AA HIGHER DIMENSIONAL HILBERT IRREDUCIBILITY THEOREM
GIULIO BRESCIANIA
BSTRACT . Assuming the weak Bombieri-Lang conjecture, we prove that a generalization of Hilbert’sirreducibility theorem holds for families of geometrically mordellic varieties (for instance, families ofhyperbolic curves). As an application we prove that, assuming Bombieri-Lang, there are no polynomialbijections Q × Q → Q . C ONTENTS
1. Introduction 12. Pulling families to maximal Kodaira dimension 33. Higher dimensional HIT 94. Polynomial bijections Q × Q → Q NTRODUCTION
Serre reformulated Hilbert’s irreducibility theorem as follows [Ser97, Chapter 9].
Theorem (Hilbert’s irreducibility, Serre’s form) . Let k be finitely generated over Q , and let f : X → P be a morphism with X a scheme of finite type over k. Suppose that the generic fiber is finite, and that thereare no generic sections Spec k ( P ) → X. Then X ( k ) → P ( k ) is not surjective. Recall that the weak Bombieri-Lang conjecture states that, if X is a positive dimensional varietyof general type over a field k finitely generated over Q , then X ( k ) is not dense in X .A variety X over a field k is geometrically mordellic , or GeM , if every subvariety of X ¯ k is of generaltype. This generalizes to defining a scheme X as geometrically mordellic, or GeM, if it is of finitetype over k and every subvariety of X ¯ k is of general type. If the weak Bombieri-Lang conjectureholds and k is a field finitely generated over Q , then the set of rational points of a GeM scheme over k is finite, since its Zariski closure cannot have positive dimension.Assuming Bombieri-Lang, we prove that Hilbert’s irreducibility theorem generalizes to mor-phisms whose generic fiber is GeM. Theorem A.
Let k be finitely generated over Q , and let f : X → P be a morphism with X a scheme of finitetype over k. Suppose that the generic fiber is GeM, and that there are no generic sections Spec k ( P ) → X. The author is supported by the DFG Priority Program "Homotopy Theory and Algebraic Geometry" SPP 1786. a r X i v : . [ m a t h . N T ] F e b GIULIO BRESCIANI
Assume either that the weak Bombieri-Lang conjecture holds in every dimension, or that it holds upto dimension equal to dim
X and that there exists an N such that | X v ( k ) | ≤ N for every rational pointv ∈ P ( k ) . Then X ( k ) → P ( k ) is not surjective. There is a version of Hilbert’s irreducibility theorem over non-rational curves, and the same istrue for the higher dimensional generalization.
Theorem B.
Assume that the weak Bombieri-Lang conjecture holds in every dimension. Let k be finitelygenerated over Q , and let f : X → C be a morphism with X any scheme of finite type over k and C ageometrically connected curve. Assume that the generic fiber is GeM, and that there are no generic sections
Spec k ( C ) → X. Then X ( h ) → C ( h ) is not surjective for some finite extension h / k. As an application of Theorem A we give an answer to a long-standing Mathoverflow question[Mat19] which asks whether there exists a polynomial bijection Q × Q → Q , conditional on theweak Bombieri-Lang conjecture. Theorem C.
Assume that the weak Bombieri-Lang conjecture for surfaces holds, and let k be a field finitelygenerated over Q . There are no polynomial bijections k × k → k. We remark that B. Poonen has proved that, assuming the weak Bombieri-Lang conjecture forsurfaces, there are polynomials giving injective maps Q × Q → Q , see [Poo10].In 2019, T. Tao suggested on his blog [Tao19] a strategy to try to solve the problem of polynomialbijections Q × Q → Q conditional on Bombieri-Lang, let us summarize it. Given a morphism A → A and a cover c : A (cid:57)(cid:57)(cid:75) A , denote by P c the pullback of A . If P c is of general type,by Bombieri-Lang P c ( Q ) is not dense in P c and hence by Hilbert irreducibility a generic section A (cid:57)(cid:57)(cid:75) P c exists. If P c is of general type for "many" covers c , one might expect this to force theexistence a generic section A (cid:57)(cid:57)(cid:75) A , which would be in contradiction with the bijectivity of A ( Q ) → A ( Q ) .The strategy had some gaps, though. There were no results showing that the pullback P c is ofgeneral type for "many" covers c , and it was not clear how this would force a generic section of A → A . Tao started a so-called "polymath project" in order to crowdsource a formalization. Theproject was active for roughly one week in the comments section of the blog but didn’t reach aconclusion. Partial progress was made, we cite the two most important contributions. W. Sawinshowed that A ( Q ) → A ( Q ) can’t be bijective if the generic fiber has genus 0 or 1. H. Pastenshowed that, for some morphisms A → A with generic fiber of genus at least 2, the base changeof A along the cover z − b : A → A is of general type for a generic b .Theorem A is far more general than Theorem C, but it is possible to extract from the proof ofthe former the minimal arguments needed in order to prove the latter. These minimal argumentsare a formalization of the ideas described above, hence as far as Theorem C is concerned we haveessentially filled in the gaps in Tao’s strategy. Acknowledgements.
I would like to thank Hélène Esnault for reading an earlier draft of the paperand giving me a lot of valuable feedback, and Daniel Loughran for bringing to my attention theproblem of polynomial bijections Q × Q → Q . HIGHER DIMENSIONAL HILBERT IRREDUCIBILITY THEOREM 3
Conventions.
A variety over k is a geometrically integral scheme of finite type over k . A smooth,projective variety is of general type if its Kodaira dimension is equal to its dimension: in particular,a point is a variety of general type.We say that a variety is of general type if it is birational to a smooth, projective variety of generaltype. More generally, we define the Kodaira dimension of any variety X as the Kodaira dimensionof any smooth projective variety birational to X .Curves are assumed to be smooth, projective and geometrically connected. Given a variety X (resp. a scheme of finite type X ) and C a curve, a morphism X → C is a family of varieties ofgeneral type (resp. of GeM schemes) if the generic fiber is a variety of general type (resp. a GeMscheme). Given a morphism f : X → C , a generic section of f is a morphism s : Spec k ( C ) → X (equivalently, a rational map s : C (cid:57)(cid:57)(cid:75) X ) such that f ◦ s is the natural morphism Spec k ( C ) → C (equivalently, the identity C (cid:57)(cid:57)(cid:75) C ).2. P ULLING FAMILIES TO MAXIMAL K ODAIRA DIMENSION
This section is of purely geometric nature, thus we may assume that k is algebraically closedof characteristic 0 for simplicity. The results then descend to non-algebraically closed fields withstandard arguments.Given a family f : X → P of varieties of general type and c : P → P a finite covering, let f c : X c → P be the fiber product and, by abuse of notation, c : X c → X the base change of c .The goal of this section is to obtain sufficient conditions on c such that X c is of general type. Thisgoal will be reached in Corollary 2.13, which contains all the geometry we’ll need for arithmeticapplications.Let us say that X → P is birationally trivial if there exists a birational morphism X (cid:57)(cid:57)(cid:75) F × P which commutes with the projection to P . If f is birationally trivial, then clearly our goal isunreachable, since X c will have Kodaira dimension − ∞ no matter which cover c : P → P wechoose. We will show that this is in fact the only exception.Assume that X is smooth and projective (we can always reduce to this case), then the relativedualizing sheaf ω f exists [Kle80, Corollary 24]. First, we show that for every non-birationally trivialfamily there exists an integer m such that f ∗ ω mf has some positivity 2.10. Second, we show that if f ∗ ω mf has enough positivity, then X is of general type 2.11. We then pass from "some" to "enough"positivity by base changing along a cover c : P → P .2.1. Positivity of f ∗ ω mf . There are two cases: either there exists some finite cover c : C → P suchthat X d → C is birationally trivial, or not. Let us say that f : X → P is birationally isotrivial in thefirst case, and non-birationally isotrivial in the second case.The non-birationally isotrivial case has been extensively studied by Viehweg and Kollár, wedon’t need to do any additional work. Proposition 2.1 (Kollár, Viehweg [Kol87, Theorem p.363]) . Let f : X → P be a non-birationallyisotrivial family of varieties of general type, with X smooth and projective. There exists an m > such that,in the decomposition of f ∗ ω mf in a direct sum of line bundles, each factor has positive degree. (cid:3) GIULIO BRESCIANI
We are thus left with studying the positivity of f ∗ ω mf in the birationally isotrivial, non-birationallytrivial case. We’ll have to deal with various equivalent birational models of families, not alwayssmooth, so let us first compare their relative pluricanonical sheaves.2.1.1. Morphisms of pluricanonical sheaves.
In this subsection, fix a base scheme S . If a morphism to S is given, it is tacitly assumed to be flat, locally projective, finitely presentable, with Cohen-Macauleyequidimensional fibers of dimension n . For such a morphism f : X → S , the relative dualizingsheaf ω f exists and is coherent, see [Kle80, Theorem 21]. Recall that ω f satisfies the functorialisomorphism f ∗ Hom X ( F , ω f ⊗ X f ∗ N ) (cid:39) Hom S ( R n f ∗ F , N ) for every quasi-coherent sheaf F on X and every quasi-coherent sheaf N on S . Write ω ⊗ mf for the m -th tensor power, we may drop the superscript _ ⊗ and just write ω mf if ω f is a line bundle.Every flat, projective map f : X → S of smooth varieties over k satisfies the above, see [Kle80,Corollary 24], and in this case we can compute ω f as ω X ⊗ f ∗ ω − S , where ω X and ω S are the usualcanonical bundles. Moreover, the relative dualizing sheaf behaves well under base change alongmorphisms S (cid:48) → S , see [Kle80, Proposition 9.iii].Given a morphism g : Y → X over S and a quasi-coherent sheaf F over Y , then R n f ∗ ( g ∗ F ) is the E n ,02 term of the Grothendieck spectral sequence ( R p f ◦ R q g )( F ) ⇒ R p + q ( f ◦ g )( F ) , thus there is anatural morphism R n f ∗ ( g ∗ F ) → R n ( f g ) ∗ F . This induces a natural mapHom Y ( F , ω f g ) = Hom S ( R n ( f g ) ∗ F , O S ) → Hom S ( R n f ∗ ( g ∗ F ) , O S ) = Hom X ( g ∗ F , ω f ) . Definition 2.2. If g : Y → X is a morphism over S , define g (cid:52) , f : g ∗ ( ω f g ) → ω f as the sheafhomomorphism induced by the identity of ω f g via the homomorphismHom Y ( ω f g , ω f g ) → Hom X ( g ∗ ω f g , ω f ) given above for F = ω f g . With an abuse of notation, call g (cid:52) , f the induced sheaf homomorphism g ∗ ( ω ⊗ mf g ) → ω ⊗ mf for every m ≥
0. If there is no risk of confusion, we may drop the subscript _ f and just write g (cid:52) .The following facts are straightforward, formal consequences of the definition of g (cid:52) , we omitproofs. Lemma 2.3.
Let g : Y → X be a morphism over S and s : S (cid:48) → S any morphism, f (cid:48) : X (cid:48) → S (cid:48) ,g (cid:48) : Y (cid:48) → X (cid:48) the pullbacks to S (cid:48) . By abuse of notation, call s the morphisms Y (cid:48) → Y, X (cid:48) → X, too. Theng (cid:48) (cid:52) = g (cid:52) | X (cid:48) ∈ Hom X (cid:48) ( g (cid:48)∗ ω f (cid:48) g (cid:48) , ω f (cid:48) ) = Hom X (cid:48) ( s ∗ g ∗ ω f g , s ∗ ω f ) . (cid:3) Lemma 2.4.
For every quasi-coherent sheaf F on Y, the natural map
Hom Y ( F , ω f g ) → Hom X ( g ∗ F , ω f ) constructed above is given by ϕ (cid:55)→ g (cid:52) ◦ g ∗ ϕ : g ∗ F → g ∗ ω f g → ω f . (cid:3) HIGHER DIMENSIONAL HILBERT IRREDUCIBILITY THEOREM 5
Corollary 2.5.
Let h : Z → Y, g : Y → X be morphisms over S. Then, for every m ≥ ,g (cid:52) ◦ g ∗ h (cid:52) = ( gh ) (cid:52) : gh ∗ ω ⊗ mf gh → g ∗ ω ⊗ mf g → ω ⊗ mf . (cid:3) Corollary 2.6.
Let g : Y → X be a morphism over S. Suppose that a group H acts on Y , X , S and g , f are H-equivariant. Then g ∗ ω ⊗ mf g , ω ⊗ mf are H-equivariant sheaves and g (cid:52) : g ∗ ω ⊗ mf g → ω ⊗ mf is H-equivariant. (cid:3) Lemma 2.7.
Let g : Y → X be a morphism over S. Assume that Y , X are smooth varieties over a field k,and that g is birational. Then g (cid:52) is an isomorphism.Proof.
We have ω f = ω X ⊗ f ∗ ω − S and ω f g = ω Y ⊗ ( f g ) ∗ ω − S . Moreover, ω Y = g ∗ ω X ⊗ O Y ( R ) where R is some effective divisor whose irreducible components are contracted by g , hence ω f g = g ∗ ω f ⊗ O Y ( R ) . Since g ∗ O Y ( mR ) (cid:39) O X , we have a natural isomorphism g ∗ ( ω mf g ) (cid:39) ω mf byprojection formula. This is easily checked to correspond to g (cid:52) , which is then an isomorphism asdesired. (cid:3) Birationally isotrivial families.
Let C be a smooth projective curve and f : X → C a birationallyisotrivial family of varieties of general type, and let F / k be a smooth projective variety such thatthe generic fiber of f is birational to F . Let H be the finite group of birational automorphisms of F .The scheme of fiberwise birational isomorphisms Bir ( X / C , F ) → C restricts to an H -torsor on somenon-empty open subset V of C . The action of H on Bir ( X / C , F ) | V is transitive on the connectedcomponents, thus they are all birational. Definition 2.8.
In the situation above, define b : B f → C as the smooth completion of any connectedcomponent of Bir ( X / C , F ) | V , and G f ⊆ H as the subgroup of elements mapping B f to itself. Let uscall B f → C and G f the monodromy cover and the monodromy group of f respectively.We have that B f → C is a G f -Galois covering characterized by the following universal property:if C (cid:48) is a smooth projective curve with a finite morphism c : C (cid:48) → C , then X c → C (cid:48) is birationallytrivial if and only if there exists a factorization C (cid:48) → B f → C . Proposition 2.9.
Let f : X → C be a birationally isotrivial family of varieties of general type, with Xsmooth and projective. If p ∈ B f is a ramification point of the monodromy cover b : B f → C, then for somem there exists an injective sheaf homomorphism O B f ( p ) → f b ∗ ω mf b .Proof. The statement is equivalent to the existence of a non-trivial section of ω mf b which vanisheson the fiber X b , p . Let F be as above, G f acts faithfully with birational maps on F . By equivariantresolution of singularities, we may assume that G f acts faithfully by isomorphisms on F . We havethat X is birational to ( F × B f ) / G f where G f acts diagonally.By resolution of singularities, let X (cid:48) be a smooth projective variety with birational morphisms X (cid:48) → X , X (cid:48) → ( F × B f ) / G f : thanks to Lemma 2.7 we may replace X with X (cid:48) and assume we havea birational morphism X → ( F × B f ) / G f . By equivariant resolution of singularities again, we mayfind a smooth projective variety Y with an action of G f , a birational morphism g : Y → X b and abirational, G f -equivariant morphism y : Y → F × B f . Call π : F × B f → B f the projection. GIULIO BRESCIANI
Y X b XF × B f ( F × B f ) / G f B f C y g b π b π y f b Recall that we are trying to find a global section of ω mf b that vanishes on X b , p , where p is aramification point of b . Thanks to Lemma 2.7, we have that π y ∗ ω m π y (cid:39) π ∗ ω m π (cid:39) O B f ⊗ H ( F , ω mF ) ,thus H ( Y , ω m π y ) = H ( F , ω mF ) = H ( Y p , ω mY p ) .The sheaf homomorphism g (cid:52) = g (cid:52) , f b : g ∗ ω m π y → ω mf b induces a linear map g (cid:52) ( p ) : H ( Y p , ω mY p ) = H ( Y , ω m π y ) g (cid:52) −→ H ( X b , ω mf b ) •| p −→ H ( X b , p , ω mX b , p ) where the last map is the restriction to the fiber. Let V ⊆ B f be the étale locus of b : B f → C . Since X b | V is smooth, then g (cid:52) restricts to an isomorphism on X b | V thanks to Lemma 2.7 and thus themap H ( Y , ω m π y ) → H ( X b , ω mf b ) is injective.We want to show that the restriction map H ( X b , ω mf b ) → H ( X b , p , ω mX b , p ) is not injective forsome m , it is enough to show that g (cid:52) ( p ) is not injective. Thanks to Lemma 2.3, we have that g (cid:52) ( p ) = g p , (cid:52) : H ( Y p , ω mY p ) → H ( X b , p , ω mX b , p ) .Recall now that G f acts on Y . Let G f , p be the stabilizer of p ∈ B f , it is a non-trivial group since p is a ramification point. Thanks to Corollary 2.6, the stabilizer G f , p acts naturally on H ( Y p , ω mY p ) ,H ( F , ω mF ) , H ( X b , p , ω mX b , p ) , and the maps y p , (cid:52) : H ( Y p , ω mY p ) (cid:39) H ( F , ω mF ) , g p , (cid:52) : H ( Y p , ω mY p ) → H ( X b , p , ω mX b , p ) are G f , p -equivariant. Moreover, the action on H ( X b , p , ω mX b , p ) is trivial since theaction on X b , p is trivial. It follows that g (cid:52) ( p ) is G f , p -invariant, and hence to show that it is notinjective for some m it is enough to show that the action of G f , p on H ( F , ω mF )) = H ( Y p , ω mY p ) isnot trivial for some m .Since F is of general type, F (cid:57)(cid:57)(cid:75) P ( H ( F , ω mF )) is generically injective for some m , fix it. Sincethe action of G f , p on F is faithful, for every non-trivial g ∈ G f , p there exists a section s ∈ H ( F , ω mF ) and a point v ∈ F such that s ( v ) = s ( g ( v )) (cid:54) =
0, in particular the action of G f , p on H ( F , ω mF ) is not trivial and we conclude. (cid:3) Corollary 2.10.
Let f : X → P be a non-birationally trivial family of varieties of general type, with Xsmooth and projective. Then there exists an m with an injective homomorphism O ( ) → f ∗ ω mf .Proof. If f is not birationally isotrivial, apply Proposition 2.1. Otherwise, f is birationally isotrivialand not birationally trivial, thus the monodromy cover b : B f → P is not trivial. Since P hasno non-trivial étale covers, we have that B f → P has at least one ramification point p . Let m be HIGHER DIMENSIONAL HILBERT IRREDUCIBILITY THEOREM 7 the integer given by Proposition 2.9, and write f ∗ ω mf = (cid:76) i O P ( d i ) . Since O B f ( p ) ⊆ f b ∗ ω mf b and ω f b = b ∗ ω f , see [Kle80, Proposition 9.iii], there exists an i with d i > (cid:3) Pulling families to maximal Kodaira dimension.
Now that we have established a positivityresult for f ∗ ω mf of any non-birationally trivial family f : X → P , let us use this to pull families tomaximal Kodaira dimension. Proposition 2.11.
Let f : X → P be a family of varieties of general type, with X smooth and projective.Then X is of general type if and only if there exists an injective homomorphism O P ( ) → f ∗ ω m X , orequivalently O P ( m + ) → f ∗ ω m f , for some m > .Proof. By resolution of singularities, there exists a birational morphism g : X (cid:48) → X with X (cid:48) smooth and projective such that the generic fiber of X (cid:48) → P is smooth and projective. Wehave ω X (cid:48) = g ∗ ω X ⊗ O X (cid:48) ( R ) where R is some effective divisor whose irreducible components arecontracted by g , hence g ∗ ω mX (cid:48) = ω mX ⊗ g ∗ O ( mR ) = ω mX for every m ≥
0. We may thus replace X with X (cid:48) and assume that the generic fiber is smooth. This guarantees that rank f ∗ ω mX = rank f ∗ ω mf has growth O ( m dim X − ) .If there are no injective homomorphisms O P ( ) → f ∗ ω mX for every m >
0, then h ( ω mX ) ≤ rank f ∗ ω mX = rank f ∗ ω mf , and this has growth O ( m dim X − ) .On the other hand, let O P ( ) → f ∗ ω m X be an injective homomorphism for some m >
0. Inparticular, X has Kodaira dimension ≥ m , the closure Y of the image of X (cid:57)(cid:57)(cid:75) P ( H ( X , ω mm X )) has dimension equal to theKodaira dimension of X and k ( Y ) is algebraically closed in k ( X ) , see [Iit71, §3]. If X (cid:48) is a smoothprojective variety birational to X , then there is a natural isomorphism H ( X , ω mm X ) = H ( X (cid:48) , ω mm X (cid:48) ) ,see [Har77, Ch. 2, Theorem 8.19]. Thus, up to replacing X with some other smooth, projectivevariety birational to X , we may assume that X (cid:57)(cid:57)(cid:75) Y ⊆ P ( H ( X , ω mm X )) is defined everywhereand has smooth, projective generic fiber Z by resolution of singularities. Iitaka has then shown that Z has Kodaira dimension 0, see [Iit71, Theorem 5]. This is easy to see in the case in which ω mm X isbase point free, since then ω mm X is the pullback of O ( ) and thus ω mm Z = ω mm X | Z is trivial.Let us recall briefly Grothendieck’s convention that, if V is a vector bundle, then P ( V ) is the set(or scheme) of linear quotients V → k up to a scalar. A non-trivial linear map W → V thus inducesa rational map P ( V ) (cid:57)(cid:57)(cid:75) P ( W ) by restriction. If L is a line bundle with non-trivial global sections,the rational map X (cid:57)(cid:57)(cid:75) P ( H ( X , L )) is defined by sending a point x ∈ X outside the base locusto the quotient H ( X , L ) → L x (cid:39) k . If L embeds in another line bundle M , then there is a naturalfactorization X (cid:57)(cid:57)(cid:75) P ( H ( X , M )) (cid:57)(cid:57)(cid:75) P ( H ( X , L )) , and any point of X outside the support of M / L and outside the base locus of L maps to the locus of definition of P ( H ( X , M )) (cid:57)(cid:57)(cid:75) P ( H ( X , L )) .Let F ⊆ X be the fiber over any rational point of P . The injective homomorphism O P ( ) → f ∗ ω m X induces an injective homomorphism O P ( m ) → f ∗ ω mm X , choose any embedding O P ( ) →O P ( m ) , these induce an injective homomorphism O X ( F ) → ω mm X . Since O X ( F ) induces themorphism f : X → P , the composition X → Y ⊆ P ( H ( X , ω mm X )) (cid:57)(cid:57)(cid:75) P coincides with f . Observe that the right arrow depends on the choice of the embedding O X ( F ) → ω mm X , but the composition doesn’t.Let ξ be the generic point of P , U ⊆ Y an open subset such that U → P is defined, Y ξ theclosure of U ξ in Y . Then the generic fiber Z of X → Y is the generic fiber of X ξ → Y ξ , too. Byhypothesis, X ξ is of general type, thus by adjunction ω X ξ | Z = ω Z is big and hence Z is of generaltype.Since Z is a variety of general type of Kodaira dimension 0 over Spec k ( Y ) , then Z = Spec k ( Y ) ,the morphism X → Y is generically injective and thus X is of general type. (cid:3) Remark 2.12.
We don’t actually need the precision of Proposition 2.11: for our purposes it isenough to show that, if f ∗ ω m X has a positive enough sub-line bundle for some m , then X is ofgeneral type. This weaker fact has a more direct proof, let us sketch it.First, let us mention an elementary fact about injective sheaf homomorphisms. Let P , Q be vectorbundles on P and M , N vector bundles on X , with P of rank 1. Suppose we are given injectivehomomorphisms m ∈ Hom ( P , f ∗ M ) , n ∈ Hom ( Q , f ∗ N ) . Then m a ⊗ n ∈ Hom ( P ⊗ a ⊗ Q , f ∗ ( M ⊗ a ⊗ N )) is injective for every a >
0: this can be checked on the generic point of P and thus on thegeneric fiber X k ( P ) , where the fact that P has rank 1 allows us to reduce to the fact that the tensorproduct of non-zero sections of vector bundles is non-zero on an integral scheme.Assume we have an injective homomorphism O P ( m ) → f ∗ ω m X , or equivalently O P ( m ) → f ∗ ω m f , we want to prove that X is of general type. Let r ( m ) be the rank f ∗ ω mm f for every m . Sincethe generic fiber is of general type, up to replacing m by a multiple m (cid:48) we may assume that thegrowth of r ( m ) is O ( m dim X − ) . The induced morphism O P ( m (cid:48) ) → f ∗ ω m (cid:48) f is injective thanks tothe above.Thanks to [Vie83, Theorem III], every line bundle in the factorization of f ∗ ω mm f has non-negativedegree, we may thus choose an injective homomorphism O r ( m ) P → f ∗ ω mm f . Taking the tensorproduct with the m -th power of the homomorphism given by hypothesis, we get an homomorphism O P ( mm ) r ( m ) → f ∗ ω mm f which is injective thanks to the above.Since f ∗ ω mm X = f ∗ ω mm f ⊗ O P ( − mm ) , we thus have an injective homomorphism O P ( mm ) r ( m ) → f ∗ ω mm X .In particular, we have h ( ω mm X ) ≥ ( mm + ) r ( m ) which has growth O ( m n ) , hence X is of generaltype. Corollary 2.13.
Let f : X → P be a non-birationally trivial family of varieties of general type. Then thereexists an integer d and a non-empty open subset U ⊆ P such that, for every finite cover c : P → P with deg c ≥ d and such that the branch points of c are contained in U, we have that X c is of general type. If Xis smooth and projective, U can be chosen as the largest open subset such that f | f − ( U ) is smooth.Proof. By resolution of singularities, we may assume that X is smooth and projective. By genericsmoothness, there exists an open subset U ⊆ P be such that f | X U is smooth. We have that X c issmooth for every c : P → P whose branch points are contained in U since each point of X c issmooth either over X or over P . HIGHER DIMENSIONAL HILBERT IRREDUCIBILITY THEOREM 9
Let m be the integer given by Corollary 2.10, we have an injective homomorphism O ( ) → f ∗ ω m f . Set d = m +
1, for every finite cover c of degree deg c ≥ d = m + O ( m + ) → f c ∗ ω m f c and thus O ( ) → f c ∗ ω m X c . It follows that X c is ofgeneral type thanks to Proposition 2.11. (cid:3)
3. H
IGHER DIMENSIONAL
HIT3.1.
Pulling fat sets.
Recall that Serre [Ser97, Chapter 9] defined a subset S of P ( k ) as thin if thereexists a morphism f : X → P with X of finite type over k , finite generic fiber and no genericsections Spec k ( P ) → X such that S ⊆ f ( X ( k )) . It’s immediate to check that a subset of a thin setis thin, and a finite union of thin sets is thin. Serre’s form of Hilbert’s irreducibility theorem saysthat, if k is finitely generated over Q , then P ( k ) is not thin. Definition 3.1.
A subset S ⊆ P ( k ) is fat if the complement P ( k ) \ S is thin.Given a subset S ⊆ P ( k ) , a finite set of finite morphisms D = { d i : D i → P } i each of degree > D i smooth, projective and geometrically connected is a scale for S if S ∪ (cid:83) i d i ( D i ( k )) = P ( k ) .The set of branch points of the scale D is the union of the sets of branch points of d i .Using the fact that a connected scheme with a rational point is geometrically connected [Sta20,Lemma 04KV], it’s immediate to check that a subset of P is fat if and only if it has a scale. The setof branch points of a scale gives valuable information about a fat set. Lemma 3.2.
Let S ⊆ P be a fat set, and let D = { d i : D i → P } i be a scale for S. Let c : P → P be amorphism such that the sets of branch points of c and D are disjoint. Then c − ( S ) is fat.Proof. Let d (cid:48) i : D (cid:48) i → P be the base change of d i along c : P → P . By construction, c − ( S ) ∪ (cid:83) i d (cid:48) i ( D (cid:48) i ( k )) = P ( k ) . Since the sets of branch points of c and d i are disjoint, we have that D (cid:48) i isgeometrically connected, see for instance [Str20, Lemma 2.8]. Moreover, D (cid:48) i is smooth since eachpoint of D (cid:48) i is étale either over P or D i . It follows that d (cid:48) i has degree > { d (cid:48) i : D (cid:48) i → P } i is ascale for c − ( S ) , which is thus fat. (cid:3) Decreasing the fiber dimension.
Let us now prove Theorem A. Using Hilbert’s irreducibility,it’s easy to check that Theorem A is equivalent to the following statement.If the generic fiber of f : X → P is GeM and f ( X ( k )) is fat, there exists a section Spec k ( P ) → X .We prove this statement by induction on the dimension of the generic fiber. If the generic fiberhas dimension 0, this follows from the definition of fat set. Let us prove the inductive step.We define recursively a sequence of closed subschemes X i + ⊆ X i with X = X and such that f ( X i ( k )) ⊆ P k is fat. • Define X (cid:48) i as the closure of X i ( k ) with the reduced scheme structure, f ( X (cid:48) i ( k )) = f ( X i ( k )) ⊆ P k is fat. • Define X (cid:48)(cid:48) i as the union of the irreducible components of X (cid:48) i which dominate P , f ( X (cid:48)(cid:48) i ( k )) ⊆ P k is fat since f ( X (cid:48) i ( k )) \ f ( X (cid:48)(cid:48) i ( k )) is finite. • Write X (cid:48)(cid:48) i = (cid:83) j Y i , j as union of irreducible components, Y i , j → P is dominant for every j . For every j , there exists a finite cover C i , j → P with C i , j smooth projective and arational map Y i , j (cid:57)(cid:57)(cid:75) C i , j with geometrically irreducible generic fiber. If C i , j → P is anisomorphism, define Z i , j = Y i , j . Otherwise, there exists a non-empty open subset V i , j ⊆ Y i , j such that Y i , j (cid:57)(cid:57)(cid:75) C i , j is defined on V i , j . In particular, f ( V i , j ( k )) ⊆ P ( k ) is thin. Define Z i , j = Y i , j \ V i , j and X i + = (cid:83) j Z i , j ⊆ X i . By construction, f ( X i + ( k )) ⊆ P ( k ) is fat since f ( X (cid:48)(cid:48) i ( k )) \ f ( X i + ( k )) is thin.By noetherianity, the sequence is eventually stable, let r be such that X r + = X r . Since X r + = X r ,then X r ( k ) is dense in X r , thus every irreducible component is geometrically irreducible, see [Sta20,Lemma 0G69]. Moreover, every irreducible component of X r dominates P with geometricallyirreducible generic fiber. Replace X with X r and write X = (cid:83) j Y j as union of irreducible components,we may assume that Y j → P is a family of GeM varieties for every j and Y j ( k ) is dense in Y j .If Y j → P is birationally trivial for some j , since Y j ( k ) is dense in Y j and a generic fiber of Y j → P has a finite number of rational points, then dim Y j = Y j → P is birational and weconclude. Otherwise, thanks to Corollary 2.13, there exists an integer d and a non-empty opensubset U ⊆ P such that, for every finite cover c : P → P with deg c ≥ d such that the branchpoints of c are contained in U , we have that Y j , c is of general type for every j .Let D = { d l : D l → P } be a scale for f ( X ( k )) . Up to shrinking U furthermore, we mayassume that the set of branch points of D is disjoint from U . Since we are assuming that the weakBombieri-Lang conjecture holds up to dimension dim X , the dimension of Y j , c ( k ) ⊆ Y j , c is strictlysmaller than dim Y j for every j . Moreover, we have that f c ( X c ( k )) = m − c ( f ( X ( k ))) is fat thanks toLemma 3.2. It follows that, by induction hypothesis, there exists a generic section Spec k ( P ) → X c for every finite cover c as above. There are a lot of such covers: let us show that we can choose themso that the resulting sections "glue" to a generic section Spec k ( P ) → X .3.3. Gluing sections.
Choose coordinates on P so that 0, ∞ ∈ U , let p be any prime numbergreater than d . For any positive integer n , let m n : P → P be the n -th power map. Wehave shown above that there exists a rational section P (cid:57)(cid:57)(cid:75) X m p for every prime p ≥ d , call s p : P (cid:57)(cid:57)(cid:75) X m p → X the composition.We either assume that there exists an integer N such that, for every rational point v ∈ P ( k ) ,we have | X v ( k ) | ≤ N or that the Bombieri-Lang conjecture holds in every dimension. In thesecond case, the uniform bound N exists thanks to a theorem of Caporaso-Harris-Mazur andAbramovich-Voloch [CHM97, Theorem 1.1] [AV96, Theorem 1.5] [Abr97]. Choose N + p , . . . , p N greater than d , for each one we have a rational section X P P fm p s p HIGHER DIMENSIONAL HILBERT IRREDUCIBILITY THEOREM 11
Let Q = ∏ Ni = p i , for every i =
0, . . . , N , we get a rational section S p i by composition with s p i : X P P P fm Q / pi S pi m Q m pi s pi Let V ⊆ P be an open subset such that S p i is defined on V for every i . For every rational point v ∈ V ( k ) , we have | X v ( k ) | ≤ N and thus there exists a couple of different indexes i (cid:54) = j such that S p i ( v ) = S p j ( v ) for infinitely many v ∈ V ( k ) , hence S p i = S p j . Let Z ⊆ X be the image S p i = S p j , byconstruction we have k ( P ) = k ( t ) ⊆ k ( Z ) ⊆ k ( t − p i ) ∩ k ( t − p j ) ⊆ k ( t − Q ) .Using Galois theory on the cyclic extension k ( t − Q ) / k ( t ) , it is immediate to check that k ( t − p i ) ∩ k ( t − p j ) = k ( t ) ⊆ k ( t − Q ) since p i , p j are coprime, thus k ( Z ) = k ( t ) and Z → P is birational. Thisconcludes the proof of Theorem A.3.4. Non-rational base.
Let us show how Theorem A implies Theorem B. Let C be a geometricallyconnected curve over a field k finitely generated over Q , and let f : X → C be a morphism of finitetype whose generic fiber is a GeM scheme. Assume that there exists a non-empty open subset V ⊆ C such that X | V ( h ) → V ( h ) is surjective for every finite extension h / k . We want to prove thatthere exists a generic section C (cid:57)(cid:57)(cid:75) X . It’s easy to reduce to the case in which C is smooth andprojective, so let us make this assumption.Observe that, up to replacing X with an affine covering, we may assume that X is affine.Choose C → P any finite map: since X is affine, the Weil restriction R C / P ( X ) → P exists[BLR90, §7.6, Theorem 4]. Recall that R C / P ( X ) → P represents the functor on P -schemes S (cid:55)→ Hom C ( S × P C , X ) .If L / k ( C ) / k ( P ) is a Galois closure and Σ is the set of embeddings σ : k ( C ) → L as k ( P ) extensions, the scheme R C / P ( X ) L is isomorphic to the product ∏ Σ X × Spec k ( C ) , σ Spec L and henceis a GeM scheme, see [Bre20, Lemma 3.3]. It follows that the generic fiber R C / P ( X ) k ( P ) is a GeMscheme, too.Let U ⊆ P be the image of V ⊆ C . The fact that X | V ( h ) → V ( h ) is surjective for every finiteextension h / k implies that R C / P ( X ) | U ( k ) → U ( k ) is surjective. By Theorem A, we get a genericsection P (cid:57)(cid:57)(cid:75) R C / P ( X ) , which in turn induces generic section C (cid:57)(cid:57)(cid:75) X by the universal propertyof R C / P ( X ) . This concludes the proof of Theorem B.4. P OLYNOMIAL BIJECTIONS Q × Q → Q Let us prove Theorem C. Let k be finitely generated over Q , and let f : A → A be anymorphism. Assume by contradiction that f is bijective on rational points.First, let us show that the generic fiber of f is geometrically irreducible. This is equivalent tosaying that Spec k ( A ) is geometrically connected over Spec k ( A ) , or that k ( A ) is algebraically closed in k ( A ) . Let k ( A ) ⊆ L ⊆ k ( A ) a subextension algebraic over k ( A ) . Let C → A be afinite cover with C regular and k ( C ) = L . The rational map A (cid:57)(cid:57)(cid:75) C is defined in codimension1, thus there exists a finite subset S ⊆ A and an extension A \ S → C . Since the composition A \ S ( k ) → C ( k ) → A ( k ) is surjective up to a finite number of points, by Hilbert’s irreducibilitytheorem we have that C = A , i.e. L = k ( A ) .This leaves us with three cases: the generic fiber is a geometrically irreducible curve of geometricgenus 0, 1, or ≥
2. The first two have been settled by W. Sawin in the polymath project [Tao19],while the third follows from Theorem A. Let us give details for all of them.4.1.
Genus 0.
Assume that the generic fiber of f has genus 0. By generic smoothness, there existsan open subset U ⊆ A such that f | U is smooth. For a generic rational point u ∈ U ( k ) , the fiber f − ( f ( u )) is birational to a Brauer-Severi variety of dimension 1 and has a smooth rational point,thus it is birational to P and f − ( f ( u ))( k ) is infinite. This is absurd.4.2. Genus 1.
Assume now that the generic fiber has genus 1. By resolution of singularities, thereexists an open subset V ⊆ A , a variety X with a smooth projective morphism g : X → V whosefibers are smooth genus 1 curves and a compatible birational map X (cid:57)(cid:57)(cid:75) A . Up to shrinking V ,we may suppose that the fibers of f | V are geometrically irreducible. Let U be a variety with openembeddings U ⊆ X , U ⊆ A , replace V with g ( U ) ⊆ V so that g | U is surjective.The morphism X \ U → V is finite, let N be its degree. Since the fibers of U → V have at mostone rational point, it follows that | X v ( k ) | ≤ N + v ∈ V ( k ) .Every smooth genus 1 fibration is a torsor for a relative elliptic curve (namely, its relative Pic ),thus there exists an elliptic curve E → V such that X is an E -torsor. Moreover, every torsor for anabelian variety is torsion, thus there exists a finite morphism π : X → E over V induced by the n -multiplication map E → E for some n .If v ∈ V ( k ) is such that X v ( k ) is non-empty, then | X v ( k ) | = | E v ( k ) | ≤ N +
1. This means that, upto composing π with the ( N + ) ! multiplication E → E , we may assume that π ( X ( k )) ⊆ V ( k ) ⊆ E ( k ) , where V → E is the identity section. In particular, X ( k ) ⊆ π − ( V ( k )) is not dense. This isabsurd, since X is birational to A .4.3. Genus ≥ . Thanks to Theorem A, there exists an open subset V ⊆ A and a section s : V → A . It follows that A | V ( k ) = s ( V ( k )) , which is absurd since s ( V ) is a proper closed subset and A | V ( k ) is dense. R EFERENCES [Abr97] D. Abramovich. “A high fibered power of a family of varieties of general type dominatesa variety of general type”. In:
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DOI : . F REIE U NIVERSITÄT B ERLIN , A
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