Rational points of varieties with ample cotangent bundle over function fields of positive characteristic

Let $K$ be the function field of a smooth curve over an algebraically closed field $k$. Let $X$ be a scheme, which is smooth and projective over $K$. Suppose that the cotangent bundle $\Omega_{X/K}$ is ample. Let $R:={\rm Zar}(X)(K)\cap X)$ be the Zariski closure of the set of all $K$-rational points of $X$, endowed with its reduced induced structure. We prove that there is a projective variety $X_0$ over $k$ and a finite and surjective $K^{\rm sep}$-morphism $X_{0,K^{\rm sep}}\to R_{K^{\rm sep}}$, which is birational when ${\rm char}(K)=0$.


Introduction
Let K be the function field of a smooth curve U over a finite field k of characteristic p > 0. Let X be a scheme, which is smooth and projective over K. Suppose that the cotangent bundle Ω 1 X := Ω 1 X/K is ample. This aim of this note is to prove the following theorem. Theorem 1.1. If X(K) is Zariski dense in X then there is a smooth and projective variety X ′ 0 overk and a finite and surjective K sep -morphism X ′ 0,K sep → X K sep .

The geometry of the compactifications of torsors under vector bundles
We shall need the following results from [4].
Let S be a scheme, which is of finite type over a field k 0 .
If V a locally free sheaf over S, we shall write P(V ) for the S-scheme representing the functor on S-schemes T → {iso. classes of surjective morphisms of O T -modules V T → Q, where Q is locally free of rank 1 By definition, P(V ) comes with a universal line bundle O P (1). Let now be an exact sequence of locally free sheaves over S. Consider the S-group scheme F := Spec(Sym(F )) representing the group functor on S-schemes T → F ∨ T (T ). Let R E be the functor from S-schemes to sets given by There is an obvious (group functor-)action of F on R E .
(1) The natural morphism P(F ) → P(E) is a closed immersion and there is an isomorphism of line bundles O(P(F )) ≃ O P (1).
(2) The complement P(E)\P(F ) represents the functor R E . The isomorphism of functors on S-schemes R E → P(E)\P(F ) can be described as follows. There is a natural transformation of functors viewed as a morphism from E T onto a locally free sheaf of rank 1 (the latter being the trivial sheaf). This gives a morphism of schemes R E → P(E), which is an open immersion onto P(E)\P(F ).

Thus
(3) the scheme R E with its F-action is an S-torsor under F.
Further, by (1): (4) if E is ample then the scheme P(E)\P(F ) is affine (point (4) will actually not be used in the text).
Let us now suppose until the end of this section that F is ample.
(5) if Z ֒→ R E is a subscheme, which is closed in P(E), then the induced map Z → S is finite and has only a finite number of non-finite fibres; in particular, it is generically radicial; (6) for sufficiently large n ∈ N, the line bundle O P (n) is generated by its global sections and the induced k 0 -morphism φ n : P(E) → P(Γ(O P (n))) is generically finite; (7) the positive-dimensional fibres of the morphism φ are disjoint from P(E).
From the fact that fibre dimension is upper semi-continuous (see [3,IV,13.1.5]) and (7) we deduce that (8) the union I φn of the positive dimensional fibres of φ n is closed in P(E) and is contained in R E .
We endow I φn with its reduced-induced structure. From (5) we deduce that (9) the map I φn → S is finite and has only a finite number of non-finite fibres; in particular, it is generically radicial.
In particular, if S is irreducible of dimension > 0 and I φ → S is surjective then I φ has a single irreducible component of dimension > 0 and this irreducible component is of dimension dim(S).
We shall also need the (10) Every torsor under F is isomorphic to a torsor R E for some exact sequence E as in (1). The class in under the connecting map in the long exact sequence associated with the dual exact sequence E ∨ .

Torsors under vector bundles and purely inseparable base-change
Let now S be a scheme, which is smooth and projective over a field k 0 of characteristic p. Let V /S be a locally free sheaf. In [7, exp. 2, Prop. 1], the following result is shown: Here F S : S → S is the absolute Frobenius morphism.
In particular, if H 0 (S, F * S V ⊗Ω 1 S/k 0 ) = 0 then a non-trivial torsor under the vector bundle V is not trivialized by a base-change by F S . Proposition 2.2. Suppose that dim(S) > 0. Let V be an ample vector bundle of rank r over S. Let W be any vector bundle over S. Then we have H 0 (S, F n, * S (V ∨ ) ⊗ W ) = 0 for n large enough.
Proof. By induction on the dimension d 1 of S.
We may suppose without restriction of generality that k 0 is algebraically closed. Consider a pencil of hypersurfaces in S and let b : S → S the total space of the pencil, so that we are given a birational morphism m : . This is possible by the induction hypothesis and because (b * V ∨ ) η is ample. The fact that (b * V ∨ ) η is ample is a consequence of the fact that the restriction of V to any closed fibre of m is ample and of the fact that ampleness on the fibre of m is a constructible property. Now if we had H 0 (S, F n, * S (V ∨ ) ⊗ W ) = 0 for some n n 0 then the pull-back b * (F n, * would have a non-zero section. This section would be non-vanishing on S η , which is a contradiction. Thus we are reduced to prove the statement for d = 1. In this case, V is cohomologically p-ample (see [5,Rem. 6), p. 91]). Furthermore, using Serre duality, we may compute This completes the proof.
In other words, if T is not trivial, then it cannot be trivialized by a purely inseparable proper morphism.
Proof. (of Corollary 2.3). Let H be the function field of S and let H ′ |H be the (purely inseparable) function field extension given by φ. Let ℓ be sufficiently large so that there exist extensions H p −ℓ |H ′ |H. We may suppose that S ′ is a normal scheme, since we may replace S ′ by its normalization without restriction of generality. On the other hand the morphism F ℓ 0 S : S → S gives a presentation of S as its own normalization in H p −ℓ . Thus there is a natural factorization ). Hence the torsor T is not trivialized by F ℓ 0 S and thus cannot be trivialized by φ.
3 Proof of Theorem 1.1 In this section, the hypotheses of Theorem 1.1 are supposed to hold. In particular, we suppose that X(K) is dense in X. Recall that we use the notation of the introduction.
We shall use the jet schemes introduced in [6, sec. 2].
We suppose that X/K extends to a (not necessarily proper) smooth scheme π : X → U. We are then provided with an infinite tower of jet schemes · · · → J 2 (X /U) Here the bundle TX (resp. Sym i (Ω 1 U/k )) is implicitly pulled back from X (resp. U) to the scheme J i−1 (X /U).
The jet scheme J i (Y/U) exists for any smooth scheme Y over U and J i (·/U) is covariantly functorial for U-morphisms. The functor J i (·/U) preserves closed immersions and smooth morphisms. In particular, for any i ∈ N, there is a natural map and these maps are compatible with the morphisms Λ i .
We shall divide the proof into steps. The main part of the proof will take place over K and will not make use of the model X of X.
By a compactificationT of an S-scheme T , we shall mean a proper schemeT → S, which comes with an open immersion T ֒→T with dense image.
LetŪ → Spec k be a smooth compactification of U (this exists because k is perfect).
We suppose without restriction of generality that X(K) = ∅.
First choose any projective compactificationX 0 of X viewed as a scheme over U . By applying Néron desingularization toX 0 (see [1, chap. 3, th. 2])), we obtain another projective compactificationX 00 of X overŪ, with the property that the injectionX sm 00 (Ū) ֒→X 00 (Ū) = X(K) is a bijection. Here X sm 00 ⊂ X 00 is the largest open subset X sm 00 of X 00 , such that X sm 00 →Ū is smooth. This shows that there exists a model of X overŪ (the modelX sm 00 ), which is smooth and surjective ontoŪ , since X(K) = ∅.
Thus, we may (and do) suppose that U =Ū and that X is surjective (and smooth) overŪ . We letX be any compactification of X over U.
We now choose specific compactificationsJ i (X /U) over U for the jet schemes J i (X /U).
For i = 0, we letJ 0 (X /U) =X and we define them inductively for i 0.
So suppose that the compactificationJ i (X /U) has already been constructed. As said above, the J i (X /U)-scheme J i+1 (X /U) is a torsor under F ∨ i , where F i := (TX ⊗ Sym i+1 (Ω 1 U/k )) ∨ (viewed as a vector bundle over J i (X /U)). We shall denote this torsor by T i . Let be an extension (unique up to non-unique isomorphism) associated with the class of T i in H 1 (J i (X /U), F ∨ i ). It was mentionned in (2) section 2.1 that the J i (X /U)scheme J i+1 (X /U) can be realized as the complement P(E i )\P(F i ). We now define the compactificationJ i+1 (X /U) to be someJ i (X /U)-compactification of P(E i ), the latter being viewed as a scheme overJ i (X /U) via the open immersion J i (X /U) ֒→J i (X /U). We callΛ i+1 :J i+1 (X /U) →J i (X /U) the corresponding morphism.
The following diagram summarizes the resulting geometric configuration: Here the hooked horizontal arrows are open immersions and the square on the right is cartesian. Recall the following key properties. The scheme U is proper over k and the schemesJ i (X /U) are proper over U. The morphismsJ i+1 (X /U) → J i (X /U) are proper. The schemes J i (X /U) and P(E i ) are smooth and surjective onto U. By the valuative criterion of properness, there are natural maps λ i : U →J i (X /U) extending the maps λ i : U → J i (X /U). By unicity, the mapλ i is none other than the map λ i composed with the open immersion Step II. The schemes Z i ֒→J i (X/K).
DefineJ i (X/K) :=J i (X /U) K . We shall inductively construct closed integral subschemes Z i ֒→J i (X/K). The schemes Z i have the following properties. They are sent onto each other by the morphismsΛ i . The morphism Z i+1 → Z i is finite and generically radicial. Furthermore the image of X (U) = X(K) byλ i,K in J i (X/K) meets Z i in a dense set and Z i ⊆ J i (X/K).
The schemes Z i are defined via the following inductive procedure.
To define Z i+1 from Z i notice that by the Step I, we have an identification Notice also that F i,Z i is ample (over K) since Z i → X is finite. Thus by (6) in section 2.1, we are given a K-morphism for some n i ∈ N and φ n i . Call I φn i the union of the positive dimensional fibres of φ n i . Write H i ⊆ P n i K for a hyperplane such that φ −1 n i (H i ) = P(F i,Z i ). This exists by (1) in section 2.1.
We define Z i+1 as the union of the irreducible components of positive dimension of I φn i .
According to (7) in section 2.1, So it remains to show that Z i+1 is irreducible, finite and generically radicial over Z i to complete Step II.
For this, consider a section σ ∈ X (U) and suppose that λ i,K (σ) ∈ Z i (K).
LetP(E i,Z i ) be the Zariski closure of P(E i,Z i ) inJ i+1 (X /U) and letP(F i,Z i ) be the Zariski closure of P(F i,Z i ) inJ i+1 (X /U).

Then we haveλ
The latter follows from the fact thatλ i+1 (σ) ∈ J i+1 (X /U) = P(E i )\P(F i ) and from the fact that we have a set-theoretic identityP

Now choose a proper birational U-morphism
, which is an isomorphism over K and such that there exists a proper U-morphism be the section obtained from the valuative criterion of properness.
In this set-up, we then have thatλ ′ i+1 (σ) ∩P ′ (F i,Z i ) = ∅ (by equation (5)) and thus there exists a constant β i+1 0 such that Here ∩ refers to the scheme-theoretic intersection. This can be seen as follows.
Notice that φ * i (H i ) has a finite number of irreducible components andλ ′ i+1 (σ) meets only the horizontal irreducible components (over U) among those. Furthermore, the intersection multiplicity ofλ ′ i+1 (σ) with a fixed horizontal component is bounded independently of σ. Now (6) follows from the equality Finally we have Proof. Since length(·) has values in N, it is sufficient to prove the lemma with "=" rather than " " in the statement. Let L i be a U-ample line bundle on P n i U . Let χ i (m) be a polynomial with integer coefficients. According to [2], there is a scheme M i , which is of finite type over k, which represents the functor T → {µ ∈ P n i (U T )|χ((µ * L) ⊗m t ) = χ i (m) for all t ∈ T and all m ∈ Z} on locally noetherian k-schemes T /k. Now set L i = O(H i ) and Then we have a canonical inclusion Since M i (k) is finite (because k is finite), we may conclude.
We conclude from Lemma 3.1 that almost all the sections σ ∈ X (U) such that λ i,K (σ) ∈ Z i (K) have the property thatλ i+1,K (σ) ∈ I φn i (K). Call Σ i ⊆ X (U) = X(K) the set of these sections. Sinceλ i,K (Σ i ) is dense in Z i by assumption, we deduce from (9) in section 2.1 that I φn i has a single irreducible component of positive dimension, which is finite and generically radicial over Z i .
We have shown that Z i+1 = I φn i has all the required properties.
We now choose n 0 large enough so that for all n n 0 (this is possible by Proposition 2.2). Write F n 0 , * X Z i for the basechange of Z i → X by F n 0 X . Let is injective. Set X 0 := X U 0 =: Z 0 . The torsor F n 0 , * and since both schemes are projective over U u 0 , Grothendieck's GAGA theorem shows that this morphism of formal schemes comes from a unique morphism of schemes ι : X u 0 × k U u 0 → X Uu 0 .
By construction the morphism ι specializes to F n 0 Xu 0 at the closed point u 0 of U u 0 . Since the set of points of X Uu 0 , where the fibres of ι are of dimension 0 is open, we see that the morphism ι is finite over the generic point of U u 0 .
Let K be the function field of U u 0 . Since k is an excellent field, we know that the field extension K|K is separable. On the other hand the just constructed finite and surjective morphism X u 0 × k K → X K is defined over a finitely generated subfield K ′ (as a field over K) of K. The field extension K ′ |K is then still separable, so that by the theorem on separating transcendence bases, there exists a variety U ′ /K, which is smooth over K and whose function field is K ′ . Furthermore, possibly replacing U ′ by one of its open subschemas, we may assume that the morphism X u 0 × k K ′ → X K ′ extends to a finite and surjective morphism α : Let P ∈ U ′ (K sep ) be a K sep -point over K (the set U ′ (K sep ) is not empty because U ′ is smooth over K). The morphism P * α is the morphism advertised in Theorem 1.1.