) (cid:98) S Lb S ( B ) CB Figure 3.
Producing the multisection C of the corresponding singular fibers over B . As a consequence, the familyover C is a standard model over C in the sense of 3.1.By Lemma 5.1, there is a section ˜ s C of the family over C , which agreeswith (cid:98) s (cid:48) i to order N and otherwise lies in the smooth locus of the fibration.Then by Lemma 4.5, there is a section σ C of the new family which agreeswith both (cid:98) s (cid:48) i and the restriction of s to the formal fibers over c i to order N .See Figure 4 for an illustration of the situation.The section σ C gives a degree 2 multisection, still denoted by σ C ( C ), ofthe original family π : X → B , which agrees with s and (cid:98) s (cid:48) i to order N inthe formal neighborhood of the points b i . Take the family of lines spannedby the degree 2 multisection σ C ( C ). This family corresponds to a line ˜ L defined over the generic fiber, i.e. the function field C ( B ). The line ˜ L agreeswith the line L to order N over the points b i by construction. We have athird intersection point of ˜ L with the cubic surface, necessarily defined over C ( B ). The section corresponding to this rational point will agree with theformal sections (cid:98) s i to order N by construction, thus completing the proof(c.f. Figure 5). (cid:3) EAK APPROXIMATION FOR CUBIC HYPERSURFACES 29 (cid:98) S (cid:48) S | Spec (cid:98) O b ,B (cid:98) S S C ( C ) σ C ( C ) CC C (cid:48) Figure 4.
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