Weak Approximation over Function Fields of Curves over Large or Finite Fields
aa r X i v : . [ m a t h . AG ] J a n Weak Approximation over Function Fieldsof Curves over Large or Finite Fields
Yong HU ∗ Abstract
Let K = k ( C ) be the function field of a curve over a field k and let X be a smooth, projective, separably rationally con-nected K -variety with X ( K ) = ∅ . Under the assumption that X admits a smooth projective model π : X → C , we prove the fol-lowing weak approximation results: (1) if k is a large field, then X ( K ) is Zariski dense; (2) if k is an infinite algebraic extensionof a finite field, then X satisfies weak approximation at places ofgood reduction; (3) if k is a nonarchimedean local field and R -equivalence is trivial on one of the fibers X p over points of goodreduction, then there is a Zariski dense subset W ⊆ C ( k ) suchthat X satisfies weak approximation at places in W . As appli-cations of the methods, we also obtain the following results overa finite field k : (4) if | k | >
10, then for a smooth cubic hyper-surface
X/K , the specialization map X ( K ) −→ Q p ∈ P X p ( κ ( p ))at finitely many points of good reduction is surjective; (5) ifchar k = 2 , | k | >
47, then a smooth cubic surface X over K satisfies weak approximation at any given place of goodreduction. Contents R -equivalence . . . . . . . . . . . . . . . 6 ∗ Math´ematiques, Bˆatiment 425, Universit´e Paris-Sud, 91405, Orsay Cedex,France, e-mail: [email protected] Proofs of Results over Large Fields 125 Proofs of Results over Finite Fields 14
References 22Convention.
In this paper, a variety means a separated geometri-cally integral scheme of finite type over a field. Somewhat inconsis-tently, a curve means a 1-dimensional separated geometrically con-nected and geometrically reduced scheme of finite type over a field.The words “ vector bundle ” will be used interchangeably as “locallyfree sheaf of finite constant rank”.
Given a variety over a topological field F , the topology of F inducesa natural topology on the set of rational points. We call this topologythe F -topology . If K ⊆ F is a subfield and X is a K -variety, the set X ( F ) of F -points carries an F -topology via the natural identification X ( F ) = X F ( F ), where X F = X × K F is the F -variety obtained bybase extension of the K -variety X . Let K be a number field or afunction field of a curve. The completion K v at each place v of K isa topological field containing K . The K v -topologies will be also called v -adic topologies . Given a set Ω of places of K , we say X satisfies weak approximation at places in Ω if the image of the diagonalmap X ( K ) → Q v ∈ Ω X ( K v ) is dense, where Q v ∈ Ω X ( K v ) is given theproduct topology of the v -adic topologies. The classical approximationtheorem claims that the projective line P satisfies weak approximationat all places. In higher dimension and over the function field of acurve defined over an algebraically closed field, it is expected that weakapproximation holds for separably rationally connected varieties.Roughly speaking, these are varieties containing plenty of geometricrational curves that may be deformed “very freely”. (See Definition 2for the precise definition.)In this paper, we are interested in the following problem. Problem 1.
Let k be a field, C a smooth projective (geometricallyconnected) curve over k , K = k ( C ) the function field of C , and X asmooth, projective, separably rationally connected K -variety. Assumethat X has a smooth projective model, i.e., there is a surjective mor-phism π : X → C from a smooth projective k -variety X to C withgeneric fiber X/K . In this context, rational points of X correspond to2ections of π : X → C and weak approximation amounts to the exis-tence of sections passing through prescribed points in the fibers withprescribed jets.Consider the following questions:(EX) (Existence of sections) Is X ( K ) nonempty? Or equivalently,does π admit a section σ : C → X ?(SP) (Surjectivity of specialization map) Suppose that π has a sec-tion σ : C → X . Let P = { p , . . . , p n } be a finite set of closed pointsof C such that each fiber X i = X p i is smooth and separably rationallyconnected (a point or a place with this property will be called of goodreduction ). Given rational points r i ∈ X i ( κ ( p i )) in the fibers, is therea section s : C → X of π such that s ( p i ) = r i for i = 1 , . . . , n ?(WA) (Weak approximation) Suppose that X ( K ) = ∅ . Does thegeneric fiber X satisfy weak approximation at a certain collection ofplaces?When k is algebraically closed, a series of results have been estab-lished. Firstly, Koll´ar, Miyaoka and Mori [22] proved that question (SP)has a positive answer in this case. Later, question (EX) was solved byGraber, Harris and Starr [12] in characteristic 0 and soon generalizedto positive characteristic by de Jong and Starr [6]. As for question(WA), Colliot-Th´el`ene and Gille [3] proved that if X belongs to one ofsome special classes of varieties, including conic bundles and del Pezzosurfaces of degree ≥
4, then X satisfies weak approximation at allplaces. For arbitrary separably rationally connected varieties, Hassettand Tschinkel [15] proved weak approximation at places of good reduc-tion, generalizing the case of cubic surfaces proved earlier by Madore[25]. In the same paper, they conjectured that weak approximation atplaces of bad reduction also holds. This conjecture is open even forcubic surfaces. Hassett and Tschinkel [16] confirmed weak approxima-tion for cubic surfaces with not too bad reductions. A similar resultwas obtained by Knecht [17] for del Pezzo surfaces of degree 2. Alsoinspired by the methods of Hassett and Tschinkel [16], Xu [33] provedweak approximation for del Pezzo surfaces of degree ≤ ≤ k is not algebraically closed, particular resultsfor some special fields (e.g., the reals R ) have been established withsomewhat different methods (cf. [2], [29], [10]).In this paper, we study the case where the ground field k is a largefield (cf. Definition 3) or a finite field. The approach we follow wassuggested by Colliot-Th´el`ene and Hassett in 2005. It heavily relies onthe method of Hassett and Tschinkel [15] as well as on some results ofKoll´ar and Szab´o ([19], [20], [23]).Our main results are the following.3 heorem 1. With notation as in Problem 1, let k be an infinite alge-braic extension of a finite field. Assume that X ( K ) = ∅ .Then the K -variety X satisfies weak approximation at places of goodreduction. Theorem 2.
With notation as in Problem 1, let k be a nonarchimedeanlocal field. Suppose that π : X → C has a section.If for some point p ∈ C ( k ) , R -equivalence ( cf. Definition ) on therational points of the fiber X p is trivial, then there is a subset W ⊆ C ( k ) of rational points which is Zariski dense in C such that the K -variety X satisfies weak approximation at places in W . Applying the methods used in the proofs, we also obtain the follow-ing theorems.
Theorem 3.
With notation as in Problem 1, let k be a large field. If X ( K ) = ∅ , then X ( K ) is Zariski dense in the K -variety X . Theorem 4.
Let k = F be a finite field, C a smooth projective ( geometricallyconnected ) curve over k , K = k ( C ) its function field, and X ⊆ P Nk × k C a smooth closed subvariety such that the canonical projection π : X → C is a surjective morphism whose generic fiber is a smooth cubic hyper-surface X ⊆ P NK of dimension ≥ . Suppose that X ( K ) = ∅ .Then for any finite set P of closed points of good reduction on C with | κ ( p ) | > , ∀ p ∈ P , the specialization map X ( K ) −→ Y p ∈ P X p ( κ ( p )) (1) is surjective. In particular, if | k | > , then question (SP) has a positiveanswer in this situation. In proving the above theorem, we use some ideas of Swinnerton-Dyer, who proved an analogous result for cubic surfaces over numberfields (cf. [31], Theorem 5). In particular, his method of descent froma tower of quadratic extensions of the ground field finds a significantapplication in our proof. Yet another important ingredient is a resultof Koll´ar on the geometry of cubic hypersurfaces over finite fields ([21],Lemma 9.4).As a consequence, we obtain a weak approximation theorem forcubic surfaces, which is analogous to a result of Swinnerton-Dyer ([31],Corollary to Theorem 5).
Theorem 5.
Let k = F q be a finite field of characteristic not divid-ing and of cardinality q > , C a smooth projective ( geometrically onnected ) curve over k , K = k ( C ) its function field, and X ⊆ P k × k C a smooth closed subvariety such that the canonical projection π : X → C is a surjective morphism whose generic fiber is a smooth cubic surface X ⊆ P K with X ( K ) = ∅ . Let v be a place of good reduction correspond-ing to a closed point p ∈ C .Then X ( K ) is v -adically dense in X ( K v ) . More precisely, for any x ∈ X p ( κ ( p )) there is an R -equivalence class C on X ( K ) such that C is v -adically dense in the inverse image of x under the specializationmap X ( K v ) → X p ( κ ( p )) . Definition 1 ([7], Definition 4.5) . Let X be a smooth variety and r ≥
0. An r -free rational curve on X is a nonconstant morphism f : P → X such that f ∗ T X ∼ = O P ( a ) ⊕ · · · ⊕ O P ( a n )with a ≥ · · · ≥ a n ≥ r . (By a theorem of Grothendieck ([14], p.384,Exercise 2.6), every vector bundle over P is isomorphic to a direct sum ⊕ O P ( a i ) and the sequence a ≥ · · · ≥ a n is uniquely determined by thevector bundle.) We shall say free (resp. very free ) instead of 0-free(resp. 1-free). Definition 2 ([18], IV.3.2) . Let k be a field and ¯ k an algebraic closureof k . A smooth proper k -variety X is called separably rationallyconnected if it has the following property:(1) There is a separated integral algebraic k -scheme Z and a mor-phism g : P × k Z → X such that the morphism g (2) : P × k P × k Z −→ X × k X ; ( t , t , z ) ( g ( t , z ) , g ( t , z ))is dominant and smooth at the generic point.When dim X >
0, this is equivalent to the following two properties:(2) There is a very free rational curve P k → X ¯ k defined over ¯ k .(3) For any finite collection of closed points x , . . . , x m ∈ X ¯ k andany tangent directions ξ i at the points x i , there is a very free curve f : P k → X ¯ k defined over ¯ k such that f is an unramified morphismwith image containing all the x i as smooth points and having tangentdirections ξ i at these points.In property (3) of the above definition, one can take f to be a closedembedding if dim X ≥
3. For a proof of the above equivalences, see[18] IV.3.7 and 3.9, and [8], Th´eor`eme 2.2.5 xample 1. (1) A smooth k -variety X is separably rationally connected if itis geometrically separably unirational (i.e., over ¯ k there is a separabledominant rational map from some P n ¯ k to X ¯ k ).(2) In characteristic 0, smooth Fano varieties are separably ratio-nally connected (Campana, [1]). In particular, a smooth hypersurfacein P n of degree d ≤ n is separably rationally connected.(3) In characteristic p >
0, smooth cubic hypersurfaces of dimen-sion ≥ ≥
3, see [5]Remarque 2.34.) R -equivalence Definition 3 ([11], Chapter 10 and [27]) . A field k is said to be(1) pseudo-algebraically closed (PAC), if every (geometricallyintegral) k -variety has a rational point;(2) large , if for every (geometrically integral) k -variety X having asmooth k -point, X ( k ) is Zariski dense in X . Example 2. (1) PAC fields are large fields. Algebraic extensions of PAC (resp.large) fields are PAC (resp. large). (cf. [11], Chapter 10 and [27],Proposition 1.2.)(2) Infinite algebraic extensions of finite fields are PAC. (cf. [11],Chapter 10.)(3) The class of large fields contains all local fields (i.e., finite exten-sions of R , Q p or the field of Laurent series F p (( t )) over the finite field F p ). More generally, a field that is complete with respect to a nontrivialabsolute value is large.(4) All real closed fields and all p -adic closed fields (i.e., fields withabsolute Galois group isomorphic to that of R or that of a p -adic field)are large (cf. [28], Theorem 7.8). Definition 4 (Manin) . Let X be an algebraic scheme over a field k .Two rational points x, y ∈ X ( k ) are said to be directly R -equivalent if there is a k -rational map f : P k X such that f (0) = x and f ( ∞ ) = y . The R -equivalence relation on X ( k ) is the equivalencerelation generated by direct R -equivalence. We will write X ( k ) /R forthe set of R -equivalence classes on X ( k ). When X ( k ) /R has only oneelement, we write X ( k ) /R = 1 and say that R -equivalence is trivial on X . R -equivalence is closely related to separably rational connectedness.In [19], Koll´ar proved the finiteness of R -equivalence classes on separa-bly rationally connected varieties over local fields. Later he generalizedthe main theorem in [19] as follows.6 heorem 6 (Koll´ar, [20], Theorem 23) . Let k be a large field and let X be a smooth, projective, separably rationally connected k -variety ofdimension ≥ . Let n ≥ and x , . . . , x n ∈ X ( k ) . Then the followingtwo conditions are equivalent: (i) There is a closed embedding f : P → X that is ( n + 1) -free suchthat x , . . . , x n ∈ f ( P ( k )) . (ii) All the points x i ∈ X ( k ) are in the same R -equivalence class. We remark that in general it may happen that there is no smoothrational curve passing through a given finite set of rational points on avariety X as in the above theorem. A sufficient condition for this is thatthe ground field k is PAC (cf. [23], Theorem 19 and [32], Corollaire 3.2).If k = F is a finite field, we have the following result. Theorem 7 (Koll´ar and Szab´o, [23], Theorems 2 and 7) . (i) There is a function
Φ : N → N having the following property:Let X ⊆ P N be a smooth, projective, separably rationally connectedvariety over a finite field F of dimension ≥ , and let S ⊆ X be azero-dimensional smooth subscheme. If | F | > Φ(deg X, dim X, deg S ) ,then there is a closed embedding f : P → X that is (deg S + 1) -freewith image containing S . (ii) There is a function
Ψ : N → N having the following property:Let X ⊆ P N be a smooth, projective, separably rationally connectedvariety over a finite field F . If | F | > Ψ(deg X, dim X ) , then X ( F ) /R =1 . In the above two theorems the hypothesis that dim X ≥ f to be a closed embedding. If dim X =2, we can apply the theorem to X × k P to obtain a rational curvepassing through the given points. Remark . In Theorem 7 (ii), if one restricts to the case where X ⊆ P N is a smooth cubic hypersurface of dimension ≥
2, then [21], Theorem 1.1gives an even better lower bound for | F | : one has X ( F ) /R = 1 whenever | F | ≥ ∗ To establish weak approximation we use the method initiated by Has-sett and Tschinkel in [15]. In this section we briefly recall their method.
Let K = k ( C ) be the function field of a smooth projective curve andlet X be any smooth proper variety over K . A result of Nagata ([26]) ∗ The lower bound | F | ≥ | F | ≥
11. See alsothe footnote attached to Lemma 4. X always admits a proper model , i.e., an algebraicspace π : X → C that is flat and proper over C with generic fiber X .Rational points of X correspond bijectively in a natural way to sectionsof π : X → C .In this paper we will assume for simplicity that X is projective andadmits a smooth projective model π : X → C as in Problem 1.For each closed point p ∈ C , let X p = X × C Spec( κ ( p )) be the fiberover p . Let b O C, p be the completed local ring at p , with maximal ideal b m C, p . If v denotes the place of K corresponding to p , then K v is thefield of fractions of b O C, p . Let b C p = Spec( b O C, p ), X b C p = X × C b C p and b π p : X b C p → b C p the natural projection. Sections of b π p correspond bijectivelyto K v -valued points of X . Suppose a point M v ∈ X ( K v ) correspondsto a section b s p : b C p → X b C p . Then basic v -adic open neighborhoods of M v consist of those sections of b π p which agree with b C p, N := Spec( b O C, p / b m N +1 C, p ) ֒ → b C p b s p −→ X b C p when restricted to b C p, N ⊆ b C p , for some N ∈ N .For N ∈ N , we say a morphism j : b C p, N = Spec( b O C, p / b m N +1 C, p ) −→ X b C p = X × C Spec( b O C, p )is an N -jet at the closed point p ∈ C if b π p ◦ j coincides with thenatural inclusion b C p, N ֒ → b C p = Spec( b O C, p ). A section of b π p determinesnaturally an N -jet.Thus weak approximation over function fields has a geometric re-formulation as follows: for any finite set of closed points { p i } in C , anysections b s i of b π p i and any N ∈ N , there exists a section s of π such thatfor each i , the N -jets determined by s and b s i coincide. We keep notation as above. To develop the method of Hassett andTschinkel over an arbitrary ground field k , we assume that places inthe consideration below are given by separable points (i.e. points withresidue fields separable over k ).Let { p i } ⊆ C be a finite number of separable closed points of goodreduction and let J = { j i } be N -jets at these points given by formalsections b s i of b π p i : X b C pi → b C p i . By considering the iterated blowup associated to J , whose construction we will recall below, Hassett andTschinkel reduce finding a section passing through given points withprescribed jet data to simply finding a section through given points onthe iterated blowup. 8ere is the construction of the iterated blowup β ( J ) : X ( J ) → X associated to J . It is obtained by the following sequence of blowups: X ( J ) = X ( N ) → X ( N − → · · · → X (1) → X where X (1) → X is the blowup of X at { b s i ( p i ) } and for each n =1 , . . . , N − X ( n +1) → X ( n ) is the blowup at the points where theproper transform of b s i meets the fiber over p i .Write d = dim X . For each i the fiber X ( J ) p i decomposes intoirreducible components: X ( J ) p i = E i, ∪ · · · ∪ E i, N where(i) E i, is the proper transform of X p i , isomorphic to the blowup of X p i at the κ ( p i )-point r i, := b s i ( p i );(ii) E i, n for n = 1 , . . . , N − P dκ ( p i ) at the κ ( p i )-point r i, n where the proper transform of b s i meets the fiber over p i ofthe n -th blowup;(iii) E i, N ∼ = P dκ ( p i ) .Let r i ∈ E i, N \ E i, N − denote the point where the proper transformof b s i meets E i, N . Sections s ′ of X ( J ) → C with s ′ ( p i ) = r i yield sectionsof π : X → C with N -jets j i at the points p i . So we have the followingcriterion. Proposition 1 ([15], Proposition 11) . Let
X, C and π : X → C beas above. Consider all data ( { p i } , J = { j i } , { r i } ) consisting of afinite collection of separable closed points { p i } ⊆ C of good reduction, N -jets { j i } at these points and κ ( p i ) -points { r i } on the correspondingiterated blowup X ( J ) with r i ∈ E i, N \ E i, N − .If for every datum there exists a section s ′ of X ( J ) → C with s ′ ( p i ) = r i for all i , then X satisfies weak approximation at placesgiven by separable closed points of good reduction. The main steps in constructing sections through prescribed points in-volve techniques for smoothing combs.
Definition 5 ([15], Definition 18) . Let k be any field and let ¯ k be analgebraic closure of k . A comb with m ( reducible ) teeth over k is aprojective curve C ∗ over k with a distinguished irreducible component D ⊆ C ∗ defined over k such that the following conditions hold:(1) D is a smooth (geometrically connected) curve over k , called the handle of the comb;(2) C ∗ = C ∗ × k ¯ k is the union of D = D × k ¯ k and m other curves T , . . . , T m which are called the teeth of the comb;93) each T j is a chain of P ’s;(4) the T j are disjoint with one another and each of them meets D transversally at a single point.Let X, C , π : X → C , { p i } and so on be as in Proposition 1. Recallthat we are assuming that the total space X is smooth projective andthat the p i are separable points. Since X satisfies weak approximationif and only if X × K P K does, without loss of generality we may assumethat d = dim X ≥ σ : C → X ( J ). Let I ′ (resp. I ′′ )denote the set of indices with q i := σ ( p i ) = r i (resp. q i = r i ).Working over an algebraically closed ground field k , the most tech-nical part of [15] shows that a section C → X ( J ) passing through r i can be found so long as the following two tasks (A) and (B) may beaccomplished:(A). In the case N = 0, find a comb C ∗ with handle D = σ ( C ) andsmooth teeth T , . . . , T m and a closed embedding f : C ∗ → X such thatthe following properties hold:(i) the teeth T j are very free rational curves contained in distinctgeneral fibers of π ;(ii) for each i ∈ I ′ , there is a tooth T i with image containing r i as asmooth point;(iii) if r denotes the set of points on C ∗ mapping to the r i , i ∈ I = I ′ ∪ I ′′ and N f = N C ∗ / X denotes the normal bundle of C ∗ in X , thenthe restriction of N f ⊗ O C ∗ ( − r ) to every irreducible component of C ∗ is generated by global sections and has no higher cohomology.(B). In the case N ≥
1, assume q i ∈ E i, N \ E i, N − . Find a comb C ∗ with handle D = σ ( C ) and reducible teeth T , . . . , T m and a closedembedding f : C ∗ → X ( J ) having the following properties:(i) for each i ∈ I ′ , there is a tooth T i mapped into X ( J ) p i andcontaining r i ;(ii) for each i ∈ I ′ , f ( C ∗ ) contains r i as a smooth point, so there isa unique irreducible component T i, N ⊆ C ∗ with image containing r i ;(iii) the remaining teeth T j , j = | I ′ | + 1 , . . . , m are very free rationalcurves contained in distinct general fibers of X ( J ) → C ;(iv) if r denotes the set of points on C ∗ mapping to the r i , i ∈ I = I ′ ∪ I ′′ and N f = N C ∗ / X ( J ) denotes the normal bundle of C ∗ in X ( J ),then the restriction of N f ⊗ O C ∗ ( − r ) to every irreducible component of C ∗ is generated by global sections and has no higher cohomology.In [15], (A) and (B) are done in Lemmas 25 and 26, assumingof course the generic fiber X is separably rationally connected. Thedesired section is a smooth deformed curve of the comb C ∗ , which isobtained by applying [15], Proposition 24.10he key condition here is the vanishing of the higher cohomologyof the twisted normal sheaf N f ⊗ O C ∗ ( − r ). We consider this conditionon each irreducible component of C ∗ . For a component in a tooth T j ,the condition is easier to verify since such a curve is usually a very freecurve. The difficult part is the restriction to the handle D = σ ( C ).Fortunately, the sheaf N f ⊗ O D has a nice interpretation as a kind ofelementary transform of N σ , where N σ denotes the normal sheaf of D in X ( J ). Let D c be the closure of C ∗ \ D in C ∗ and S = D ∩ D c the locus where the handle D meets the teeth. For each q i ∈ S , let ξ i ⊆ N σ ⊗ κ ( q i ) denote the subspace determined by the tangent direction T D c , q i . It turns out that N f ⊗ O D is the so-called sheaf of rationalsections of N σ having at most a simple pole at each q i in thedirection ξ i and regular elsewhere (cf. [12], Lemma 2.6).A key lemma needed is the following. Lemma 1 ([12], Lemma 2.5) . Let C be a smooth curve over an al-gebraically closed field, V a vector bundle on C and let l ≥ be apositive integer. Then for sufficiently large m there exist general points q , . . . , q m ∈ C and one-dimensional subspaces ξ i ⊆ V ⊗ κ ( q i ) such thatfor any l points w , . . . , w l ∈ C the sheaf V ′ ( − w −· · ·− w l ) is generatedby global sections and H ( C, V ′ ( − w − · · · − w l )) = 0 , where V ′ denotes the sheaf of rational sections of V that have at mosta simple pole at each q i in the direction ξ i and regular elsewhere. Note that the smallest value for m depends on the number l andthe vector bundle V .Lemma 1 applied to the normal bundle V = N σ on D shows thatthe vanishing of higher cohomology required in the tasks (A) and (B)may be guaranteed by adding sufficiently many teeth to the comb C ∗ .Now we want to deal with the case where k is not necessarily alge-braically closed and we want to produce a deformed curve defined over k . We face two problems.(D.1) We should be able to construct the comb C ∗ over k .First consider task (A). Suppose that we want to add teeth in thefibers over separable closed points p i ∈ C . If we are able to find in eachfiber a very free curve T i defined over κ ( p i ), then the union ∪ i T i ∪ σ ( C )considered as a k -scheme gives us a comb defined over k . Unlike the caseof algebraically closed ground field, the existence of a very free curvedefined over the ground field in a separably rationally connected fiberis not a priori evident, and if we moreover require the very free curveto pass through two prescribed points, this may be even impossible. Inthe next section we shall use the results of Koll´ar and Szab´o on thetriviality of R -equivalence to solve this problem.11he situation for (B) is essentially the same. The difference is thatthe fiber X ( J ) p i becomes a bit more complicated. However, by a care-ful inspection on the construction of reducible teeth in [15], we see thatfor each reducible tooth T i all but one of its components T i, , . . . , T i, N are just lines connecting two κ ( p i )-points and the last one T i, is avery free curve passing through one κ ( p i )-point in the piece E i, , whichis the blowup of X p i at a κ ( p i )-point. Separably rational connected-ness and the set of R -equivalence classes are both birationally invari-ant for smooth projective varieties (cf. [18], IV.3.3 and [4], p.195,Proposition 10). So if a property determined by separably rationalconnectedness or R -equivalence holds for X p i , then the same shouldhold for E i, . In particular, if the same thing can be done in X p i , thenthe curve T i, in E i, may be constructed over κ ( p i ). So it makes noessential difference in fulfilling tasks (A) and (B).(D.2) We should be able to find deformations of C ∗ that are definedover k .Having constructed a comb C ∗ defined over k with good propertiesas stated in (A) or (B), by applying the same argument as in [15] wesee that the parameter space M of curves in X ( J ) passing throughthe separable points { r i } is smooth at the k -point [ C ∗ ] and there arenearby deformed curves defined over ¯ k that are smooth. The final stepis to show the existence of a k -rational point in the parameter space.When the field k is large or finite, we have some tools to do this as weshall now see. Proposition 2.
With notation as in Problem 1, let k be a large field.Consider question (SP) . Suppose that the closed points p , . . . , p n areseparable and that the point r i ∈ X i ( κ ( p i )) lies in the same R -equivalenceclass as q i = σ ( p i ) ∈ X i ( κ ( p i )) for every i = 1 , . . . , n .Then there exists a section s : C → X of π such that s ( p i ) = r i for i = 1 , . . . , n .Proof. In each fiber X i = X p i , the two rational points r i and q i = σ ( p i )are R -equivalent by hypothesis. Thus Koll´ar’s theorem (Theorem 6)implies that there is a very free curve f i : P → X i defined over κ ( p i )such that q i , r i ∈ f i ( P ( κ ( p i ))). On the other hand, the smooth curve C has sufficiently many separable closed points. So we can pick asmany good points as we need and construct teeth in the correspondingfibers to get a comb C ∗ defined over k satisfying all the properties intask (A). The comb C ∗ then has smooth curves as nearby deformations,and since [ C ∗ ] is a smooth k -point in the parameter space, the propertyof large fields implies that there exists a deformed smooth curve that is12efined over k . The problems (D.1) and (D.2) described in § s passing through the points r i . Proof of Theorem 1.
Let U ⊆ C be the open subset consisting of pointsof good reduction. (Such an open subset exists by [18], IV.3.11.) Notethat k is perfect so that every point is separable. Let p , . . . , p n beclosed points lying in U .The theorem of Koll´ar and Szab´o (Theorem 7 (ii)) implies that R -equivalence on a smooth separably rationally connected variety overan infinite algebraic extension of a finite field is trivial. So we have X p i ( κ ( p i )) /R = 1. A similar argument as in the proof of Proposition 2shows that a comb C ∗ defined over k may be constructed satisfying allthe required properties in task (A) or (B). Since k is a large field, theproblem (D.2) may also be solved. This means that the comb C ∗ admitssmooth deformed curves defined over k , which give rise to sections onthe iterated blowup with the required property. Weak approximationat places in U is thus proved for the generic fiber X . Proof of Theorem 2.
By a result of Koll´ar ([20], Theorem 3), the func-tion C ( k ) −→ N , c
7→ |X c ( k ) /R | is upper semi-continuous for the k -topology. So there is a nonemptyopen subset W of C ( k ) (for the k -topology) such that X c ( k ) /R = 1 forall c ∈ W . For any nonempty Zariski open subset U of C , U ( k ) is densein C ( k ) for the k -topology. Hence W ∩ U ( k ) = ∅ , showing that W isZariski dense in C . The same argument as in the proof of Theorem 1proves weak approximation at places in W . Remark . In Theorem 2, the hypothesis on the existence of a point p ∈ C ( k ) with X p ( k ) /R = 1 is satisfied if there is a point p ∈ C ( k )such that X p has a smooth separably rationally connected reductionover the residue field F of k and if the residue field F has sufficientlylarge cardinality (cf. [23], Theorem 8).The following lemma is an easy consequence of a theorem of Koll´ar. Lemma 2.
Let X be a smooth, projective, separably rationally con-nected variety over a large field k . Suppose that X ( k ) = ∅ . Then forevery x ∈ X ( k ) , there is a subset W ⊆ X ( k ) that is Zariski dense in X with the following property: for every y ∈ W there is a very free curve f xy : P → X defined over k such that x, y ∈ f xy ( P ( k )) .Proof. In [19], Theorem 1.4, Koll´ar proved that there exists a very freecurve f x : P → X defined over k sending a rational point 0 ∈ P to x .Let Hom( P , X ; 0 x ) be the Hom-scheme parametrizing morphisms13 → X sending 0 to x . By [7], p.91, Proposition 4.8, the evaluationmap e : P × Hom( P , X ; 0 x ) −→ X , ( t, g ) g ( t )is smooth at any rational point ( t, [ f x ]) with t = 0. In particular, e isdominant. Therefore, for every nonempty open subset U of X , we canfind a very free curve f : P → X defined over k sending a rationalpoint of P into U with f (0) = x . Proof of Theorem 3.
Let σ : C → X be a section of π : X → C . Let Z be a proper closed subset of X . Its closure in X is a proper closedsubset Z of X . There is a nonempty open subset U of C such that forevery c ∈ U , the fiber X c is smooth separably rationally connected and X c ∩ Z 6 = X c . Choose a separable closed point p ∈ U . By Lemma 2,there is a point r ∈ X p ( κ ( p )) \ Z which is directly linked to q = σ ( p ) bya very free rational curve defined over κ ( p ). Proposition 2 implies thatthere is a section s : C → X such that s ( p ) = r . This section gives a K -rational point of X which does not lie in Z . Since Z is arbitrary, itfollows that X ( K ) is Zariski dense in X . Proposition 3.
With notation as in Problem 1, let k = F be a finitefield. Let S ⊆ C be the closed subset consisting of points of bad reductionfor the model π : X → C . Assume that π has a section σ : C → X .We fix closed embeddings C ֒ → P k , X ֒ → P Nk × k C ֒ → P Nk × k P k ֒ → P Mk where P Nk × k P k ֒ → P Mk is the Segre embedding. We denote by deg C the degree of C in P k , deg σ ( C ) and deg X the degrees of σ ( C ) and X in P Mk , and deg X the degree of X with respect to the embedding X = X × C K ֒ → P NK = ( P Nk × k C ) × C K .
There is a lower bound B = B ( N σ , | S | , deg C P, deg C, deg σ ( C ) , deg X, dim X ) for the cardinality | F | , depending on the normal bundle N σ of the section σ , the cardinality | S | , the number deg C P = P p ∈ P [ κ ( p ) : k ] and thegeometric invariants deg C, deg σ ( C ) , deg X and dim X , such that theanswer to question (SP) is yes whenever | F | > B . roof. According to the method we described in §
3, we want to firstconstruct a comb C ∗ such that the conditions in task (A) hold. When | F | is greater than a lower bound in terms of deg X and dim X , by thetheorem of Koll´ar and Szab´o (Theorem 7 (i)), in any fiber of X → C over a point p ∈ C of good reduction there always exists a very freecurve defined over κ ( p ) and passing through two given κ ( p )-points. Thisimplies that the comb C ∗ may be constructed over F when | F | is largeenough.In addition to the teeth given by very free curves in fibers over somepoints of P , many other teeth have to be added to the comb C ∗ suchthat the condition on the vanishing of higher cohomology in task (A)holds. Let m be the number of these teeth. The smallest value for m depends on the normal bundle N σ and the number deg C P , as weremarked after the statement of Lemma 1. By the Lang-Weil estimate([24], Theorem 1), we may assume that | F | is greater than a lower bounddepending on deg C , deg C P and | S | so that we can find m F -rationalpoints outside P ∪ S . We then construct the m teeth in fibers over thesepoints. The total number of teeth of the comb C ∗ is at most deg C P + m .By [23], Theorem 16 and the Lang-Weil estimate (cf. [23], p.258, Proofof Theorem 16 implies Theorem 2), we may choose each tooth withdegree bounded from above in terms of dim X , deg X . Thus the degree d of C ∗ is bounded from above in terms of deg C P , m , dim
X , deg X and deg σ ( C ).In the parameter space M d of degree d curves in X containing { r i } ,the comb C ∗ corresponds to a smooth F -rational point [ C ∗ ]. Choose asmooth geometrically irreducible curve T in M d passing through [ C ∗ ]such that deformations given by points in W = T \ { [ C ∗ ] } are allsmooth curves. Then W is contained in a unique geometrically irre-ducible component W ′ of the subspace H d parametrizing smooth degree d curves in X containing { r i } . By [23], Proposition 20, the basic pro-jective invariants of W ′ are bounded in terms of dim X , deg X , d anddeg C P . In our situation, dim X and deg X can be expressed in termsof dim X , deg C and deg X . Taking into account all these things, wefinally get, using the Lang–Weil theorem, a lower bound B = B ( N σ , | S | , deg C P , deg
C , deg σ ( C ) , deg X , dim X )having the property that W ′ ( F ) = ∅ whenever | F | > B . Let C w bea deformed curve corresponding to a rational point of W ′ . Deformedcurves given by points of W have the same numerical properties as C ∗ ,so does the curve C w . In particular, the intersection number of C w witha general fiber of X → C is one. Therefore, the curve C w gives a section s : C → X as we want.A drawback of the result in Proposition 3 is that the lower bound B is ineffective and depends on many non-intrinsic objects. However,15rom the above proof we see that the dependence of B on the normalbundle N σ relies only on its cohomological property, so the value of B will not change when we go over to an extension of the ground field F .Let us now start the proof of Theorem 4. Basically, it amounts toshowing that it is really possible to come down to the ground field F from a big enough extension F n over which the result may be guaranteedby Proposition 3. Using geometry of cubic hypersurfaces we will do thiswith F n / F a tower of quadratic extensions. The basic ideas go back toSwinnerton-Dyer, who obtained similar results for cubic surfaces overnumber fields (cf. [31], Theorem 5 and its corollary).Let V ⊆ P Nk be a cubic hypersurface over a field k and let k /k be aquadratic extension. Denote by σ the nontrivial element in the Galoisgroup Gal( k /k ). We define a “dashed arrow” V ( k ) V ( k ) (2)as follows: For a point v ∈ V ( k ), the line ℓ ( v , σ v ) joining v and itsconjugate point σ v will generally intersect V at only one more point v , which is k -rational. Whenever this is well-defined, we associate to v ∈ V ( k ) the third intersection point v ∈ V ( k ) of the line ℓ ( v , σ v )and V . In what follows, we will always mean the dashed arrow is definedat v and sends it to v when we write “ v v ”. Lemma 3.
Let A be a discrete valuation ring with field of fractions K and residue field F . Let X K ⊆ P NK be a smooth cubic hypersurface and X A ⊆ P NA its scheme-theoretic closure in P NA . Suppose that the specialfiber X F ⊆ P NF is a smooth cubic hypersurface. Let F /F be a quadraticextension and K /K an unramified quadratic extension having residuefield F /F . Then the diagram X K ( K ) (cid:15) (cid:15) / / X F ( F ) (cid:15) (cid:15) X K ( K ) / / X F ( F ) is commutative in the following sense:If x ∈ X K ( K ) specializes to e x ∈ X F ( F ) and e x e x ∈ X F ( F ) ,then the dashed arrow X K ( K ) X K ( K ) is defined at x and theimage x ∈ X K ( K ) of x specializes to e x ∈ X F ( F ) .Proof. Let σ be the nontrivial element of the Galois group Gal( K /K ) =Gal( F /F ). The hypothesis e x e x means that the F -line L F = ℓ ( e x , σ e x ) joining e x and σ e x is not contained in X F and the intersec-tion ℓ ( e x , σ e x ) ∩ X F has exactly 3 points, the third being e x ∈ X F ( F ).Since K /K is unramified, it is easily verified that σ x specializes to σ e x .Let L K = ℓ ( x , σ x ) be the K -line joining x and σ x . Let L A ⊆ P NA be its scheme-theoretic closure in P NA .16 laim. L A is an A -line in P NA .We leave aside the proof of the claim for a moment and continuethe proof of the lemma.The special fiber L A × A F is thus a line containing e x and σ e x ,whence L A × A F = L F = ℓ ( e x , σ e x ). We have L K * X K since L F * X F . Hence L K ∩ X K = { x , σ x , x } with x ∈ X K ( K ), showingthat x x . To prove x specializes to e x , it suffices to show thatthe scheme-theoretic closure of Z K := L K ∩ X K ⊆ P NK in P NA is equalto Z A = L A ∩ X A ⊆ P NA . By [13], (IV.2.8.5), we need only show that Z A is A -flat. Note that Z A → Spec A is proper and both the genericfiber Z K /K and the special fiber Z F = L F ∩ X F /F are finite. ByChevalley’s theorem, Z A is finite over A . Since the generic fiber Z K /K and the special fiber Z F /F have the same length, Z A is flat over A by[14], p.174, Lemma 8.9.We finish by giving the proof of our claim. This is essentially aneasy consequence of [13], (IV.2.8.1.1).The embedding L K ⊆ P NK corresponds to a surjective K -homomorphism ϕ : K N +1 → K . Let M be the image of the composite map A N +1 ֒ → K N +1 ϕ −→ K and let ϕ A : A N +1 → M be the induced map. Then thesurjection ϕ A : A N +1 → M corresponds to the embedding L A ⊆ P NA .We need prove that M is free of rank 2 over A . In fact, there is aninduced commutative diagram with exact rows A N +1 (cid:15) (cid:15) ϕ A / / M (cid:15) (cid:15) / / K N +1 ϕ / / K / / A -submodule of K , M is torsion-free and hence free over thediscrete valuation ring A . The natural homomorphism M ⊗ A K → K is bijective, whence rank A M = 2. This completes the proof.We shall now make use of the following lemma due to Koll´ar. Lemma 4 ([21], Lemma 9.4) . Let F q be a finite field of cardinality q ≥ . † Then for any smooth cubic hypersurface X ⊆ P N defined over F q with N ≥ , the dashed arrow X ( F q ) X ( F q ) is surjective, i.e., for every p ∈ X ( F q ) , there is a point x ∈ X ( F q ) \ X ( F q ) such that the line ℓ ( x , σ x ) joining x and its conjugate point σ x satisfies ℓ ( x , σ x ) ∩ X = { p , x , σ x } . † In [21], the proof of Lemma 9.4 holds only for q ≥ roof of Theorem 4. We consider the case where P = { p } consists of asingle point. The general case may be treated in the same way withoutessential difference.Let κ = κ ( p ) be the residue field of the point. We fix an algebraicclosure F of F and let F n , κ n , n ≥ F determinedby the following conditions: F = F , κ = κ , F n ⊆ κ n , and [ F n +1 : F n ] = [ κ n +1 : κ n ] = 2 , for all n ≥
0. Let K n = F n ( C F n ) be the function field of the curve C F n = C × F F n defined over F n . Then for each n the following diagramis commutative in the sense described as in Lemma 3: X ( K n +1 ) (cid:15) (cid:15) / / X p ( κ n +1 ) (cid:15) (cid:15) X ( K n ) / / X p ( κ n )Suppose that | κ ( p ) | > φ ( N ). Then, by Lemma 4, starting from anypoint r ∈ X p ( κ ) we can find successively a sequence of points r i ∈X p ( κ i ) such that for each i , r i +1 r i via the dashed arrow X p ( κ i +1 ) X p ( κ i ). Due to Proposition 3, we may choose n big enough so that thespecialization map X ( K n +1 ) → X p ( κ n +1 ) is surjective. Pick a point s n +1 ∈ X ( K n +1 ) that specializes to r n +1 ∈ X p ( κ n +1 ). By Lemma 3, weobtain points s i ∈ X ( K i ) for i = n + 1 , n . . . , s n +1 s n · · · · · · s s and each s i specializes to r i . In particular, there exists a point s ∈ X ( K ) which specializes to r ∈ X p ( κ ( p )). Our proof of Theorem 5 follows the method of Swinnerton-Dyer. Inparticular, the following result due to him will be needed.
Theorem 8 ([31], Theorem 4) . Let K be a global field, K v the comple-tion of K at a nonarchimedean place v and k the residue field of v . Let V ⊆ P K be a smooth cubic surface whose reduction e V ⊆ P k at v is alsosmooth. Let e P ∈ e V ( k ) be the reduction of a point P ∈ V ( K ) .Suppose that there is a point R ∈ V ( K ) whose reduction e R ∈ e V ( k ) has the following properties: (1) the line e P e R intersects e V at exactly three distinct points; (2) no geometric line on e V passes through e R ; (3) letting γ = T e R e V ∩ e V , there exist two distinct points t , t ∈ γ ( k ) such that for each i = 1 , , t i = e R and the line e P t i intersects e V at threedistinct points and T t γ ∩ T t γ ∩ T e P e V = ∅ . hen there is an R -equivalence class C in V ( K ) such that for everypoint Q ∗ ∈ V ( K v ) that specializes to e P ∈ e V ( k ) and every v -adic openneighborhood U v of Q ∗ , one has C ∩ U v = ∅ . Swinnerton-Dyer stated the above theorem for a number field K .But his proof works for function fields as well. (In his paper [31],Swinnerton-Dyer deduced this theorem from his Lemma 8 and Theorem 3.In proving these two results the key ingredients have been nonarchimedeananalysis plus some geometric arguments involving tangent spaces andintersection theory, which do not actually depend on special propertiesof number fields as one may verify with more or less patience.)In the proof of the next result, some geometric arguments have beenused in [31] and [21]. The idea being very similar, here we still includea proof because of a slightly greater generality and some useful explicitestimates. Proposition 4.
There is a function φ : N → N having the followingproperty:Let F q be a finite field, N ≥ , and X ⊆ P N a smooth cubic hyper-surface over F q . If q = | F q | > φ ( N ) , then for every p ∈ X ( F q ) , there isa point x ∈ X ( F q ) such that the line ℓ ( x, p ) joining x and p intersects X at exactly three distinct F q -points.Proof. Fix p ∈ X ( F q ). Set B = { x ∈ X ( F q ) | x = p and ℓ ( x, p ) * T p X } , where T p X ⊆ P N denotes the tangent hyperplane to X at p . For a line ℓ ⊆ P N defined over F q not contained in X and a point x ∈ ( ℓ ∩ X )( F q ),we denote by ( ℓ · X ) x the intersection multiplicity of ℓ with X at thepoint x . Then B = B ∪ B , where for i = 1 , B i consistsof points x ∈ B such that ( ℓ ( x, p ) · X ) x = i . Let b = B and b i = B i , i = 1 ,
2. We have b = b + b = X ( F q ) \ { p } ) − { F q -line ℓ ⊆ T p X | p ∈ ℓ and ( ℓ · X ) p = 2 } whence X ( F q ) − − q N − − q − ≤ b = b + b ≤ X ( F q ) − . (3)We want to show that for q large enough, X ( F q ) − − q N − − q − > b
19o that B = ∅ . Observe that B = { x ∈ X ( F q ) | x / ∈ T p X , p ∈ T x X } . Suppose that X is defined by a cubic form ϕ ∈ F q [ T , · · · , T N ] and put ϕ ′ = ∂ϕ∂T . We may assume that p = (1 : 0 : · · · : 0). Letting Y ⊆ P N denote the subscheme defined by the equations ϕ = ϕ ′ = 0, we haveclearly B ⊆ Y ( F q ), whence b ≤ Y ( F q ).By the Lang-Weil theorem, X ( F q ) = q N − + O ( q N − / )where the term O ( q N − / ) depends only on N and q . Using [9], Th´eor`eme8.1, we can get an even more explicit estimate: (cid:12)(cid:12)(cid:12)(cid:12) X ( F q ) − q N − q − (cid:12)(cid:12)(cid:12)(cid:12) ≤ β N · q N − where β N denotes the ( N − P N C . In view of (3), it is now sufficient to show that Y ( F q ) = O ( q N − ) (4)with the term O ( q N − ) depending only on N and q . An easy verificationshows that the form ϕ ′ cannot be identically zero on X since X issmooth and geometrically irreducible. It follows that dim Y < dim X = N − Y ( F q ). Forinstance, [30], Chapter 4, Lemma 3.3 gives Y ( F q ) ≤ q N − q − . (5)The proposition is thus proved. Remark . According to the above proof (cf. (3) and (5)), for afunction φ : N → N to have the property in Proposition 4, one sufficientcondition is12 q N − q − < q N − − − β N q N − , ∀ N ≥ , ∀ q > φ ( N ) . So one can compute explicitly at least one possible value of φ ( N ) foreach N . For instance, using β = 7 one verifies easily that one maytake φ (3) = 20. Remark . Letting L N be the set of F q -lines ℓ ⊆ P N such that ℓ ∩ X = { p , x , σ x } , with x ∈ X ( F q ) \ X ( F q ) , our proof of Proposition 4 can also give a lower bound depending on N (and q ) for the cardinality L N . But as a sufficient condition for L N = ∅ , Koll´ar’s lemma (Lemma 4) is obviously better.20 emma 5. Let e V be a smooth cubic surface over a finite field k ofcharacteristic not dividing . If q = | k | > , then for any e P ∈ e V ( k ) ,one can find a point e R ∈ e V ( k ) satisfying all the conditions (1) , (2) and (3) of Theorem 8.Proof. Swinnerton-Dyer has noticed this fact as he remarked in [31],p.379, lines 7–10. However, an explicit proof seems not included there.So we give our own proof here.Let B be the set of points e R ∈ e V ( k ) for which the condition (1)holds. We know from the proof of Proposition 4 (cf. (3) and (5)) thatthe cardinality b = B satisfies b ≥ q − q − − β q − (cid:18) q − q − (cid:19) = q − q − q − q − . Since the union of all geometric lines on e V contains at most 27( q + 1) k -points, the subset B ′ ⊆ B consisting of points e R ∈ e V ( k ) which haveboth the properties (1) and (2) has cardinality b ′ = B ′ ≥ b − q + 1) ≥ q − q − . For q >
47, we can always find a point e R ∈ e V ( k ) having the properties(1) and (2). Then the intersection γ = T e R e V ∩ e V is a geometricallyirreducible plane cubic curve. Such a curve has at least q − t ∈ e V ( k ) such that e P ∈ T t e V all lie on aquadratic Y and the intersection γ ∩ Y hat at most 6 points. Thus,when q >
47, we can always find points t = t ∈ γ ( k ) with t i = e R suchthat the line e P t i intersects e V at 3 distinct points for each i = 1 ,
2. Itremains to show that t , t may be so chosen that the intersection T t γ ∩ T t γ ∩ T e P e V = T t e V ∩ T t e V ∩ (cid:16) T e R e V ∩ T e P e V (cid:17) is empty. This is a consequence of the Lemma 6 below. Lemma 6.
Let F be any field of characteristic not dividing , γ ⊆ P F a geometrically irreducible plane cubic curve having a nonsmooth point R ∈ γ ( F ) . Then for any S ∈ P ( F ) , the set T := { t ∈ γ ( F ) | the line St joining S and t is tangent to γ at t } has cardinality ≤ .Proof. Take coordinates such that R = (1 : 0 : 0). Since γ passesthrough R and is not smooth there, the equation of γ has the followingform ϕ = T q ( T , T ) + c ( T , T )21here q is a quadratic form and c is a cubic form. We may assume S = R and take coordinates such that S = (0 : 1 : 0). Then T is theset of rational points of the subscheme Z ⊆ P F defined by ϕ = ∂ϕ∂T = 0 . Using the hypothesis on the characteristic of F , one concludes by aneasy computation that ∂ϕ∂T is not identically zero because of the geomet-rical irreducibility of γ . It then follows that T = Z ( F ) ≤ Proof of Theorem 5.
We apply Theorem 8 with V equal to the genericfiber X and e V equal to the special fiber X p . By Theorem 4, the hy-pothesis implies that lifting a point in e V ( k ) to a point in V ( K ) isalways possible. The result then follows easily from Theorem 8 andLemma 5. Acknowledgements.
The author wishes to thank the referee for valu-able comments, especially for pointing out the possibility of generalizingthe first version of Theorem 4 using the reference [21]. The author alsothanks Prof. Jean-Louis Colliot-Th´el`ene for many valuable discussionsand comments. Some of the problems solved here arose out of conver-sations which Jean-Louis Colliot-Th´el`ene and Brendan Hassett had in2005, and the approach we follow is also inspired by their discussions.Thanks also go to Prof. J´anos Koll´ar for useful discussions about cubichypersurfaces over finite fields.
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