Intermediate Jacobians and rationality over arbitrary fields
aa r X i v : . [ m a t h . AG ] D ec INTERMEDIATE JACOBIANS AND RATIONALITYOVER ARBITRARY FIELDS
OLIVIER BENOIST AND OLIVIER WITTENBERG
Abstract.
We prove that a three-dimensional smooth complete intersectionof two quadrics over a field k is k -rational if and only if it contains a linedefined over k . To do so, we develop a theory of intermediate Jacobians forgeometrically rational threefolds over arbitrary, not necessarily perfect, fields.As a consequence, we obtain the first examples of smooth projective varietiesover a field k which have a k -point, and are rational over a purely inseparablefield extension of k , but not over k . Introduction
Let k be a field and let Γ k := Aut( k/k ) be its absolute Galois group. Our mainresult answers positively a conjecture of Kuznetsov and Prokhorov. Theorem A (Theorem 4.7) . Let X ⊂ P k be a smooth complete intersection of twoquadrics. Then X is k -rational if and only if it contains a line defined over k . The question of the validity of Theorem A goes back to Auel, Bernardara andBolognesi [ABB14, Question 5.3.2 (3)], who raised it when k is a rational functionfield in one variable over an algebraically closed field.Using the fact that varieties X as in Theorem A are k -unirational if and onlyif they have a k -point (see Theorem 4.8), we obtain new counterexamples to theLüroth problem over non-closed fields. Theorem B (Theorem 4.11) . For any algebraically closed field κ , there exists athree-dimensional smooth complete intersection of two quadrics X ⊂ P κ (( t )) whichis κ (( t )) -unirational, κ (( t )) -rational, but not κ (( t )) -rational. When κ has characteristic 2, Theorem B yields the first examples of smoothprojective varieties over a field k which have a k -point and are rational over theperfect closure of k , but which are not k -rational (see Remark 4.12 (iii)).Theorem A may be compared to the classical fact that a smooth quadric over k is k -rational if and only if it has a k -point. However, although it is easy to check that asmooth projective k -rational variety has a k -point, the fact that a k -rational three-dimensional smooth complete intersection of two quadrics X necessarily contains a k -line is highly non-trivial. To prove it, we rely on obstructions to the k -rationalityof X arising from a study of its intermediate Jacobian.Such obstructions go back to the seminal work of Clemens and Griffiths [CG72]:the intermediate Jacobian of a smooth projective threefold over C that is C -rationalis isomorphic, as a principally polarized abelian variety over C , to the Jacobian of a Date : December 11th, 2019. (not necessarily connected) smooth projective curve. This implication was used in[CG72] to show that smooth cubic threefolds over C are never C -rational, and waslater applied to show the irrationality of several other classes of complex threefolds(see for instance [Bea77]).The arguments of Clemens and Griffiths were extended by Murre [Mur73] toalgebraically closed fields of any characteristic.More recently, we implemented them over arbitrary perfect fields k [BW19].That the intermediate Jacobian may be isomorphic to the Jacobian of a smoothprojective curve over k while not being so over k allowed us to produce new examplesof varieties over k that are k -rational but not k -rational.Hassett and Tschinkel [HT19a] subsequently noticed that over a non-closedfield k , the intermediate Jacobian carries further obstructions to k -rationality: if X is a smooth projective k -rational threefold, not only is its intermediate Jacobianisomorphic to the Jacobian Pic ( C ) of a smooth projective curve C over k , butfor an appropriate choice of C , the Pic ( C )-torsors associated with Γ k -invariantalgebraic equivalence classes of codimension 2 cycles on X are also of the form Pic α ( C ) for some Γ k -invariant class α in the Néron–Severi group of C k . When X is a smooth three-dimensional complete intersection of two quadrics, they usedthese obstructions in combination with the natural identification of the variety oflines of X with a torsor under the intermediate Jacobian of X , and with the workof Wang [Wan18], to prove Theorem A when k = R [HT19a, Theorem 36] (andlater [HT19b] over subfields of C ).The aim of the present article is to extend these arguments to arbitrary fields.Applications to k -rationality criteria for other classes of k -rational threefoldsappear in the work of Kuznetsov and Prokhorov [KP19].So far, we have been imprecise about what we call the intermediate Jacobian ofa smooth projective threefold X over k .If k = C , one can use Griffiths’ intermediate Jacobian J X constructed bytranscendental means. This is the original path taken by Clemens and Griffiths[CG72]. The algebraic part of Griffiths’ intermediate Jacobian has been shownto descend to subfields k ⊂ C by Achter, Casalaina-Martin and Vial [ACMV17,Theorem B]; the resulting k -structure on J X is the one used in [HT19b].Over algebraically closed fields k of arbitrary characteristic, a different construc-tion of an intermediate Jacobian Ab ( X ), based on codimension 2 cycles, wasprovided by Murre [Mur85, Theorem A p. 226] (see also [Kah18]). This cycle-theoretic approach to intermediate Jacobians had already been applied by him torationality problems (see [Mur73]). Over a perfect field k , the universal propertysatisfied by Murre’s intermediate Jacobian Ab ( X k ) induces a Galois descent datumon Ab ( X k ), thus yielding a k -form Ab ( X ) of Ab ( X k ) [ACMV17, Theorem 4.4].It is this intermediate Jacobian Ab ( X ), which coincides with J X when k ⊂ C ,that we used in [BW19].Over an imperfect field k , one runs into the difficulty that Murre’s definitionof Ab ( X k ) does not give rise to a k/k -descent datum on Ab ( X k ). Achter,Casalaina-Martin and Vial still prove, in [ACMV19], the existence of an algebraicrepresentative Ab ( X ) for algebraically trivial codimension 2 cycles on X (see §1.2of op. cit. for the definition). However, when k is imperfect, it is not knownwhether Ab ( X ) k is isomorphic to Ab ( X k ). For this reason, we do not know NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 3 how to construct on Ab ( X ) the principal polarization that is so crucial to theClemens–Griffiths method.To overcome this difficulty and prove Theorem A in full generality, we provide,over an arbitrary field k , an entirely new construction of an intermediate Jacobian.Our point of view is inspired by Grothendieck’s definition of the Picard scheme(for which see [FGA], [BLR90, Chapter 8], [Kle05]). With any smooth projective k -rational threefold X over k , we associate a functor CH X/k, fppf : (Sch /k ) op → (Ab)endowed with a natural bijection CH ( X k ) ∼ −→ CH X/k, fppf ( k ) (see Definition 2.9and (3.1)). The functor CH X/k, fppf is an analogue, for codimension 2 cycles, of thePicard functor Pic
X/k, fppf .Too naive attempts to define the functor CH X/k, fppf on the category of k -schemes,such as the formula “ T CH ( X T )”, fail as Chow groups of possibly singularschemes are not even contravariant with respect to arbitrary morphisms: one wouldneed to use a contravariant variant of Chow groups (see Remark 3.2 (ii)). To solvethis issue, we view Chow groups of codimension ≤ K -theoryby means of the Chern character, and we define CH X/k, fppf as an appropriatesubquotient of (the fppf sheafification of) the functor T K ( X T ). That thisprocedure gives rise to the correct functor, even integrally, is a consequence of theRiemann–Roch theorem without denominators [Jou70].We show that CH X/k, fppf is represented by a smooth k -group scheme CH X/k (Theorem 3.1 (i)). Our functorial approach is crucial for this, as it allows us toargue by fppf descent from a possibly inseparable finite extension l of k such that X is l -rational. By construction, there is a natural isomorphism CH X l /l ≃ ( CH X/k ) l for all field extensions l of k .The k -group scheme that we use as a substitute for the intermediate Jacobianof X is then the identity component ( CH X/k ) of CH X/k , which is an abelianvariety (Theorem 3.1 (ii)). We hope that this functorial perspective on intermediateJacobians may have other applications (to intermediate Jacobians in families, todeformations of algebraic cycles).Establishing an identification ( CH X/k ) k ≃ Ab ( X k ) (Theorem 3.1 (vi)) andusing the principal polarization on Ab ( X k ) constructed in [BW19], we endow( CH X/k ) with a canonical principal polarization, which paves the way for applica-tions to rationality questions. Let us now state the most general obstruction to the k -rationality of a smooth projective threefold that we obtain by analyzing CH X/k . Theorem C (Theorem 3.1 (vii)) . Let X be a smooth projective k -rational threefoldover k . Then there exists a smooth projective curve B over k such that the k -groupscheme CH X/k can be realized as a direct factor of
Pic
B/k in a way that respectsthe canonical principal polarizations.
In Theorem 3.10, we deduce from Theorem C more concrete obstructions tothe k -rationality of X , pertaining to the Néron–Severi group NS ( X k ) of algebraicequivalence classes of codimension 2 cycles on X k , to the principally polarizedabelian variety ( CH X/k ) , and to the ( CH X/k ) -torsors that are of the form( CH X/k ) α for some α ∈ (cid:0) CH X/k / ( CH X/k ) (cid:1) ( k ) = NS ( X k ) Γ k .The principle of the proof of Theorem C goes back to Clemens and Griffiths.Since X is k -rational, it can be obtained from P k by a composition of blow-ups of OLIVIER BENOIST AND OLIVIER WITTENBERG regular curves and of closed points, followed by a contraction. The curve B whoseexistence is predicted by Theorem C is roughly the union of the blown up curves.This works perfectly well if k is perfect. If k is imperfect, however, some of theblown up curves may be regular but not smooth over k . It is nevertheless veryimportant, in view of the application to Theorem A, that the curve B appearingin the statement of Theorem C be smooth over k . To prove Theorem C as stated,one thus has to show that the contributions of these non-smooth regular curves arecancelled out by the final contraction. This non-trivial fact relies on a completeunderstanding of which Jacobians of proper reduced curves over k split as theproduct of an affine group scheme and of an abelian variety over k (Theorem 1.7).The text is organized as follows. Sections 1 and 2 gather preliminaries,concerning respectively group schemes and K -theory. Section 2 contains inparticular the definition of the above-mentioned functor CH X/k, fppf associated witha smooth projective k -rational threefold X over k (Definition 2.9). In Section 3,we prove that this functor is representable and study the k -group scheme CH X/k that represents it (Theorem 3.1). A number of obstructions to the k -rationalityof X are then derived (Theorem 3.1 (vii) and Theorem 3.10). Section 4 is devotedto applications to three-dimensional smooth complete intersections of two quadrics:we compute CH X/k entirely (Theorem 4.5), deduce the irrationality criterion that isour main theorem (Theorem 4.7), and apply the criterion to examples over Laurentseries fields (Theorem 4.11).
Acknowledgements.
Over a perfect field, the results of this article were obtainedduring the “Rationality of Algebraic Varieties” conference held on SchiermonnikoogIsland in April 2019 after reading the arguments of Hassett and Tschinkel [HT19a]over the reals. We are grateful to Jean-Louis Colliot-Thélène and to SashaKuznetsov for encouraging us to write them up. Finally, we thank Asher Auel,Marcello Bernardara and Michele Bolognesi for their comments and their interest.
Notation and conventions.
We fix a field k and an algebraic closure k of k .Let k p be the perfect closure of k in k and Γ k := Gal( k/k p ) be the absolute Galoisgroup of k p . If X and T are two k -schemes, we set X T := X × k T . A variety over k is a separated scheme of finite type over k ; a curve is a variety of puredimension 1. If G is a group scheme locally of finite type over k , we denoteby G its identity component. If X is a smooth proper variety over k , we letCH c ( X k ) alg ⊂ CH c ( X k ) be the subgroup of algebraically trivial codimension c cycle classes and define NS c ( X k ) := CH c ( X k ) / CH c ( X k ) alg .We use qcqs as a shorthand for quasi-compact and quasi-separated. We denote by(Sch /k ) the category of qcqs k -schemes and by (Ab) the category of abelian groups.If X is a commutative k -group scheme, we let Φ X : (Sch /k ) op → (Ab) be the functorgiven by Φ X ( T ) = Hom k ( T, X ). The functor Φ : X Φ X , from the category ofcommutative k -group schemes to the category of functors (Sch /k ) op → (Ab), isfully faithful, by Yoneda’s lemma and since all schemes are covered by affine (henceqcqs) open subschemes. We say that a functor (Sch /k ) op → (Ab) is representable if it is isomorphic to Φ X for some commutative k -group scheme X , which need notbe qcqs. The functor Z : (Sch /k ) op → (Ab) sending T ∈ (Sch /k ) to the group Z ( T )of locally constant maps T → Z is represented by the constant k -group scheme Z . NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 5
We refer to [SGA6, VII, Définition 1.4] for the definition of a regular immersion of schemes that we use, based on the Koszul complex. A closed immersion of regularschemes is always regular. A morphism of schemes f : X → Y is said to be a localcomplete intersection or lci if it can be factored, locally on X , as the compositionof a regular immersion and of a smooth morphism [SGA6, VIII, Définition 1.1].If ℓ is a prime number and M is a Z -module, we let M { ℓ } ⊂ M be the ℓ -primarytorsion subgroup of M , and T ℓ ( M ) := Hom( Q ℓ / Z ℓ , M ) and V ℓ ( M ) := T ℓ ( M )[1 /ℓ ]be the ℓ -adic Tate modules of M . If G is a commutative k -group scheme, we set T ℓ G := T ℓ ( G ( k )) and V ℓ G := V ℓ ( G ( k )).1. Group schemes
We first collect miscellaneous information concerning group schemes. The mainnew result of this section is Theorem 1.7.1.1.
Chevalley’s theorem. If G is a connected smooth group scheme over k , thereis a unique short exact sequence(1.1) 0 → L ( G k p ) → G k p → A ( G k p ) → k p , where L ( G k p ) is smooth, connected and affine and where A ( G k p ) is an abelian variety. This statement was proved by Chevalley [Che60], see[Con02, Theorem 1.1] for a modern proof.1.2. Principal polarizations.
Recall that a principal polarization on an abelianvariety A over k p is an ample class θ ∈ NS( A k ) Γ k whose associated isogeny A k → ˆ A k is an isomorphism, that a principally polarized abelian variety A over k p is aproduct of indecomposable principally polarized abelian varieties over k p and thatthe factors of this decomposition are unique as subvarieties of A (see [BW19, §2.1]).We define a polarization (resp. a principal polarization ) of a smooth commutativegroup scheme G over k to be a polarization (resp. a principal polarization) of theabelian variety A ( G k p ) over k p . If G is endowed with a polarization θ , we say thata smooth subgroup scheme H ⊂ G is a polarized direct factor of G if there exists asubgroup scheme H ′ ⊂ G such that the canonical morphism ι : H × H ′ ∼ −→ G is anisomorphism and such that ι ∗ θ = π ∗ θ | A ( H k p ) + π ′∗ θ | A ( H ′ k p ) , where π and π ′ denotethe projections of A ( H k p ) × A ( H ′ k p ) onto A ( H k p ) and A ( H ′ k p ). If in addition θ is aprincipal polarization, then so are π ∗ θ | A ( H k p ) and π ′∗ θ | A ( H ′ k p ) ; in this case, we alsospeak of a principally polarized direct factor .1.3. Locally constant functions.
For a variety X over k , consider the functor Z X/k : (Sch /k ) op → (Ab) T Z ( X T ).Equivalently Z X/k is the push-forward of the constant sheaf Z by the structuralmorphism X → Spec( k ) (and hence is an fpqc sheaf). Let π ( X/k ) denote the étale k -scheme of connected components of X , defined in [DG70, §I.4, Définition 6.6].The Weil restriction of scalars Res π ( X/k ) /k Z exists as a k -scheme by [BLR90, 7.6/4]. Proposition 1.1.
Let X be a variety over k . Then the functor Z X/k is canonicallyrepresented by the Weil restriction of scalars
Res π ( X/k ) /k Z . OLIVIER BENOIST AND OLIVIER WITTENBERG
Proof.
Recall that there is a canonical faithfully flat morphism q X : X → π ( X/k )whose fibres are geometrically connected [DG70, §I.4, Propositions 6.5 and 6.7].For any T ∈ (Sch /k ), the resulting morphism X T → π ( X/k ) T is surjective, hasconnected fibres, and is open [EGA42, Théorème 2.4.6]; therefore q X induces abijection between the sets of connected components of X T and of π ( X/k ) T . As aconsequence, the homomorphism Z ( π ( X/k ) T ) → Z ( X T ) is an isomorphism, andhence so is the morphism of functors Res π ( X/k ) /k Z → Z X/k that it induces when T varies. (cid:3) Remarks . (i) We will usually denote by Z X/k , rather than by Res π ( X/k ) /k Z ,the group scheme that Z X/k represents.(ii) Proposition 1.1 shows that a morphism p : X ′ → X of varieties over k thatinduces a bijection between the sets of connected components of X k and of X ′ k givesrise to an isomorphism Z X/k ∼ −→ Z X ′ /k .1.4. Picard schemes.
The absolute Picard functor of a proper variety X over k isPic X/k : (Sch /k ) op → (Ab) T Pic( X T ) . Beware that our notation differs slightly from that of [Kle05, §9.2].If τ ∈ { Zar , ´et , fppf } , we denote the sheafification of Pic X/k for the corresponding(Zariski, étale or fppf) topology by Pic
X/k,τ . The functors Pic
X/k, ´et and Pic
X/k, fppf are equal [BLR90, 8.1 p. 203] and are represented by a group scheme locally of finitetype over k [BLR90, 8.2/3] which we denote by Pic
X/k : the
Picard scheme of X .These two functors contain Pic X/k,
Zar and T Pic( X T ) / Pic( T ) as subfunctors if H ( X, O X ) = k , and they coincide with them if in addition X ( k ) = ∅ (see [Kle05,Theorem 9.2.5] and [EGA32, Proposition 7.8.6]), for instance if X is connected andreduced and k = k .1.4.1. Picard schemes of blow-ups.
In §1.4.1, we consider the following situation.We fix a regular closed immersion i : Y → X of qcqs schemes of pure codimension c ≥
2, we let p : X ′ → X be the blow-up of X along Y with exceptional divisor Y ′ ,and we let p ′ : Y ′ → Y and i ′ : Y ′ → X ′ be the natural morphisms. The morphism p ′ : Y ′ → Y is a projective bundle of relative dimension c − ≥ X ) × Z ( Y ) → Pic( X ′ )( L , ψ ) p ∗ L ⊗ O X ′ (cid:16) − X n ∈ Z n h p − ( ψ − ( n )) i(cid:17) . (1.2) Proposition 1.3.
Under the hypotheses of §1.4.1, the map (1.2) is bijective.Proof.
By absolute noetherian approximation [TT90, Theorem C.9] and the limitarguments of [EGA43, §8], we may assume that X is noetherian. If N ∈
Pic( X ′ ),the function ψ : Y → Z such that N | X ′ y ≃ O X ′ y ( ψ ( y )) for all y ∈ Y is locallyconstant. (Indeed, for n ≫
0, the O Y -module p ′∗ (( N | Y ′ )( n )) is locally free and itsformation commutes with base change, by [Har77, III, Theorems 8.8 and 12.11].)Since O X ′ ( − Y ′ ) | X ′ y ≃ O X ′ y (1) for all y ∈ Y (see [Tho93, §1.2]), it follows that N ⊗ O X ′ ( P n ∈ Z n [ p − ( ψ − ( n ))]) is trivial on the fibers of p , and that ψ is theunique function with this property. It remains to show that p ∗ : Pic( X ) → Pic( X ′ ) NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 7 is injective with image the subgroup of isomorphism classes of line bundles that aretrivial on the fibers of p . The injectivity follows from [Tho93, Lemme 2.3 (a)], andthe description of the image from Lemma 1.4 (iii) below. (cid:3) Lemma 1.4.
Under the hypotheses of §1.4.1, assume that X is noetherian and let N be a line bundle on X ′ such that N | X ′ y ≃ O X ′ y for all y ∈ Y . For any integer n ,set N ( n ) = N ⊗ O X ′ ( − nY ′ ) . Then:(i) For all j ≥ and n ≥ , the sheaf R j p ∗ ( N ( n )) vanishes.(ii) For all n ≥ , the natural morphism p ∗ p ∗ ( N ( n )) → N ( n ) is surjective.(iii) The sheaf p ∗ N is invertible and p ∗ p ∗ N → N is an isomorphism.Proof.
By [Har77, III, Theorem 8.8 (c)], assertion (i) holds for n ≫
0. To prove (i)for all n ≥ n , we consider for j ≥ R j p ∗ N ( n ) → R j p ∗ N ( n − → R j p ∗ ( N ( n − | Y ′ )and note that R j p ′∗ ( N ( n − | Y ′ ) = 0 for n ≥ n ≫ n ≥ n , we consider the natural commutativediagram with exact rows p ∗ p ∗ N ( n ) / / (cid:15) (cid:15) p ∗ p ∗ N ( n − / / (cid:15) (cid:15) p ∗ p ∗ ( N ( n − | Y ′ ) (cid:15) (cid:15) / / / / N ( n ) / / N ( n − / / N ( n − | Y ′ / / , in which the exactness of the upper row follows from the vanishing of R p ∗ ( N ( n ))proved in (i), and note that p ∗ p ∗ ( N ( n − | Y ′ ) → N ( n − | Y ′ is surjective for n ≥ p is surjective bycohomology and base change [Har77, III, Theorem 12.11].To prove (iii), we work Zariski-locally around a point y ∈ X . In view of (ii) for n = 0, we can assume after shrinking X the existence of a section σ ∈ H ( X ′ , N )that does not vanish identically on X ′ y . Since N | X ′ y ≃ O X ′ y and X ′ y is a projectivespace, the section σ vanishes nowhere on X ′ y and, after shrinking X again, it inducesan isomorphism σ : O X ′ ∼ −→ N . Assertion (iii) now follows from the fact that thenatural morphism O X → p ∗ O X ′ is an isomorphism [SGA6, VII, Lemme 3.5]. (cid:3) Corollary 1.5.
Under the hypotheses of §1.4.1, if X is moreover a proper varietyover k , the formula (1.2) induces an isomorphism of functors (1.3) Pic X/k × Z Y/k ∼ −→ Pic X ′ /k .Proof. Since the formation of the blow-up of a regular closed immersion commuteswith flat base change (see [SGA6, VII, Propositions 1.5 et 1.8 i)]), we can applyProposition 1.3 to the morphisms i T : Y T → X T for T ∈ (Sch /k ). (cid:3) Picard schemes of curves. If C is a proper curve over k , then Pic
C/k issmooth over k by [BLR90, 8.4/2]. Moreover, letting D := g C red k p be the normalizationof the reduction of C k p , which is a smooth proper curve over k p , the pull-backmorphism Pic C k p /k p → Pic D/k p induces an isomorphism(1.4) A ( Pic C k p /k p ) ∼ −→ Pic D/k p , OLIVIER BENOIST AND OLIVIER WITTENBERG as [BLR90, 9.2/11] shows. The principal polarization of
Pic D/k p given by the thetadivisor thus induces a canonical principal polarization on Pic
C/k in the sense of §1.2.If C is irreducible, then so is D [EGA42, Proposition 2.4.5] and the principallypolarized abelian variety Pic D/k p over k p is thus indecomposable if non-zero (see[BW19, §2.1]). Proposition 1.6.
Let C be a proper curve over k and let C ′ := g C red be thenormalization of its reduction. Then there is a short exact sequence → Pic C/k → Pic
C/k → Z C ′ /k → . (1.5) Proof.
Both Z C ′ /k and Pic
C/k / Pic C/k are étale group schemes over k . Theyare thus isomorphic if and only if so are their base changes G := Z C ′ k p /k p and H := Pic C k p /k p / Pic C k p /k p to k p . Letting D := g C red k p be the normalization of thereduction of C k p , which is a smooth proper curve over k p , one has G = Z D/k p byRemark 1.2 (ii) and [EGA42, Proposition 2.4.5], and H = Pic
D/k p / Pic D/k p by[BLR90, 9.2/11]. That G ≃ H now follows from the fact that G ( k ) and H ( k ) areboth isomorphic, as Γ k -modules, to Z ( D k ). (cid:3) When do Jacobians split?
We now provide, in Theorem 1.7, a criterion forthe Jacobian of a proper reduced curve to be the product of an abelian variety andof an affine group scheme. We will use Theorem 1.7 in Lemma 3.8, which plays akey role in the proof of Theorem C.1.5.1.
Statement.
Let us introduce some notation. Whenever D is a smooth properintegral curve over k , the genus of D is the dimension of the abelian variety Pic D/k .We note that D has genus 0 if and only if the irreducible components of D k (whichare all isomorphic) are rational. Given a proper reduced curve C over k , we denoteby C rat (resp. C irrat ) the union of the irreducible components B of C such that thenormalization of ( B k p ) red has genus 0 (resp. has genus ≥ C rat and C irrat as reduced closed subschemes of C . We define a strict cycle of components of C k to be a sequence of pairwise distinct irreducible components B , . . . , B n of C k forsome integer n ≥
2, such that there exist pairwise distinct points x , . . . , x n of C k with x i ∈ B i ∩ B i +1 for all i ∈ { , . . . , n − } and x n ∈ B n ∩ B . Finally, we recallthat a reduced curve over k is seminormal if it is étale locally isomorphic to theunion of the coordinate axes in an affine space over k [Kol96, Chapter I, 7.2.2]. Theorem 1.7.
Let C be a proper reduced curve over k . The group scheme Pic C/k is the product of an abelian variety and of an affine group scheme over k if andonly if the following conditions all hold:(i) the scheme ( C irrat ) k is reduced and seminormal and its irreducible componentsare smooth;(ii) any strict cycle of components of C k is contained in ( C rat ) k ;(iii) for every connected component B of C rat , either the scheme B ∩ C irrat is étaleover k , or it is of the form Spec( k ′ ) for some field k ′ and the restriction map H ( B, O B ) → k ′ is bijective.In this case, the natural map Pic
C/k → Pic C rat /k × Pic C irrat /k is an isomorphismand Pic C rat /k is affine while Pic C irrat /k is an abelian variety. NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 9
Condition (iii) holds if ( C rat ∩ C irrat ) k is reduced, and it implies, in turn, that C rat ∩ C irrat is reduced. The reverse implications are true if k is perfect.When C is integral and geometrically locally irreducible (for instance when C isintegral and geometrically unibranch), Theorem 1.7 takes on a particularly simpleform, which we now state. In the sequel, we shall only apply Theorem 1.7 to normalcurves, through Corollary 1.8. Let us recall that normal varieties are geometricallyunibranch [EGA42, Proposition 6.15.6]. Corollary 1.8.
Let C be a proper integral curve over k . Assume that the connectedcomponents of C k are irreducible; such is the case, for instance, if C is geometricallyunibranch. Then the group scheme Pic C/k is the product of an abelian variety andof an affine group scheme over k if and only if at least one of the following twoconditions holds:(i) C is smooth over k (in which case Pic C/k is an abelian variety);(ii) the normalization D of ( C k p ) red has genus (in which case Pic C/k is affine).Proof.
Our assumptions imply that C rat = ∅ or C irrat = ∅ , that there is no strictcycle of components of C k and that C is smooth over k if and only if C k is reducedwith smooth irreducible components. Thus, the conditions of Theorem 1.7 are allmet if and only if at least one of (i) and (ii) holds. (cid:3) Remark . Corollary 1.8 applies to integral curves that may not be geometricallyreduced. It would however fail in general for irreducible but non-reduced curves,as the following example shows. Let E be an elliptic curve over k , and let L be anample line bundle on E . Define C := Proj E ( O E ⊕ L ), where sections of L squareto 0. The natural closed immersion i : E = C red → C then induces an isomorphism i ∗ : Pic C/k ∼ −→ Pic E/k ≃ E . Indeed, in view of (1.4), it suffices to show that thekernel of ( i ∗ ) k p has trivial tangent space at the identity, which follows from thefact that the pull-back i ∗ : H ( C, O C ) ∼ −→ H ( E, O E ) is an isomorphism. We notethat C is geometrically unibranch since C red is normal.1.5.2. A few general lemmas.
We establish a series of lemmas on which the proofof Theorem 1.7 will rely. The first one is due to Tanaka, see [Tan18, Lemma 3.3].
Lemma 1.10.
Let F be a field extension of k . If F/k is not separable, there existfinite purely inseparable field extensions k ⊂ k ′ ⊂ k ′′ and a k ′ -linear embedding k ′′ ֒ → F ⊗ k k ′ such that F ⊗ k k ′ is a field and k ′′ = k ′ .Proof. Let k ′′ be a minimal finite purely inseparable field extension of k such that F ⊗ k k ′′ fails to be reduced. Let p denote the characteristic of k . As k ′′ /k isfinite, purely inseparable and non-trivial, there exists a subextension k ′ /k suchthat [ k ′′ : k ′ ] = p . Fix x ∈ k ′ such that k ′′ = k ′ ( x /p ). By the minimality of k ′′ , thefinite connected non-zero F -algebra F ′ = F ⊗ k k ′ is reduced, hence is a field. Onthe other hand, as F ⊗ k k ′′ = F ′ [ t ] / ( t p − x ) is not reduced, we see that x must bea p -th power in F ′ , so that k ′′ embeds k ′ -linearly into F ′ . (cid:3) Lemma 1.11.
Let f : G → G ′ be a surjective morphism between connected smoothgroup schemes over k such that the kernel of A ( f k p ) : A ( G k p ) → A ( G ′ k p ) is smoothand connected. If G is isomorphic to the product of an abelian variety and of anaffine group scheme over k , then so is G ′ . Proof.
Suppose G = L × A with L affine and A an abelian variety. Let K = Ker( f ).Let p : G → A be the projection, let i : A → G be the inclusion, and let p ( K ) denotethe scheme-theoretic image of K by p ; it is a subgroup scheme of A [SGA31, VI A ,Proposition 6.4]. We note that A ( G k p ) = A k p and that p ( K ) k p ⊂ Ker( A ( f k p )). Themorphism Ker( A ( f k p )) → G ′ k p induced by f ◦ i factors through L ( G ′ k p ) and hencevanishes since Ker( A ( f k p )) is an abelian variety and L ( G ′ k p ) is affine. Therefore themorphism p ( K ) → G ′ induced by f ◦ i also vanishes. We deduce that i ( p ( K )) ⊂ K ,hence K = ( K ∩ L ) × i ( p ( K )). On the other hand, the closed immersion G/K → G ′ induced by f is an isomorphism at it is surjective and G ′ is reduced. It follows that G ′ = ( L/ ( K ∩ L )) × ( A/p ( K )), and the lemma is proved. (cid:3) Lemma 1.12.
Let C be a proper reduced curve over k and B be a geometricallyconnected reduced closed subscheme of C of pure dimension . View the union of theirreducible components of C not contained in B as a reduced closed subscheme B ′ of C . If the natural morphism B ∩ B ′ → Spec( H ( B ′ , O B ′ )) is étale and if anystrict cycle of components of C k is contained in B ′ k , then the natural morphism Pic
C/k → Pic
B/k × Pic B ′ /k is an isomorphism.Proof. If i : B T ֒ → C T , i ′ : B ′ T ֒ → C T , i ′′ : B T ∩ B ′ T ֒ → C T denote the inclusions,the short exact sequence 1 → G m → i ∗ G m × i ′∗ G m → i ′′∗ G m → C T induces, for any T ∈ (Sch /k ), an exact sequence of groups(1.6) 1 → G m ( C T ) → G m ( B T ) × G m ( B ′ T ) → G m ( B T ∩ B ′ T ) → Pic( C T ) → Pic( B T ) × Pic( B ′ T ) → Pic( B T ∩ B ′ T ).The natural morphism B ∩ B ′ → Spec( H ( B ′ , O B ′ )) is an étale morphism betweenfinite k -schemes that induces an injection on k -points, in view of the assumptionabout strict cycles of components. (Note that Spec( H ( B ′ , O B ′ ))( k ) is the setof connected components of B ′ k .) It is therefore an open and closed immersion[SGA1, I, Théorème 5.1]. As such, it admits a retraction, and hence so does theinclusion B ∩ B ′ ֒ → B ′ . Thus, the restriction map G m ( B ′ T ) → G m ( B T ∩ B ′ T ) isonto. Noting that Pic B ∩ B ′ /k = 0, the conclusion of the lemma now results from theexact sequence of fppf sheaves obtained by sheafifying (1.6) with respect to T . (cid:3) Lemma 1.13.
Let C be a proper integral curve over a finite purely inseparableextension k ′ of k . If Pic C/k is the product of an abelian variety and of an affinegroup scheme over k , then k ′ = k or C = C rat .Proof. For all T ∈ (Sch /k ), pull-back induces an equivalence between the categoriesof étale T -schemes and of étale T k ′ -schemes [SGA1, IX, Théorème 4.10]. It followsat once that the natural morphismPic C/k = Res k ′ /k (Pic C/k ′ ) → Res k ′ /k (Pic C/k ′ , ´et )of functors (Sch /k ) op → (Ab) becomes an isomorphism after étale sheafification.We get an isomorphism Pic
C/k = Res k ′ /k ( Pic
C/k ′ ), which restricts to an isomor-phism Pic C/k = Res k ′ /k ( Pic C/k ′ ) in view of [CGP15, Proposition A.5.9]. Write Pic C/k = L × A with L affine and A an abelian variety. Applying [SGA32, XVII,Appendice III, Proposition 5.1] with U = L (or [CGP15, Proposition A.7.8]) showsthat if k ′ = k , then Pic C/k = L , so that A (( Pic C/k ) k p ) = 0 and C = C rat . (cid:3) NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 11
Lemma 1.14.
Let D be the disjoint union of smooth proper geometrically integralcurves D , . . . , D n over k . Let C be a proper curve over k such that H ( C, O C ) = k and C ( k ) = ∅ . Let ν : D → C be a morphism. If ν ∗ : Pic
C/k → Pic
D/k admits asection, then there exist morphisms ρ i : C → Pic D i /k for i ∈ { , . . . , n } such that ρ i ◦ ν | D i is the canonical morphism for all i while ρ i ◦ ν | D j is constant for all j = i .Proof. We let ν i : D i → C be the restriction of ν , fix a section σ : Pic
D/k → Pic
C/k of ν ∗ and, noting that Pic
D/k = Q ni =1 Pic D i /k , let σ i : Pic D i /k → Pic
C/k be therestriction of σ , so that ν ∗ i ◦ σ j = δ ij . Let ι i : D i → Pic D i /k be the canonicalmorphism. Then ν ∗ j maps σ i ◦ ι i ∈ Pic
C/k ( D i ) to ι i ∈ Pic D i /k ( D i ) if j = i , to0 ∈ Pic D j /k ( D i ) otherwise. As H ( C, O C ) = H ( D i , O D i ) = k and C ( k ) = ∅ ,there are a canonical bijection Pic( C T ) / Pic( T ) ∼ −→ Pic
C/k ( T ) and a canonicalinjection Pic(( D i ) T ) / Pic( T ) ֒ → Pic D i /k ( T ) for all T . We deduce, for each i , theexistence of α i ∈ Pic( C × k D i ) such that ( ν j × ∗ α ∈ Pic( D j × k D i ) is the class of thediagonal for j = i and comes from pull-back from Pic( D i ) for all j = i . Switchingthe factors, the class α i gives rise to the desired element ρ i ∈ Pic D i /k ( C ). (cid:3) Lemma 1.15.
Let C be a proper curve over k . Let ν : C ′ → C be the normalizationof C red . The pull-back morphism ν ∗ : Pic C/k → Pic C ′ /k is surjective.Proof. We may assume that k is separably closed. We then claim that the map ν ∗ : Pic
C/k ( k ) → Pic C ′ /k ( k ) is onto. As Pic C ′ /k is smooth over k , it will followthat the morphism ν ∗ : Pic
C/k → Pic C ′ /k is dominant, by [BLR90, 2.2/13], so thatthe morphism ν ∗ : Pic C/k → Pic C ′ /k is dominant, by Proposition 1.6, and hencesurjective, by [SGA31, VI A , Corollaire 6.2 (i)]. To prove the claim, we may assumethat C is reduced, by [BLR90, 9.2/5]. Letting C ⊂ C be a dense open normalsubset, we then remark that ν ∗ : Pic( C ) → Pic( C ′ ) is onto since any divisor on C ′ is linearly equivalent to a divisor supported on ν − ( C ). As k is separably closed,it follows that ν ∗ : Pic C/k, ´et ( k ) → Pic C ′ /k, ´et ( k ) is onto as well, as desired. (cid:3) Proof of Theorem 1.7.
As the formation of C rat and C irrat is compatiblewith separable extensions of scalars and as the assertions of the last sentenceof the theorem are of a geometric nature, we may, and will henceforth, assumethat k is separably closed. For later use, we note that thanks to this assumption,if condition (i) of Theorem 1.7 holds, then the irreducible components of C irrat ,viewed as reduced schemes, are geometrically reduced (being both reduced andgenerically geometrically reduced) and therefore smooth over k , and the non-smoothlocus of C irrat over k consists of k -points (as the intersection of any two irreduciblecomponents of C irrat is étale). Step 1 : we assume that (i)–(iii) hold and deduce the remaining assertions.From (1.4) applied to C rat , we deduce that ( Pic C rat /k ) k p is affine; hence Pic C rat /k is also affine. To prove that Pic C irrat /k is an abelian variety and that the naturalmap Pic
C/k → Pic C rat /k × Pic C irrat /k is an isomorphism, we argue by induction onthe number of irreducible components of C irrat . When C irrat = ∅ , there is nothingto prove. Otherwise, we choose an irreducible component B of C irrat . As a reducedscheme, it is smooth over k , hence Pic B/k is an abelian variety [BLR90, 9.2/3]. Toconclude the proof, we need only verify that Lemma 1.12 can be applied to C andto C irrat . To this end, let B ′ be as in its statement and fix x ∈ B ∩ B ′ . Let E be theconnected component of x in B ′ and C , . . . , C m the irreducible components of C irrat2 OLIVIER BENOIST AND OLIVIER WITTENBERG containing x , numbered so that B = C . The finite k -algebra H ( E, O E ), beingnon-zero, connected and reduced, is a field; it embeds into the residue field k ( x )of x . We have to prove that the natural morphism B ∩ E → Spec( H ( E, O E )) isétale at x .If k ( x ) = k , then m = 1 and therefore C rat ∩ C irrat is not étale over k at x . Wededuce from (iii) that E coincides with the connected component of x in C rat , that B ∩ E is reduced at x and that H ( E, O E ) = k ( x ); hence the desired result.If k ( x ) = k , then H ( E, O E ) = k and it suffices to see that B ∩ E is reduced at x .In the Zariski tangent space T x C of C at x , we have T x E = V + T x C + · · · + T x C m where V = T x C rat if x ∈ C rat and V = 0 otherwise. It follows from (iii) that C rat ∩ C irrat is reduced, so that V ∩ ( T x C + · · · + T x C m ) = 0, and from (i) that thelines T x C , . . . , T x C m are in direct sum. Hence T x ( B ∩ E ) = T x C ∩ T x E = 0, and B ∩ E is indeed reduced at x . Step 1 is complete.We now assume that Pic C/k is the product of an abelian variety and of an affinegroup scheme over k , and prove (i)–(iii) in four more steps. By Lemma 1.11, wemay assume that C is connected. In addition, we may assume that C irrat = ∅ .Let C , . . . , C n be the irreducible components of C irrat , viewed as reduced schemes.Let D i be the normalization of C i , let D be the disjoint union of the D i and let ν i : D i → C and ν : D → C be the natural morphisms. Step 2 : we prove that C , . . . , C n and C irrat are geometrically reduced over k .As C irrat = C ∪ · · · ∪ C n , it suffices to check that the C i are geometricallyreduced. Assume that some B ∈ { C , . . . , C n } is not geometrically reduced. ByLemma 1.10 applied to k ( B ), there exist subfields k ⊂ k ′ ⊂ k ′′ ⊂ k p , with k ′′ = k ′ and k ′′ /k finite, such that B k ′ is integral and such that if B ′ denotesits normalization, the natural morphism B ′ → Spec( k ′ ) factors through Spec( k ′′ ).Let ω : C ′ → C k ′ be the normalization of ( C k ′ ) red . By Lemma 1.15, the pull-back map ω ∗ : Pic C k ′ /k ′ → Pic C ′ /k ′ is surjective, and (1.4) shows that A ( ω ∗ k p )is an isomorphism. As B ′ is a connected component of C ′ , we deduce, thanks toLemma 1.11, that Pic B ′ /k ′ is the product of an abelian variety and of an affine groupscheme over k ′ . Lemma 1.13 implies that k ′′ = k ′ or B ′ = B ′ rat , a contradiction. Step 3 : assuming that D is smooth over k , we construct ρ i : C → Pic D i /k suchthat ρ i ◦ ν i is a closed immersion while ρ i ( C rat ∪ S j = i C j ) is finite, for all i .It suffices to check that the hypotheses of Lemma 1.14 are satisfied. Indeed, thecanonical morphism D i → Pic D i /k is a closed immersion if D i is a smooth properintegral curve of genus ≥ k [Mil86, Propositions 6.1 and 2.3], and ρ i ( C rat ) isfinite since any morphism from a rational curve to an abelian variety is constant.We recall that C irrat = ∅ . As C is proper, connected and reduced, the restrictionmap H ( C, O C ) → H ( C , O C ) has to be injective; as C is geometrically integral,we deduce that H ( C, O C ) = k . In addition, as k is separably closed, we have C ( k ) = ∅ [BLR90, 2.2/13], hence C ( k ) = ∅ .As D is smooth, the group scheme Pic D/k is an abelian variety and the morphism ν ∗ : Pic C/k → Pic D/k induces an isomorphism A (( Pic C/k ) k p ) ∼ −→ ( Pic D/k ) k p (see (1.4)). Our assumption on Pic C/k therefore implies that ν ∗ : Pic C/k → Pic D/k admits a section. Letting C ′ be the normalization of C red , the natural map ν ∗ : Z C ′ /k → Z D/k also admits a section. In addition, the sequence (1.5) splitsas k is separably closed and Pic C/k is smooth, and the choice of a splitting of (1.5)
NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 13 and of a section of ν ∗ : Z C ′ /k → Z D/k determines a splitting of the sequence (1.5)associated with D . Applying Proposition 1.6 to C and to D , we now conclude that ν ∗ : Pic
C/k → Pic
D/k admits a section. This completes Step 3.
Step 4 : we prove (i) and (ii) of Theorem 1.7.Step 2 ensures that (( C k ) red ) irrat = ( C irrat ) k and it follows from Lemma 1.11, inview of [BLR90, 9.2/5], that Pic C k ) red /k is the product of an abelian variety andof an affine group scheme over k . Thus, after replacing C with ( C k ) red , we may,and will until the end of Step 4, assume that k = k . In this case D is smooth over k and Step 3 becomes applicable.To complete the proof of (i), it suffices to show that for all i ∈ { , . . . , n } , thecurve C i is smooth and the scheme C i ∩ (cid:0) S j = i C j (cid:1) is reduced. We fix i . As ρ i ◦ ν i is an immersion, so is ν i ; as ν i ( D i ) = C i is a reduced closed subscheme of C , wededuce that ν i restricts to an isomorphism D i → C i , so that C i is smooth. Now therestriction of ρ i to the subscheme C i ∩ (cid:0) S j = i C j (cid:1) is both a closed immersion (sinceso is ρ i | C i ) and a morphism whose scheme-theoretic image is finite and reduced(since so is ρ i | S j = i C j ), hence this scheme is reduced.To prove (ii), we pick a strict cycle of components B , . . . , B n of C and pairwisedistinct points x i ∈ B i ∩ B i +1 for i ∈ { , . . . , n − } and x n ∈ B n ∩ B . If one ofthe B i were contained in C irrat , say B = C , then ρ ( x ) and ρ ( x n ) would haveto be distinct, since ρ | C is injective, and equal, since ρ ( B ∪ · · · ∪ B n ) is a point(being finite and connected). This is absurd. Step 5 : we prove (iii) of Theorem 1.7.We now know that (i) holds, hence D is smooth over k : Step 3 is again applicable. Lemma 1.16.
Let i ∈ { , . . . , n } . For any purely -dimensional connected reducedclosed subscheme E of C such that E ∩ C i is finite and non-empty, the restrictionmap H ( E, O E ) → H ( E ∩ C i , O E ∩ C i ) is an isomorphism of fields.Proof. The morphism ρ i | E has finite image (by Step 3), hence it factors throughan affine open Spec( R ) ⊂ Pic D i /k . To see that the map of the lemma is surjective,we note that its composition with R → H ( E, O E ) is surjective as ρ i | E ∩ C i is aclosed immersion. It is also injective, as H ( E, O E ) is a field (being a non-zero,connected, reduced, finite k -algebra) and E ∩ C i = ∅ . (cid:3) To prove (iii), let B be a connected component of C rat such that B ∩ C irrat isnot étale over k , say at a point x .After renumbering, we may assume that the C i containing x are C , . . . , C m , forsome m ∈ { , . . . , n } . If x were a k -point, the subspace T x B ∩ T x C irrat of the Zariskitangent space T x C would be non-zero. After renumbering C , . . . , C m appropriatelyand setting E = B ∪ C ∪ · · · ∪ C m , the vector space T x E ∩ T x C would be non-zero.The scheme E ∩ C would then be non-reduced; this would contradict Lemma 1.16.Thus k ( x ) = k and hence m = 1.Let E ′ be the connected component of x in C rat ∪ C ∪ · · · ∪ C n . As theevaluation map H ( E ′ ∩ C , O E ′ ∩ C ) → k ( x ) is surjective, Lemma 1.16 showsthat H ( E ′ , O E ′ ) = H ( E ′ ∩ C , O E ′ ∩ C ) = k ( x ); hence, by Step 2, there is no k -algebra morphism H ( E ′ , O E ′ ) → H ( C j , O C j ) for any j . Therefore E ′ = B and B ∩ C irrat = E ′ ∩ C . This completes Step 5 as well as the proof of Theorem 1.7. Murre’s intermediate Jacobian. If X is a smooth projective variety over k ,Murre defines an algebraic representative for algebraically trivial codimension cycles on X k to be an abelian variety Ab ( X k ) over k endowed with a morphism(1.7) φ X : CH ( X k ) alg → Ab ( X k )( k )that is initial among regular homomorphisms with values in an abelian variety (see[Mur85, Definition 1.6.1, §1.8]). It is obviously unique up to a unique isomorphism,Murre has shown its existence (see [Mur85, Theorem A p. 226] and [Kah18]),and Achter, Casalaina-Martin and Vial have shown that it descends uniquely toan abelian variety Ab ( X k p ) over k p in such a way that (1.7) is Γ k -equivariant[ACMV17, Theorem 4.4]. If X is a smooth projective k -rational threefold over k , weendow Ab ( X k p ) with the principal polarization θ X ∈ NS (Ab ( X k )) Γ k constructedin [BW19, Property 2.4, Corollary 2.8].1.7. Representability lemmas.
Here are three lemmas to be used later.
Lemma 1.17.
Let F and F ′ be two functors (Sch /k ) op → (Ab) . If F × F ′ isrepresented by a group scheme locally of finite type over k , then so is F .Proof. Let G be the group scheme that F × F ′ represents and let µ : G → G bethe morphism induced by the endomorphism ( x, y ) (0 , y ) of F × F ′ . Then F isrepresented by Ker( µ ), which is a group scheme locally of finite type over k . (cid:3) Lemma 1.18.
Let G be a commutative group scheme locally of finite type over k .Let ν : Z → G be a morphism of k -group schemes such that ν ( n ) / ∈ G ( k ) if n = 0 .Then the cokernel functor Q : (Sch /k ) op → (Ab) defined by Q ( T ) = G ( T ) /ν ( Z ( T )) is represented by a group scheme locally of finite type over k .Proof. Translation by ν ( n ) for n ∈ Z induces an action of the group Z on the set ofconnected components of G . Choose one connected component of G in each orbitof this action. Their disjoint union represents Q . (cid:3) Lemma 1.19.
Let F : (Sch /k ) op → (Ab) be a functor and k ′ be a finite extensionof k . Let τ ∈ { ´et , fppf } . Assume that k ′ /k is separable if τ = ´et . If F is a sheaffor the topology τ and the functor (Sch /k ′ ) op → (Ab) obtained by restricting F isrepresented by a group scheme locally of finite type over k ′ , then F is representedby a group scheme locally of finite type over k .Proof. Let F ′ : (Sch /k ′ ) op → (Ab) denote the restriction of F and G ′ the k -groupscheme that represents F ′ . As F ′ is the restriction of F , there is a canonical descentdatum on F ′ with respect to p : Spec( k ′ ) → Spec( k ). This descent datum inducesa descent datum on G ′ . By [Mur64, Lemma I.8.6], the latter is effective, and G ′ descends to a group scheme G locally of finite type over k . It remains to notethat F and Φ G : (Sch /k ) op → (Ab) are two τ -sheaves, that p is a τ -covering andthat there is by construction an isomorphism of τ -sheaves p ∗ F ≃ p ∗ Φ G that satisfiesthe cocycle condition; from this, it follows that F ≃ Φ G . (cid:3) K -theory functors We now define and study several functors (Sch /k ) op → (Ab) built from K -theory.Our goal, met in §2.3.2, is to define, for a smooth projective k -rational threefold X over k , a functor CH X/k, fppf : (Sch /k ) op → (Ab) that will serve as a substitute forits intermediate Jacobian. NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 15 K -theory. We follow the conventions of [SGA6, TT90].2.1.1.
Definition. If X is a qcqs scheme, we let K ( X ) be the Grothendieck groupof the triangulated category of perfect complexes of O X -modules. (This group isdenoted by K · ( X ) in [SGA6, IV, Définition 2.2] and coincides with the one definedin [TT90, §3.1], as indicated in [TT90, §3.1.1].) We endow K ( X ) with the ringstructure induced by the tensor product ([SGA6, IV, §2.7 b)], [TT90, §3.15]).If X admits an ample family of line bundles [TT90, §2.1.1] (for instance, if X isquasi-projective over an affine scheme [TT90, §2.1.2 (c)]), then K ( X ) is naturallyisomorphic to the Grothendieck group of the exact category of vector bundles on X (combine [TT90, Corollary 3.9] and [TT90, Theorem 1.11.7]).2.1.2. Functoriality.
Let f : X → Y be a morphism of qcqs schemes.The derived pull-back L f ∗ of perfect complexes along f induces a morphism f ∗ : K ( Y ) → K ( X ) ([SGA6, IV, §2.7 b)], [TT90, §3.14]).The derived push-forward R f ∗ induces a morphism f ∗ : K ( X ) → K ( Y ) ifR f ∗ preserves perfect complexes [TT90, §3.16]. This condition is satisfied if f is aproper and perfect morphism [LN07, Proposition 2.1, Example 2.2 (a)], for instanceif f is a proper lci morphism (see [SGA6, VIII, Proposition 1.7]). In this case, theprojection formula [SGA6, IV, (2.12.4)] stemming from [SGA6, III, Proposition 3.7]shows that(2.1) f ∗ ( x ⊗ f ∗ y ) = f ∗ x ⊗ y for all x ∈ K ( X ) and y ∈ K ( Y ).2.1.3. Rank and determinant.
The rank rk : K ( X ) → Z ( X ) and the determinant det : K ( X ) → Pic( X ) are group homomorphisms that are functorial with respectto pull-backs and are such that if x ∈ K ( X ) is represented by a bounded complexof vector bundles on X , then rk( x ) is the alternating sum of the ranks of its termsand det( x ) is the alternating tensor product of the determinants of its terms (see[KM76, Theorem 2 p. 42]).We define SK ( X ) to be the kernel of (rk , det) : K ( X ) → Z ( X ) × Pic( X ).2.1.4. Projective bundles and blow-ups. If X is a qcqs scheme and π : P X E → X is the projective bundle associated with a vector bundle E of rank r on X , themorphism K ( X ) r → K ( P X E ) given by the formula(2.2) ( x , . . . , x r − ) r − X j =0 π ∗ x j ⊗ [ O P X E ( − j )]is an isomorphism of K ( X )-modules ([SGA6, VI, Théorème 1.1], [TT90, Theo-rem 4.1]).If i : Y → X is a regular closed immersion of qcqs schemes of pure codimension c ≥
1, if p : X ′ → X is the blow-up of X along Y with exceptional divisor Y ′ , andif we denote by p ′ : Y ′ → Y and i ′ : Y ′ → X ′ the natural morphisms, Thomasonhas shown that the morphism K ( X ) × K ( Y ) c − → K ( X ′ ) given by the formula(2.3) ( x, y , . . . , y c − ) p ∗ x + c − X j =1 i ′∗ p ′∗ y j ⊗ [ O X ′ ( jY ′ )]is an isomorphism of K ( X )-modules [Tho93, Théorème 2.1]. Coherent sheaves. If X is a qcqs scheme, we let G ( X ) be the Grothendieckgroup of the triangulated category of pseudo-coherent complexes of O X -moduleswith bounded cohomology. (This group is denoted by K · ( X ) in [SGA6, IV,Définition 2.2], see [TT90, §3.3].)If X is noetherian, the group G ( X ) is naturally isomorphic to the Grothendieckgroup of the abelian category of coherent O X -modules [SGA6, IV, §2.4]. In thiscase, letting F d G ( X ) ⊂ G ( X ) be the subgroup generated by classes of coherentsheaves whose support has dimension ≤ d [SGA6, X, Définition 1.1.1] defines afiltration F • on G ( X ).If X is noetherian and regular, the natural morphism K ( X ) → G ( X ) is anisomorphism [TT90, Theorem 3.21]. This allows one to speak of the class ofa coherent O X -module in K ( X ), and of the filtration F • K ( X ) of K ( X ) bydimension of the support.If X is a smooth variety of pure dimension n over k , we use the notation F c K ( X ) = F n − c K ( X ) and Gr cF K ( X ) = F c K ( X ) /F c +1 K ( X ). Accordingto [Ful98, Example 15.1.5], associating with an integral closed subscheme Z ⊂ X of codimension c the class [ O Z ] ∈ K ( X ) induces a surjective morphism(2.4) ϕ c : CH c ( X ) → Gr cF K ( X ) . As explained in [Ful98, Example 15.3.6], Jouanolou’s Riemann–Roch theoremwithout denominators [Jou70] shows that ϕ , ϕ and ϕ are isomorphisms, withinverses given by the rank rk, the determinant det and the opposite − c of thesecond Chern class. In particular, F K ( X ) = SK ( X ). Lemma 2.1.
Let f : X → Y be a morphism of smooth equidimensional varietiesover k . There exists a commutative diagram (2.5) CH ( Y ) ∼ ϕ / / f ∗ (cid:15) (cid:15) Gr F K ( Y ) (cid:15) (cid:15) CH ( X ) ∼ ϕ / / Gr F K ( X ) in which the right vertical arrow is induced by f ∗ : K ( Y ) → K ( X ) .Proof. The pull-back f ∗ : K ( Y ) → K ( X ) restricts to f ∗ : SK ( Y ) → SK ( X ),hence induces f ∗ : F K ( Y ) → F K ( X ). Since the morphisms ϕ are bijectivewith inverse given by − c , and since the second Chern class is functorial, we see that f ∗ : F K ( Y ) → F K ( X ) induces a morphism f ∗ : Gr F K ( Y ) → Gr F K ( X )making the diagram (2.5) commute. (cid:3) The functor K ,X/k and its sheafifications. Definition. If X is a proper variety over k , the absolute K functor of X is K ,X/k : (Sch /k ) op → (Ab) T K ( X T ) . The rank and the determinant (see §2.1.3) give rise to morphisms of functorsrk : K ,X/k → Z X/k and det : K ,X/k → Pic
X/k . We let SK ,X/k be the kernel of(rk , det) : K ,X/k → Z X/k × Pic
X/k . If τ ∈ { Zar , ´et , fppf } , we let K ,X/k,τ (resp. SK ,X/k,τ ) be the sheafification of K ,X/k (resp. SK ,X/k ) for the corresponding(Zariski, étale, fppf) topology. NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 17
Remark . We do not know whether K ,X/k, ´et and K ,X/k, fppf coincide.2.2.2. Functoriality.
Let f : X → Y be a morphism of proper varieties over k .The pull-backs ( f T ) ∗ for T ∈ (Sch /k ) induce a natural transformation of functors f ∗ : K ,Y/k → K ,X/k .Similarly, if f : X → Y is a perfect (for instance lci) morphism of proper varietiesover k , the push-forwards ( f T ) ∗ for T ∈ (Sch /k ) (which exist by §2.1.2 since f T isperfect by [SGA6, III, Corollaire 4.7.2]) induce a natural transformation of functors f ∗ : K ,X/k → K ,Y/k , by the base change theorem [Lip09, Theorem 3.10.3] (whichcan be applied since f T : X T → Y T and Y U → Y T are Tor-independent for allmorphisms U → T in (Sch /k )). Proposition 2.3.
Let f : X → Y be a perfect birational morphism between properintegral varieties over k .(i) If Y is normal, then f ∗ restricts to a morphism f ∗ : SK ,X/k → SK ,Y/k .(ii) If X and Y are regular, then f ∗ ◦ f ∗ is the identity of K ,Y/k .Proof of (i). Let U ⊂ Y be the biggest open subset above which f is an isomor-phism. Since Y is normal, the depth of O Y,y is ≥ y ∈ Y \ U .Fix T ∈ (Sch /k ) and a class x ∈ SK ( X T ). Since (rk( x ) , det( x )) | U T is trivial,so is (rk(( f T ) ∗ x ) , det(( f T ) ∗ x )) | U T ∈ Z ( U T ) × Pic( U T ). To deduce the trivialityof (rk(( f T ) ∗ x ) , det(( f T ) ∗ x )), it suffices to show the injectivity of the restrictionmorphisms Z ( Y T ) → Z ( U T ) and Pic( Y T ) → Pic( U T ).The morphism Z ( Y T ) → Z ( U T ) is actually bijective by Remark 1.2 (ii) applied tothe injection U ֒ → Y . To show the injectivity of Pic( Y T ) → Pic( U T ), we can assumethat T is noetherian by absolute noetherian approximation [TT90, Theorem C.9]and by the limit arguments of [EGA43, §8.5]. It then suffices to combine [SGA2,XI, Lemme 3.4] and [EGA42, Proposition 6.3.1]. Proof of (ii).
Fix T ∈ (Sch /k ). Chatzistamatiou and Rülling [CR15, Theorem 1.1]have shown that the natural morphism O Y → R f ∗ O X is a quasi-isomorphism. Thebase change theorem [Lip09, Theorem 3.10.3] (which can be applied as Y T is flatover Y ) implies that the natural morphism O Y T → R( f T ) ∗ O X T is also a quasi-isomorphism. That ( f T ) ∗ ◦ ( f T ) ∗ is the identity of K ( Y T ) then follows from theprojection formula (2.1). (cid:3) Curves.
We can entirely compute K ,X/k,τ if X is a curve. Proposition 2.4. If τ ∈ { Zar , ´et , fppf } and if X is a projective variety of dimen-sion ≤ over k , then (rk , det) : K ,X/k,τ → Z X/k × Pic
X/k,τ is an isomorphism.Proof.
It suffices to prove the proposition for τ = Zar. The commutation of K and Pic with directed inverse limits of qcqs schemes with affine transition maps(see [TT90, Proposition 3.20] and [EGA43, §8.5]), applied to the system of affineneighbourhoods of a point in a qcqs k -scheme, shows that it suffices to prove thebijectivity of (rk , det) : K ( X T ) → Z ( X T ) × Pic( X T ) for any local k -scheme T .By absolute noetherian approximation [TT90, Theorem C.9], we can write T asthe limit of a directed inverse system ( T i ) i ∈ I of noetherian k -schemes with affinetransition maps. Replacing the T i by their localizations at the images of the closedpoint of T , we may assume that they are local. A limit argument as above thenshows that we may assume T to be noetherian. This case follows from Lemma 2.5below applied to the connected components of X T . (cid:3) Lemma 2.5.
Let π : Y → T be a projective morphism with T local noetherian, Y non-empty and connected, and fibers of dimension ≤ . Then the morphism (rk , det) : K ( Y ) → Z × Pic( Y ) is bijective.Proof. The surjectivity of (rk , det) is obvious and we prove its injectivity.As explained in §2.1.1, K ( Y ) is generated by classes of vector bundles on Y .Lemma 2.6 below and induction on the rank of vector bundles show that it is evengenerated by classes of line bundles on Y . Let L and M be two line bundles on Y .Applying Lemma 2.6 to the vector bundles L ⊕ M and O Y ⊕ ( L ⊗ M ) with thesame l ≫ O Y (1) on Y yields theidentity [ L ] + [ M ] = [ O Y ] + [ L ⊗ M ] ∈ K ( Y ). This identity and the fact that K ( Y ) is generated by line bundles implies that x = [det( x )] + (rk( x ) − O Y ] forall x ∈ K ( Y ). This shows at once the required injectivity. (cid:3) Lemma 2.6.
In the setting of Lemma 2.5, there exists a π -ample line bundle O Y (1) on Y with the following property. For all vector bundles E of rank ≥ on Y andall l ≫ , there exists a short exact sequence of vector bundles on Y of the form (2.6) 0 → O Y ( − l ) → E → F → .Proof. Let O Y (1) be a π -ample line bundle on Y , let t be the closed point of T ,and let A ⊂ Y t be a finite subset meeting all the irreducible components of Y t . If m ≫
0, then H ( Y t , O Y t ( m )) → H ( A, O A ( m )) is surjective. As a consequence,after replacing O Y (1) with O Y ( m ), we may assume the existence of a section α ∈ H ( Y t , O Y t (1)) that vanishes at only finitely many points.The same argument yields, for some m ≥
0, a section β ∈ H ( Y t , E Y t ( m )) thatvanishes at only finitely many points. Let B ⊂ Y t be a finite subset meeting everyirreducible component of Y t and such that αβ does not vanish at any point of B .Since H ( Y t , E Y t ( n )) → H ( B, E B ( n )) is surjective for n ≫
0, we can choose n ≥ γ ∈ H ( Y t , E Y t ( n )) such that αβ and γ are linearly dependent at onlyfinitely many points of Y t . Let C ⊂ Y t be this finite set of points.Choose any l ≥ max( m, n ) such that H ( Y t , O Y t ( l − n )) → H ( C, O C ( l − n ))and H ( Y, E ( l )) → H ( Y t , E Y t ( l )) are surjective. (Such l exist by Serre vanishing[EGA31, Théorème 2.2.1]; this is where we use the noetherianity of T .) Then thereexists δ ∈ H ( Y t , O Y t ( l − n )) such that τ := α l − m β + γδ ∈ H ( Y t , E Y t ( l )) vanishesnowhere. Lift τ to a section σ ∈ H ( Y, E ( l )). Since π is proper and T is local, σ doesnot vanish on Y , thus giving rise to a short exact sequence of the form (2.6). (cid:3) Remark . If the residue field of T is infinite, the proof of Lemma 2.6 can besimplified as one can then apply [Kle69, Corollary 3.6] on Y t to construct τ .2.2.4. Projective bundles and blow-ups. If X is a proper variety over k and if E is avector bundle of rank r on X , the formula (2.2) induces an isomorphism of functors(2.7) K r ,X/k ∼ −→ K , P X E /k . In view of the isomorphism rk : K , Spec( k ) /k, Zar ∼ −→ Z given by Proposition 2.4, wededuce that K , P r − k /k, Zar ≃ Z r and that ([ O P r − k ( − j )]) ≤ j ≤ r − forms a basis of the Z -module K , P r − k /k, Zar ( k ). The family ([ O P dk ]) ≤ d ≤ r − is another basis. For r ≥ , det) : K , P r − k /k, Zar → Z P r − k /k × Pic P r − k /k, Zar = Z yields an isomorphism SK , P r − k /k, Zar ≃ Z r − and shows that ([ O P dk ]) ≤ d ≤ r − is abasis of the Z -module SK , P r − k /k, Zar ( k ). NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 19
Let X be a proper variety over k and i : Y → X be a regular closed immersionof pure codimension c ≥
1. Define p : X ′ → X to be the blow-up of X along Y with exceptional divisor Y ′ , and p ′ : Y ′ → Y and i ′ : Y ′ → X ′ to be the naturalmorphisms. Then the formula (2.3) induces an isomorphism of functors(2.8) K ,X/k × K c − ,Y/k ∼ −→ K ,X ′ /k , in view of the functorialities described in §2.2.2.2.3. The functor CH X/k, fppf . We introduce, for a smooth proper geometricallyconnected threefold X over k with geometrically trivial Chow group of zero-cycles,the functor CH X/k, fppf . It will play for codimension 2 cycles the same role as thePicard functor Pic
X/k, fppf does for codimension 1 cycles.2.3.1.
The class of a point.
We first exhibit a canonical class ν X ∈ K ,X/k, fppf ( k ). Proposition 2.8.
Let X be a smooth proper geometrically connected varietyover k whose degree map deg : CH ( X k ) → Z is an isomorphism. Choose τ ∈ { Zar , ´et , fppf } . Assume that k = k if τ = Zar and that k = k p if τ = ´et .There exists a unique ν X ∈ K ,X/k,τ ( k ) such that for all finite extensions k ′ of k and all coherent sheaves F on X k ′ whose support has dimension , one has (2.9) [ F ] = h ( X k ′ , F ) · ν X ∈ K ,X/k,τ ( k ′ ) . Proof.
Using [BLR90, 2.2/13], choose a finite Galois extension l of k and a point x ∈ X ( l ). Let n be the dimension of X . For all field extensions l ′ of l , the definitionof the flat pull-back of a cycle [Ful98, §1.5, §1.7] and the formula [SGA6, X, (1.1.3)]show the commutativity of the natural diagram(2.10) CH ( X l ) ϕ n / / (cid:15) (cid:15) F K ( X l ) (cid:15) (cid:15) CH ( X l ′ ) ϕ n / / F K ( X l ′ ),where the morphisms ϕ n are defined in §2.1.5.Assume first that τ = fppf.Let { x , . . . , x m } be the Gal( l/k )-orbit of x . Since deg : CH ( X k ) ∼ −→ Z is anisomorphism, there exists a finite extension l ′ of l such that the x i all have thesame class in CH ( X l ′ ). Since Spec( l ′ ) → Spec( l ) is an fppf covering, we deducefrom (2.10) that ϕ n ([ x ]) ∈ K ,X/k,τ ( l ) is Gal( l/k )-invariant, hence descends to aclass ν X ∈ K ,X/k,τ ( k ) since Spec( l ) → Spec( k ) is an étale covering. Applying (2.9)with k ′ = l and F = O x shows that this is the only possible choice for ν X and provesthe uniqueness assertion of Proposition 2.8.Let us now show that ν X satisfies (2.9). Let k ′ be a finite extension of k andlet F be a coherent sheaf on X k ′ whose support has dimension 0. Let l ′ be a finiteextension of k containing both k ′ and l , with the property that all the points inthe support of F l ′ have residue field l ′ , and are rationally equivalent to x . (Suchan l ′ exists since deg : CH ( X k ) ∼ −→ Z .) The formula [SGA6, X, (1.1.3)] and thecommutativity of (2.10) show that [ F l ′ ] = h ( X l ′ , F l ′ ) · ν X ∈ K ,X/k,τ ( l ′ ). SinceSpec( l ′ ) → Spec( k ′ ) is an fppf covering, identity (2.9) follows.If τ = ´et (resp. τ = Zar), all the fppf (resp. fppf or étale) coverings that appearabove are étale (resp. Zariski) coverings, proving the proposition in these cases. (cid:3) Codimension cycles on a threefold. Let us fix in §2.3.2 a smooth propergeometrically connected threefold X over k whose degree map deg : CH ( X Ω ) → Z is an isomorphism for all algebraically closed field extensions k ⊂ Ω (an assumptionthat is satisfied if X is k -rational).Choose τ ∈ { Zar , ´et , fppf } , and assume that k = k if τ = Zar and that k = k p if τ = ´et, so that Spec( l ) → Spec( k ) is a τ -covering for any finite extension l of k .Let ν X ∈ K ,X/k,τ ( k ) be the class defined in Proposition 2.8. If x ∈ X ( l ) for somefinite extension l of k , then (rk , det)([ O x ]) = (0 , O X ) (see §2.1.5). In view of (2.9),one therefore has ν X ∈ SK ,X/k,τ ( k ) ⊂ SK ,X/k,τ ( l ).We still denote by ν X the morphism of τ -sheaves ν X : Z → SK ,X/k,τ such that ν X (1) = ν X . We view ν X as a morphism of presheaves of abelian groups. Definition 2.9.
We let CH X/k,τ : (Sch /k ) op → (Ab) be the (presheaf) cokernelof ν X : Z → SK ,X/k,τ . When CH X/k, fppf is representable, we let CH X/k be thegroup scheme over k that represents it. Remark . Let k ′ /k be a field extension. There is an obvious identification K ,X k ′ /k ′ ( T ) = K ,X/k ( T ) for all T ∈ (Sch /k ′ ). We thus obtain natural iso-morphisms K ,X k ′ /k ′ ,τ ( T ) = K ,X/k,τ ( T ), SK ,X k ′ /k ′ ,τ ( T ) = SK ,X/k,τ ( T ) andCH X k ′ /k ′ ,τ ( T ) = CH X/k,τ ( T ) for all T ∈ (Sch /k ′ ) and τ ∈ { Zar , ´et , fppf } . In partic-ular, if CH X/k, fppf is representable, so is CH X k ′ /k ′ , fppf , and CH X k ′ /k ′ = ( CH X/k ) k ′ .The following proposition justifies these definitions. Proposition 2.11.
Associating with the class [ Z ] ∈ CH ( X k ) of a codimension integral closed subvariety Z ⊂ X k the class [ O Z ] ∈ K ( X k ) of its structure sheafinduces a Γ k -equivariant isomorphism (2.11) CH ( X k ) ∼ −→ CH X k /k, Zar ( k ) = CH X/k,
Zar ( k ) .Proof. In view of (2.9), one has a natural isomorphism(2.12) SK ( X k ) /F K ( X k ) ∼ −→ CH X k /k, Zar ( k ).Precomposing (2.12) with ϕ : CH ( X k ) ∼ −→ Gr F K ( X k ) = SK ( X k ) /F K ( X k )(see §2.1.5) yields the isomorphism (2.11). It is Γ k -equivariant by construction. (cid:3) The next lemma will be used in the proof of Theorem 3.1 (iv). For the definitionof α | X t ∈ CH ( X t ) in its statement, see [Ful98, Example 5.2.1]. Lemma 2.12.
Let T be a smooth connected variety over k .(i) For all α ∈ CH ( X T ) , there exists a class β ∈ SK ( X T ) with the propertythat for all t ∈ T ( k ) , the image of α | X t by (2.11) is induced by β | X t .(ii) For all β ∈ SK ( X T ) , there exists a class α ∈ CH ( X T ) with the propertythat for all t ∈ T ( k ) , the image of α | X t by (2.11) is induced by β | X t .Proof of (i). We can assume that α is the class of an integral subvariety Z ⊂ X T of codimension 2. Define β := [ O Z ] ∈ SK ( X T ). By the Riemann–Roch theoremwithout denominators, one has α = − c ( β ) (see §2.1.5). For t ∈ T ( k ), one has α | X t = − c ( β | X t ). Applying the Riemann–Roch theorem without denominatorsagain shows that the image of α | X t by (2.11) is the class induced by β | X t . Proof of (ii).
Define α = − c ( β ) ∈ CH ( X T ) and argue as in (i). (cid:3) NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 21 Geometrically rational threefolds
In Section 3, we prove the representability of the functor CH X/k, fppf definedin §2.3.2 if X is a smooth projective k -rational threefold, and study the groupscheme CH X/k that represents it.3.1.
Main statement.
Our goal is the following theorem.
Theorem 3.1.
Let X be a smooth projective k -rational threefold over k . Then:(i) CH X/k, fppf is represented by a smooth group scheme CH X/k over k .(ii) ( CH X/k ) is an abelian variety over k .(iii) CH X k p /k p , fppf = CH X k p /k p , ´et and CH X k /k, fppf = CH X k /k, Zar .(iv) The Γ k -equivariant isomorphism (3.1) ψ X : CH ( X k ) ∼ −→ CH X/k ( k ) obtained by combining (iii) and (2.11) restricts to a Γ k -equivariant bijectiveregular homomorphism (in the sense of §1.6) (3.2) ψ X : CH ( X k ) alg ∼ −→ ( CH X/k ) ( k ) . (v) The étale group scheme CH X/k / ( CH X/k ) over k is associated with the Γ k -module NS ( X k ) , which as a Z -module is free of finite rank.(vi) The isomorphism (3.2) induces an isomorphism Ab ( X k p ) ∼ −→ ( CH X k p /k p ) (where Ab ( X k p ) denotes Murre’s intermediate Jacobian, introduced in §1.6).We endow CH X/k with the principal polarization (in the sense of §1.2) induced bythe principal polarization θ X of Ab ( X k p ) (see §1.6).(vii) If X is k -rational, there exists a smooth projective curve B over k such that CH X/k is a principally polarized direct factor of
Pic
B/k (in the sense of §1.2).
The proof of Theorem 3.1 is given in §3.2. Theorem 3.1 is complemented in §3.3by a computation of CH X/k for varieties constructed as blow-ups, and in §3.4 byan analysis of the obstructions to k -rationality arising from Theorem 3.1 (vii). Remarks . (i) Let X be a smooth projective variety over k . As recalledin §1.6, Achter, Casalaina-Martin and Vial have endowed Ab ( X k ) with a natural k p -structure. If X is moreover a k -rational threefold, Theorem 3.1 (vi) furtherendows Ab ( X k ) with a natural k -structure ( CH X/k ) . Trying to descend Ab ( X k )to k under more general hypotheses gives an incentive to define CH X/k, fppf and toprove its representability in a greater generality.(ii) For instance it would be nice to define and study a functor CH X/k, fppf for allsmooth proper varieties X over k such that CH ( X ) Q is supported in dimension 1in the sense of Bloch and Srinivas (see [BW19, Definition 2.1]). If a good enoughtheory of motivic cohomology H ∗M over a field of characteristic p > p in the coefficients, a natural choice would be the fppfsheafification of the functor T H M ( X T , Z (2)). One could also consider thefppf sheafification of the functor T H ( X T , K ), where K denotes Quillen’s K -theory sheaf [Blo73], or of the functor T A ( X T ), where A denotes Fulton’scohomological Chow group [Ful98, Definition 17.3]. (iii) Even with our definition of CH X/k, fppf , it would be interesting to showthe representability of CH X/k, fppf under the hypothesis, weaker than k -rationality,that X is a smooth, proper and geometrically connected threefold over k suchthat deg : CH ( X Ω ) ∼ −→ Z is an isomorphism for all algebraically closed fieldextensions k ⊂ Ω. We note, however, that it is the proof of representability thatwe give here, and which is specific to ¯ k -rational threefolds, that yields the crucialTheorem 3.1 (vii) and thus provides obstructions to k -rationality.(iv) We still denote by θ X the principal polarization of CH X/k induced bythat of Ab ( X k p ), as in Theorem 3.1 (vi). For the sake of completeness, weextract a characterization of θ X from [BW19, §2.3] and from the definition of theisomorphism (3.2). Let ℓ be a prime number invertible in k . Consider the diagram( CH X/k ) ( k ) { ℓ } CH ( X k ) alg { ℓ } ψ X ∼ o o λ / / H ( X k , Z ℓ (2)) ⊗ Q ℓ / Z ℓ ,where λ is Bloch’s Abel–Jacobi map (see [Blo79, §2], [BW19, (2.3)]) and ψ X is themap (3.2). Then c ( θ X ) ∈ H (( CH X/k ) k , Z ℓ (1)) = (cid:0) V H (( CH X/k ) k , Z ℓ ) (cid:1) (1)corresponds, via the identification H (( CH X/k ) k , Z ℓ ) ∨ T ℓ ( λ ◦ ( ψ X ) − ) −−−−−−−−−−→ H ( X k , Z ℓ (2)) / (torsion)(in which ∨ stands for Hom( − , Z ℓ )), to the opposite of the cup product pairing V H ( X k , Z ℓ (2)) → H ( X k , Z ℓ (4)) ∼ −→ Z ℓ (1).3.2. Proof of Theorem 3.1.
Resolution of indeterminacies.
Our main tool is a resolution of indetermi-nacies result that is due to Abhyankar [Abh98] if k is perfect, and to Cossart andPiltant [CP08] in general. Proposition 3.3.
Let X and Y be smooth projective threefolds over k . Let f : Y X be a birational map. Then there exists a diagram (3.3) X h ←− X ′ = Y N +1 → · · · → Y j +1 p j −→ Y j → · · · → Y = Y of regular projective varieties over k such that f = h ◦ p − N ◦ · · · ◦ p − , where p j isthe blow-up of an integral regular closed subscheme Z j ⊂ Y j of codimension c j andwhere h is projective and birational.Proof. This follows from [CP08], as explained in [BW19, Proposition 2.11]. Thestanding assumption that k is perfect in [BW19] is irrelevant if one really uses[CP08, Proposition 4.2] (or [CJS09, Theorem 5.9]) instead of [Abh98, (9.1.4)] inthe proof of [BW19, Proposition 2.11]. (cid:3) Remark . If k is not perfect, the subschemes Z j ⊂ Y j may not be smooth over k .3.2.2. Representability if X is k -rational. In §3.2.2, we fix τ ∈ { Zar , ´et , fppf } andassume that k = k if τ = Zar and that k = k p if τ = ´et. We also let X be a smoothprojective k -rational threefold. By Proposition 3.3, there exists a diagram (3.3)with Y = P k . Remark 1.2 (ii), Corollary 1.5 and the isomorphism (2.8) then give NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 23 canonical decompositions Z X ′ /k ∼ ←− Z P k /k ,(3.4) Pic X ′ /k ∼ ←− Pic P k /k × Y c j ≥ Z Z j /k ,(3.5) K ,X ′ /k ∼ ←− K , P k /k × Y c j =2 K ,Z j /k × Y c j =3 ( K ,Z j /k ) .(3.6)Identifying (rk , det) : K ,X ′ /k → Z X ′ /k × Pic X ′ /k in terms of these decompositionsand using the isomorphisms (rk , det) : K ,Z j /k,τ ∼ −→ Z Z j /k × Pic Z j /k,τ given byProposition 2.4 yields an isomorphism(3.7) SK ,X ′ /k,τ ∼ ←− SK , P k /k,τ × Y c j =2 Pic Z j /k,τ × Y c j =3 Z Z j /k .The morphisms p j are lci by [Tho93, §1.2], hence so is the structural morphism X ′ → Spec( k ) by [SGA6, VIII, Proposition 1.5]. Any closed embedding X ′ ֒ → P NX of the X -scheme X ′ is a regular immersion by [SGA6, VIII, Proposition 1.2],which shows that h is lci, hence perfect [SGA6, VIII, Proposition 1.7]. Thefunctors h ∗ : K ,X/k → K ,X ′ /k and h ∗ : K ,X ′ /k → K ,X/k satisfy h ∗ ◦ h ∗ = Idby Proposition 2.3 (ii). Since they restrict to h ∗ : SK ,X/k → SK ,X ′ /k andto h ∗ : SK ,X ′ /k → SK ,X/k (see Proposition 2.3 (i)), we deduce a naturaldecomposition(3.8) SK ,X/k × Ker (cid:0) h ∗ : SK ,X ′ /k → SK ,X/k (cid:1) ∼ −→ SK ,X ′ /k .The three summands of the right-hand side of (3.7) are represented by groupschemes locally of finite type over k , respectively by §2.2.4, by §1.4 and byProposition 1.1. It follows that SK ,X ′ /k,τ is represented by a group scheme locallyof finite type over k . So is SK ,X/k,τ , by (3.8) and Lemma 1.17.Let x ′ ∈ X ′ ( k ) be a general point, and let x ∈ X ( k ) and y ∈ P ( k ) be its imagesby h and by p ◦ · · · ◦ p N . Then h ∗ [ O x ] = [ O x ′ ] = p ∗ N ◦ · · · ◦ p ∗ [ O y ] ∈ SK ,X ′ /k,τ ( k ).Consequently, h ∗ ◦ ν X : Z → SK ,X ′ /k,τ and p ∗ N ◦· · ·◦ p ∗ ◦ ν P k : Z → SK ,X ′ /k,τ bothsend 1 ∈ Z ( k ) to [ O x ′ ] ∈ SK ,X ′ /k,τ ( k ). For n = 0, the class ν P k ( n ) does not belongto the identity component of SK , P k /k,τ , by §2.2.4. We deduce from the above andfrom (3.7) that n [ O x ′ ] / ∈ ( SK ,X ′ /k,τ ) ( k ), hence that ν X ( n ) / ∈ ( SK ,X/k,τ ) ( k ).Lemma 1.18 then shows that CH X/k,τ is represented by a group scheme locally offinite type over k .Applying the above with τ = fppf and combining (3.7), (3.8) and the equality h ∗ ◦ ν X = p ∗ N ◦ · · · ◦ p ∗ ◦ ν P k yields isomorphisms(3.9) CH X/k × G ∼ −→ CH X ′ /k ∼ ←− CH P k /k × Y c j =2 Pic Z j /k × Y c j =3 Z Z j /k of smooth group schemes over k , where G denotes the group scheme representing thefppf sheafification of Ker (cid:0) h ∗ : SK ,X ′ /k → SK ,X/k (cid:1) (whose representability followsfrom Lemma 1.17) and where CH X ′ /k denotes the group scheme representing thepresheaf cokernel Coker (cid:0) h ∗ ◦ ν X : Z → SK ,X ′ /k, fppf (cid:1) (whose representabilityfollows from Lemma 1.18). Representability if X is k -rational. In §3.2.3, we prove Theorem 3.1 (i)–(iii)for a smooth projective k -rational threefold X over k .Choose τ ∈ { Zar , ´et , fppf } and assume that k = k if τ = Zar and that k = k p if τ = ´et. Let l be a finite extension of k such that X is l -rational. Then SK ,X l /l,τ is represented by a group scheme locally of finite type over l , by the argumentsof §3.2.2 applied to the l -variety X l . By Lemma 1.19, it follows that SK ,X/k,τ is represented by a group scheme locally of finite type over k . As explainedin §3.2.2, the morphism ν X : Z → SK ,X/k,τ defined in §2.3.2 has the propertythat ν X ( n ) / ∈ ( SK ,X/k,τ ) ( k ) for all n = 0. It now follows from Lemma 1.18 thatCH X/k,τ is represented by a group scheme locally of finite type over k . Proof of Theorem 3.1 (i)–(iii).
Applying the above argument to the k -variety X with τ = fppf, to the k p -variety X k p with τ = ´et, and to the k -variety X k with τ = Zar shows that the three functors CH X/k, fppf , CH X k p /k p , ´et and CH X k /k, Zar are represented by group schemes locally of finite type over k , over k p and over k ,respectively. In particular, the latter two are sheaves for the fppf topology, whichproves Theorem 3.1 (iii). In addition, the arguments of §3.2.2 applied to the k -variety X k show that ( CH X k /k ) is a direct factor of a product of Jacobians ofsmooth projective curves over k (see (3.9)), hence is an abelian variety; in particular,it is smooth. As (( CH X/k ) ) k = ( CH X k /k ) , Theorem 3.1 (i)–(ii) follows. (cid:3) Relation with Murre’s work.
We now prove Theorem 3.1 (iv)–(vi).
Proof of Theorem 3.1 (iv).
That ψ X (CH ( X k ) alg ) ⊂ (( CH X/k ) ( k )), and that theresulting morphism ψ X : CH ( X k ) alg → ( CH X/k ) ( k ) is a regular homomorphismfollow at once from Lemma 2.12 (i). It remains to show that (3.2) is surjective.Since CH X k /k represents CH X k /k, Zar by Theorem 3.1 (iii), we can choose aconnected Zariski neighbourhood T of the identity in ( CH X k /k ) and a class β ∈ SK ( X T ) inducing the natural inclusion T → CH X k /k . Lemma 2.12 (ii)then implies that T ( k ) is contained in the image of (3.2). Since ( CH X/k ) ( k ) isgenerated by T ( k ) as a group, we have proved the surjectivity of (3.2). (cid:3) Proof of Theorem 3.1 (v).
That the étale k -group scheme ( CH X/k ) / ( CH X/k ) corresponds to the Γ k -module NS ( X k ) follows at once from (3.1) and (3.2).Applying the discussion of §3.2.2, and more precisely identity (3.9), to the k -variety X k , shows, in view of the isomorphism CH P k /k ≃ Z (see §2.2.4), thatNS ( X k ) is a free Z -module of finite rank, being a direct factor of such a module. (cid:3) Proof of Theorem 3.1 (vi).
The regular homomorphism (3.2) induces a morphism ι X k : Ab ( X k ) → ( CH X k /k ) . Since (3.2) is Γ k -equivariant, and in view of thedefinition of Ab ( X k p ) recalled in §1.6, this morphism descends by Galois descentto a morphism ι X k p : Ab ( X k p ) → ( CH X k p /k p ) of abelian varieties over k p . Toprove that ι X k p is an isomorphism, it suffices to prove that ι X k is an isomorphism.From now on, we may thus assume that k = k . NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 25
By Proposition 3.3, there exists a diagram (3.3). Since k = k , all the varieties Z j and Y j that appear in it are smooth over k . Consider the diagram(3.10) Ab ( P k ) × Q c j =2 Pic Z j /k ∼ / / ( ι P k , Id) (cid:15) (cid:15) Ab ( X ′ ) ι X ′ (cid:15) (cid:15) ( CH P k /k ) × Q c j =2 Pic Z j /k ∼ / / ( CH X ′ /k ) ,where the lower horizontal isomorphism is induced by (3.7) and the upper horizontalisomorphism is the one provided by [BW19, Lemma 2.10]. Since CH ( P k ) alg = 0,one has Ab ( P k ) = ( CH P k /k ) = 0 and the left vertical arrow of (3.10) is anisomorphism. We claim that (3.10) commutes. Since k = k , it suffices to verify thatit commutes at the level of k -points, which follows from unwinding the definitionsand making use of Lemma 2.1. A glance at (3.10) now shows that ι X ′ is anisomorphism.Now, consider the diagram(3.11) Ab ( X ) h + / / ι X (cid:15) (cid:15) Ab ( X ′ ) h + / / ≀ ι X ′ (cid:15) (cid:15) Ab ( X ) ι X (cid:15) (cid:15) ( CH X/k ) h ∗ / / ( CH X ′ /k ) h ∗ / / ( CH X/k ) ,whose lower horizontal arrows are induced by (3.8) and hence satisfy h ∗ ◦ h ∗ = Id,and whose upper horizontal arrows are given by the functoriality of Murre’sintermediate Jacobians (see [BW19, §2.2.1]) and satisfy h + ◦ h + = Id as aconsequence of the identity h ∗ ◦ h ∗ = Id : CH ( X ) → CH ( X ) stemming fromthe projection formula [Ful98, Proposition 8.3(c)]. To show that (3.11) commutes,it suffices to check that it commutes at the level of k -points, since k = k .This follows from Lemma 2.1 for the left-hand square, and from the fact thatthe morphisms ϕ c considered in §2.1.5 are compatible with proper push-forwards[Ful98, Example 15.1.5] for the right-hand square. A diagram chase in (3.11) showsthat ι X is an isomorphism since ι X ′ is one, which concludes the proof. (cid:3) Further analysis of k-rational varieties.
We resume the discussion of §3.2.2with τ = fppf, and keep the notation introduced there. Since CH P k /k ≃ Z by§2.2.4, identity (3.9) reads:(3.12) CH X/k × G ∼ −→ CH X ′ /k ∼ ←− Z × Y c j =2 Pic Z j /k × Y c j =3 Z Z j /k .The identity component of the right-hand side is isomorphic to Q c j =2 Pic Z j /k ,hence it carries a natural principal polarization (see §1.4.2). Via (3.12), we thusobtain a principal polarization on CH X ′ /k in the sense of §1.2. Proposition 3.5.
The isomorphism (3.12) realizes CH X/k and G as principallypolarized direct factors, in the sense of §1.2, of CH X ′ /k , and the induced polariza-tion on CH X/k coincides with the one defined in Theorem 3.1 (vi).Proof.
We fix once and for all a prime number ℓ invertible in k and start witha few recollections about (Borel–Moore) ℓ -adic étale homology. If V is a varietyover k , the i -th étale homology group of V with coefficients in Q ℓ ( j ) is defined by H i ( V, Q ℓ ( j )) = H − i ( V, Rε ! Q ℓ ( j )), where ε : V → Spec( k ) denotes the structuralmorphism (see [Lau76, Définition 2]). This group is covariantly functorial withrespect to proper morphisms ( loc. cit. , §4) and comes with a cap product operation H s ( V, Q ℓ ( t )) × H i ( V, Q ℓ ( j )) → H i − s ( V, Q ℓ ( j + t )) ( loc. cit. , p. VIII-09), and with acycle class map cl : CH i ( V ) → H i ( V, Q ℓ ( − i )) ( loc. cit. , §6), for any i , j , s , t . If V is of pure dimension n , we denote by [ V ] the fundamental cycle of V (see [Ful98,§1.5]), so that cl([ V ]) ∈ H n ( V, Q ℓ ( − n )). If in addition V is smooth, the map κ ´et V : H n − i ( V, Q ℓ ( j + n )) → H i ( V, Q ℓ ( j ))(3.13)defined by κ ´et V ( x ) = x ∩ cl([ V ]) is an isomorphism for all i , j (see [Lau76, Prop. 3.2]).Thus, given a proper morphism f : V ′ → V from a variety V ′ of pure dimension n ′ to a smooth variety V of pure dimension n over k , one can define a push-forwardin étale cohomology f ∗ : H s ( V ′ , Q ℓ ( t )) → H s +2 n − n ′ ( V, Q ℓ ( t + n − n ′ ))(3.14)as the composition ( κ ´et V ) − ◦ f ∗ ◦ κ ´et V ′ . When V is a proper variety over k of pure odddimension n , we will speak of the “cup product pairing on H n ( V, Q ℓ (( n + 1) / H n ( V, Q ℓ (( n + 1) / × H n ( V, Q ℓ (( n + 1) / → Q ℓ (1) inducedby the cup product and by the push-forward map H n ( V, Q ℓ ( n + 1)) → Q ℓ (1) alongthe structural morphism V → Spec( k ) (see (3.14)). Lemma 3.6.
Let f : D → V be a morphism between projective varieties over k ,where D has pure dimension d and V has pure dimension d + 1 , for someinteger d . Suppose that V is smooth. Let λ : Pic( D ) { ℓ } ∼ −→ H ( D, Q ℓ / Z ℓ (1)) and λ : CH ( V ) { ℓ } → H ( V, Q ℓ / Z ℓ (2)) respectively denote the Kummer isomorphismand Bloch’s ℓ -adic Abel–Jacobi map. Then the diagram V ℓ ( F K ( V )) − V ℓ ( c ) / / V ℓ (CH ( V )) V ℓ ( λ ) / / H ( V, Q ℓ (2)) V ℓ ( K ( D )) V ℓ (det) / / f ∗ O O V ℓ (Pic( D ))) V ℓ ( λ ) / / H ( D, Q ℓ (1)) f ∗ O O (3.15) commutes, where the right-hand side vertical arrow is the map (3.14) and theleft-hand side vertical arrow is induced by the composition of the canonical map K ( D ) → G ( D ) , which sends the rank subgroup of K ( D ) to F d − G ( D ) (see [SGA6, X, Corollaire 1.3.3] ), with f ∗ : F d − G ( D ) → F d − G ( V ) . We stress that D is not assumed to be reduced in Lemma 3.6. Proof.
Let f K ( D ) = Ker(rk : K ( D ) → Z ) and Gr γ K ( D ) = f K ( D ) / SK ( D ) (apiece of notation justified by the fact that SK ( D ) and f K ( D ) form the beginningof the γ -filtration on K ( D )). As the canonical map K ( D ) → G ( D ) sends f K ( D )to F d − G ( D ) and SK ( D ) to F d − G ( D ) (see [SGA6, X, Corollaire 1.3.3]), thereis an induced map f ∗ : V ℓ (Gr γ K ( D )) → V ℓ (Gr F K ( V )) and it suffices to provethe commutativity of the diagram V ℓ (Gr F K ( V )) − V ℓ ( c ) ∼ / / V ℓ (CH ( V )) V ℓ ( λ ) / / H ( V, Q ℓ (2)) V ℓ (Gr γ K ( D )) V ℓ (det) ∼ / / f ∗ O O V ℓ (Pic( D ))) O O V ℓ ( λ ) / / H ( D, Q ℓ (1)), f ∗ O O (3.16) NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 27 without the dotted arrow. We note that the leftmost horizontal arrows of (3.16) areisomorphisms; their inverses are induced by the map ϕ : CH ( V ) → Gr F K ( V )of (2.4) and by the map Pic( D ) → Gr γ K ( D ) which sends the class of a Cartierdivisor Z on D to the class of [ O D ( Z )] − [ O D ] ∈ f K ( D ).Let us complete this diagram with a dotted arrow induced by the composition ofthe canonical map Pic( D ) → CH d − ( D ) (see [Ful98, §2.1]) with the push-forward f ∗ : CH d − ( D ) → CH d − ( V ).When D is smooth, the right half of (3.16) commutes by [Blo79, Proposition 3.3,Proposition 3.6] and the left half by the description of the inverses of the horizontalarrows. Thus (3.16) commutes in this case.In general, let us choose a family ( D j ) j ∈ J of smooth projective varieties of puredimension d , and for each j ∈ J , a morphism ν j : D j → D and an element n j ∈ Z ℓ ,such that the equality of d -cycles with coefficients in Z ℓ [ D ] = X j ∈ J n j ν j ∗ [ D j ](3.17)holds. When dim( D ) ≤ D j to be desingularisations of the irreducible components of D red and the n j to be the multiplicities, in D , of these irreducible components. Inarbitrary dimension, such D j , ν j and n j exist by the Gabber–de Jong alterationtheorem [IT14, Theorem 2.1] applied to the irreducible components of D red .For j ∈ J , let f j = f ◦ ν j : D j → V . As D j is smooth, we have already seenthat the outer square of (3.16) with D and f replaced with D j and f j commutes.In order to show that the outer square of (3.16) itself commutes, it thereforesuffices, by the contravariant functoriality of the lower row of (3.16), to check theequality f ∗ = P j ∈ J n j f j ∗ ◦ ν ∗ j of maps V ℓ (Gr γ K ( D )) → V ℓ (Gr F K ( V )) and thesame equality of maps H ( D, Q ℓ (1)) → H ( V, Q ℓ (2)). Let us set Gr Fi G ( D ) = F i G ( D ) /F i − G ( D ) and denote by κ D : Gr γ K ( D ) → Gr Fd − G ( D ) the mapinduced by the canonical map K ( D ) → G ( D ). Coming back to the definitionof f ∗ in the two contexts, we now see that it is enough to check the equalities κ D = X j ∈ J n j ν j ∗ ◦ κ D j ◦ ν ∗ j : V ℓ (Gr γ K ( D )) → V ℓ (Gr Fd − G ( D ))(3.18)and κ ´et D = X j ∈ J n j ν j ∗ ◦ κ ´et D j ◦ ν ∗ j : H ( D, Q ℓ (1)) → H d − ( D, Q ℓ (1 − d )).(3.19)The K ( D )-module structure of G ( D ) induces for any i a cap product operationGr γ K ( D ) × Gr Fi G ( D ) → Gr Fi − G ( D ) (see [SGA6, X, Corollaire 1.3.3]). Letting[ O D ] denote the class of O D in Gr Fd G ( D ), we have κ D ( x ) = x ∩ [ O D ] for any x ∈ Gr γ K ( D ); moreover (3.17) implies the equality [ O D ] = P j ∈ J n j ν j ∗ [ O D j ]in Gr Fd G ( D ) ⊗ Z Z ℓ (see [Ful98, Example 15.1.5]). In view of the projectionformula [SGA6, IV, (2.11.1.2)], we deduce (3.18). Similarly, the definition of κ ´et D ,the equality obtained by applying cl to (3.17) and the projection formula [Lau76,Proposition 4.2] together imply (3.19). (cid:3) Let us finally start the proof of Proposition 3.5. As X is smooth over k , themorphism h gives rise to a push-forward map h ∗ : H ( X ′ k , Q ℓ (2)) → H ( X k , Q ℓ (2)) (see (3.14)), satisfying h ∗ ◦ h ∗ = Id on H ( X k , Q ℓ (2)), since h ∗ ( h ∗ x ∩ cl([ X ′ k ])) = x ∩ cl( h ∗ [ X ′ k ]) = x ∩ cl([ X k ]) for x ∈ H ( X k , Q ℓ (2)) (see [Lau76, Proposition 4.2]).Let K = Ker (cid:0) h ∗ : H ( X ′ k , Q ℓ (2)) → H ( X k , Q ℓ (2)) (cid:1) . We obtain a decomposition H ( X k , Q ℓ (2)) ⊕ K ∼ −→ H ( X ′ k , Q ℓ (2)).(3.20)A second decomposition of the right-hand side can be obtained using the formulafor the étale cohomology of the blow-up of a regularly immersed closed subscheme[Rio14, Proposition 2.7], which yields a canonical isomorphism M c j =2 H (( Z j ) k , Q ℓ (1)) ∼ −→ H ( X ′ k , Q ℓ (2))(3.21)even though both X ′ k and ( Z j ) k may fail to be regular.Let Z ′ j denote the normalization of ( Z j ) red k and ν j : Z ′ j → ( Z j ) k the natural mor-phism. The normality of Z j implies that ( Z j ) k is geometrically unibranch and hencethat ν j is universally bijective (see [EGA42, Proposition 6.15.6, Proposition 6.15.5]).We deduce that ν ∗ j : H (( Z j ) k , Q ℓ (1)) → H ( Z ′ j , Q ℓ (1)) is an isomorphism (see[SGA42, VIII, Corollaire 1.2]).Letting L = V ℓ ( A ( G k )), we now consider the diagram of isomorphisms H ( X k , Q ℓ (2)) ⊕ K ∼ / / H ( X ′ k , Q ℓ (2)) M c j =2 H (( Z j ) k , Q ℓ (1)) ≀ ν ∗ j (cid:15) (cid:15) ∼ o o V ℓ (( CH X/k ) k ) ⊕ L ≀ (cid:15) (cid:15) M c j =2 H ( Z ′ j , Q ℓ (1)) ≀ (cid:15) (cid:15) V ℓ ( A (( CH X/k ) k )) ⊕ L ∼ / / V ℓ ( A (( CH X ′ /k ) k )) M c j =2 V ℓ ( A (( Pic Z j /k ) k )), ∼ o o (3.22)whose upper horizontal arrows are (3.20) and (3.21), whose lower horizontal arrowsstem from (3.12), whose left vertical isomorphism results from Theorem 3.1 (ii)and whose lower right vertical isomorphism is the composition of the canonicalisomorphism V ℓ (Pic( Z ′ j )) ∼ −→ V ℓ ( A (( Pic Z j /k ) k )) coming from §1.4 and (1.4) withthe Kummer isomorphism H ( Z ′ j , Q ℓ (1)) ∼ −→ V ℓ (Pic( Z ′ j )).Let m j denote the multiplicity of ( Z j ) k p , i.e. , the length of its generic local ring,and θ j the canonical principal polarization of Pic Z j /k (see §1.4.2). Lemma 3.7. (i) Diagram (3.22) transports the opposite of the cup productpairing on H ( X ′ k , Q ℓ (2)) to the pairing on L c j =2 V ℓ ( A (( Pic Z j /k ) k )) definedas the orthogonal sum of the Q ℓ (1) -valued Weil pairings associated with thepolarizations m j θ j .(ii) Diagram (3.22) transports H ( X k , Q ℓ (2)) to V ℓ (( CH X/k ) k ) and K to L .(iii) The isomorphism H ( X k , Q ℓ (2)) = V ℓ (( CH X/k ) k ) resulting from (ii) coin-cides with the one induced by Bloch’s ℓ -adic Abel–Jacobi map and by theidentification between V ℓ (( CH X/k ) k ) and V ℓ (CH ( X k )) that stems from The-orem 3.1 (iv), (v). NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 29
Proof.
Let us consider the commutative diagram M c j =2 H (( Z j ) k , Q ℓ (1)) ≀ (cid:15) (cid:15) ν ∗ j / / M c j =2 H ( Z ′ j , Q ℓ (1)) ≀ (cid:15) (cid:15) M c j =2 V ℓ (Pic(( Z j ) k )) ν ∗ j / / M c j =2 V ℓ (Pic( Z ′ j )) M c j =2 V ℓ ( Pic Z j /k ) M c j =2 V ℓ ( Pic Z j /k ) / / ∼ o o M c j =2 V ℓ ( A (( Pic Z j /k ) k )) V ℓ ( CH X ′ /k ) V ℓ (( CH X ′ /k ) ) ∼ o o / / V ℓ ( A (( CH X ′ /k ) k )),(3.23)in which the unlabelled horizontal arrows are the obvious ones (the bottom leftwardarrow being an isomorphism in view of (3.12)), the top vertical arrows are theKummer isomorphisms, the middle vertical isomorphisms come from §1.4 and (1.4),and the bottom vertical isomorphisms are induced by (3.12).Since the top horizontal arrow of this diagram is an isomorphism, all of the mapsappearing in (3.23) have to be isomorphisms.We note that as a consequence of the projection formula [Lau76, Proposition 4.2]and of the equality of cycles [( Z j ) k ] = m j ν j ∗ [ Z ′ j ], the top horizontal isomorphismof (3.23) transports the cup product pairing on H (( Z j ) k , Q ℓ (1)) to the cup productpairing on H ( Z ′ j , Q ℓ (1)) multiplied by m j .As on the other hand (3.23) transports the Weil pairing on V ℓ ( A (( Pic Z j /k ) k )) = V ℓ ( Pic Z ′ j /k ) (see (1.4)) to the cup product pairing on H ( Z ′ j , Q ℓ (1)), we see thatLemma 3.7 (i) amounts to the assertion that (3.21) transports the orthogonal sumof the cup product pairings on H (( Z j ) k , Q ℓ (1)) to the opposite of the cup productpairing on H ( X ′ k , Q ℓ (2)). When the Z j are smooth, this is shown in [BW19, (2.7)];the same argument applies in our setting.Thus, it only remains to prove Lemma 3.7 (ii) and (iii). For this, it suffices tocheck the commutativity of the squares V ℓ ( A (( CH X ′ /k ) k )) ∼ γ ′ / / H ( X ′ k , Q ℓ (2)) V ℓ ( A (( CH X/k ) k )) h ∗ O O ∼ γ / / H ( X k , Q ℓ (2)) h ∗ O O (3.24)and V ℓ ( A (( CH X ′ /k ) k )) h ∗ (cid:15) (cid:15) ∼ γ ′ / / H ( X ′ k , Q ℓ (2)) h ∗ (cid:15) (cid:15) V ℓ ( A (( CH X/k ) k )) ∼ γ / / H ( X k , Q ℓ (2)),(3.25)where γ is the isomorphism constructed from Bloch’s ℓ -adic Abel–Jacobi map for thesmooth variety X (see the statement of Lemma 3.7 (iii)) and γ ′ is the isomorphismextracted from (3.21) and (3.23), and where the vertical arrows are those appearingin the upper and lower rows of (3.22). The square (3.25) fits into a larger diagram M c j =2 V ℓ ( K (( Z j ) k )) α (cid:15) (cid:15) det ∼ / / M c j =2 V ℓ (Pic(( Z j ) k )) M c j =2 H (( Z j ) k , Q ℓ (1)) V ℓ ( SK ( X ′ k )) h ∗ (cid:15) (cid:15) / / V ℓ ( CH X ′ /k ) h ∗ (cid:15) (cid:15) β ′ / / V ℓ ( A (( CH X ′ /k ) k )) h ∗ (cid:15) (cid:15) ∼ γ ′ / / H ( X ′ k , Q ℓ (2)) h ∗ (cid:15) (cid:15) V ℓ ( SK ( X k )) / / V ℓ ( CH X/k ) β / / V ℓ ( A (( CH X/k ) k )) ∼ γ / / H ( X k , Q ℓ (2)),(3.26)in which the map α is induced by (3.6) (see also (3.7)), the map β ′ comes from thebottom row of (3.23), the map β is constructed in the same way as β ’ (legitimatethanks to Theorem 3.1 (v)) and the isomorphisms of the square of the top rightcorner all come from (3.21) and (3.23).In order for the square in the bottom right corner to commute, it suffices thatthe outer square of the diagram commute, since the other inner squares clearlycommute. That is, fixing j such that c j = 2 and letting α j and α ´et j respectivelydenote the j -th component of α and of (3.21), we need only prove that the square V ℓ ( K (( Z j ) k )) h ∗ ◦ α j (cid:15) (cid:15) ∼ / / H (( Z j ) k , Q ℓ (1)) h ∗ ◦ α ´et j (cid:15) (cid:15) V ℓ ( SK ( X k )) / / H ( X k , Q ℓ (2)),(3.27)whose horizontal arrows are extracted from (3.26), commutes.Set D j = Z j × Y j X ′ and E j = Z j × Y j Y j +1 . Let q j : D j → Z j and p ′ j : E j → Z j denote the projections. Let ι j : Z j ֒ → Y j , ι ′ j : E j ֒ → Y j +1 and δ j : D j ֒ → X ′ bethe inclusions, so that ι j , ι ′ j , δ j are regular closed immersions of codimensions c j ,1, 1, respectively. Recall that α ´et j = ( p j +1 ◦ · · · ◦ p N ) ∗ ◦ ι ′ j ∗ ◦ p ′∗ j , where ι ′ j ∗ denotes the map given by cup product with the class of the Cartier divisor ( E j ) k in H E j ) k (( Y j +1 ) k , Q ℓ (1)) (see [Rio14, §2.1]) composed with the forgetful map H E j ) k (( Y j +1 ) k , Q ℓ (2)) → H (( Y j +1 ) k , Q ℓ (2)). As E j pulls back, as a Cartierdivisor, to D j , we deduce that α ´et j = δ j ∗ ◦ q ∗ j ,(3.28)where δ j ∗ is again defined as in loc. cit. , §2.1. Similarly, recall that α j is given by x ( p j +1 ◦· · ·◦ p N ) ∗ ( ι ′ j ∗ p ′∗ j x ⊗ [ O Y j +1 ( E j )]) = (( p j +1 ◦· · ·◦ p N ) ∗ ι ′ j ∗ p ′∗ j x ) ⊗ [ O X ′ ( D j )])(see (2.3)). Noting that the morphisms p j +1 ◦ · · · ◦ p N and ι ′ j are Tor-independent(indeed one has Tor Ai ( A/f A, B ) = 0 for any i >
0, any commutative ring A , any A -algebra B and any f ∈ A such that neither f nor its image in B is a zero divisor),the base change theorem [Lip09, Theorem 3.10.3] allows us to rewrite this as α j ( x ) = ( δ j ∗ ◦ q ∗ j )( x ) ⊗ [ O X ′ ( D j )](3.29)for any x ∈ V ℓ ( K (( Z j ) k )).Let α ′ j : V ℓ ( K (( Z j ) k )) → V ℓ ( SK ( X ′ k )) be given by α ′ j ( x ) = ( δ j ∗ ◦ q ∗ j )( x ). In viewof (3.28) and of the contravariant functoriality of the first row of (3.26), we deducefrom Lemma 3.6 applied to h ◦ δ j : D j → X that the square obtained by replacing,in (3.27), the left-hand side vertical arrow h ∗ ◦ α j with h ∗ ◦ α ′ j commutes. On theother hand, it follows from (3.29) that the map h ∗ ◦ α j − h ∗ ◦ α ′ j takes its values in NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 31 V ℓ ( F K ( X k )), since [ O X ′ ( D j )] − [ O X ′ ] ∈ f K ( X ′ ) (see [SGA6, X, Corollaire 1.3.3]).Now the lower horizontal map of (3.27) vanishes on V ℓ ( F K ( X k )) since it factorsthrough c ; we conclude that the square (3.27) itself commutes, and therefore sodoes (3.25).Let us turn to (3.24). We introduce a desingularisation π : X ′′ → X ′ k of X ′ k (which exists by Cossart and Piltant [CP09]) and consider the square V ℓ ( A (( CH X ′ /k ) k )) π ∗ (cid:15) (cid:15) ∼ γ ′ / / H ( X ′ k , Q ℓ (2)) π ∗ (cid:15) (cid:15) V ℓ ( A (( CH X ′′ /k ) )) ∼ γ ′′ / / H ( X ′′ , Q ℓ (2)),(3.30)where γ ′′ is constructed from Bloch’s ℓ -adic Abel–Jacobi map for the smoothvariety X ′′ in the same way as γ for X . One verifies the commutativity of thesquare (3.30) by proceeding exactly as we did with (3.25), that is, by reducingto Lemma 3.6 using the diagram obtained by replacing, in (3.26), all occurrencesof X with X ′′ and all occurrences of h ∗ with π ∗ , and using the equalities obtainedby replacing, in (3.28) and (3.29) and in their proofs, α j and α ´et j with π ∗ ◦ α j and π ∗ ◦ α ´et j , and D j , q j , δ j with D ′′ j , q ′′ j , δ ′′ j , where D ′′ j = Z j × Y j X ′′ and where q ′′ j : D ′′ j → Z j and δ ′′ j : D ′′ j ֒ → X ′′ denote the projections.As the Z ′ j are smooth and projective, the groups H ( Z ′ j , Q ℓ (1)) are pure ofweight − H ( X ′ k , Q ℓ (2)) is pure of weight − < −
1, asfollows from cohomological descent and Deligne’s theorem on the Weil conjectures( op. cit. , §9; proper smooth hypercoverings of X ′ k that start with π exist by [dJ96,Theorem 4.1]). Hence the right-hand side vertical map of (3.30) is injective.This injectivity, the commutativity of (3.30) and the commutativity of the square V ℓ ( A (( CH X ′′ /k ) )) ∼ γ ′′ / / H ( X ′′ , Q ℓ (2)) V ℓ ( A (( CH X/k ) k )) π ∗ ◦ h ∗ O O ∼ γ / / H ( X k , Q ℓ (2)) π ∗ ◦ h ∗ O O (3.31)together imply that (3.24) commutes. This concludes the proof of Lemma 3.7. (cid:3) We resume the proof of Proposition 3.5. Consider the diagram(3.32) ( CH X/k ) k × A ( G k ) ∼ −→ A (( CH X ′ /k ) k ) ∼ ←− Y c j =2 A (( Pic Z j /k ) k )of isomorphisms of abelian varieties stemming from (3.12) and whose ℓ -adic Tatemodules appear on the bottom line of (3.22). The product of the polarizations m j θ j on the right-hand side of (3.32) induces a polarization λ on the left-hand side( CH X/k ) k × A ( G k ) of (3.32). Let us view the Weil pairing of λ as a Q ℓ (1)-valuedpairing on H ( X ′ k , Q ℓ (2)) thanks to (3.22). By Lemma 3.7 (i), it is equal to theopposite of the cup product pairing on H ( X ′ k , Q ℓ (2)). Since, by the projectionformula [Lau76, Proposition 4.2], the decomposition (3.20) is orthogonal withrespect to the cup product, it follows from Lemma 3.7 (ii) that λ is a productpolarization on ( CH X/k ) k × A ( G k ). Since the restriction of the cup product pairing on H ( X ′ k , Q ℓ (2)) to H ( X k , Q ℓ (2)) coincides with the cup product pairing on H ( X k , Q ℓ (2)), it follows from Lemma 3.7 (iii) that the restriction of λ to ( CH X/k ) k is the canonical principal polarization defined in Theorem 3.1 (vi).By a theorem of Debarre [Deb96, Corollary 2], polarized abelian varieties canbe written in a unique way as a product of indecomposable polarized abelianvarieties. As the ( A (( Pic Z j /k ) k ) , m j θ j ) are indecomposable or trivial (since so arethe ( A (( Pic Z j /k ) k ) , θ j )), we deduce the existence of a partition { j | c j = 2 } = J ⊔ J ′ such that (3.32) induces isomorphisms Q j ∈ J A (( Pic Z j /k ) k ) ∼ −→ ( CH X/k ) k and Q j ∈ J ′ A (( Pic Z j /k ) k ) ∼ −→ A ( G k ). Since λ restricts to a principal polarization on( CH X/k ) k , we see that m j = 1 for all the j ∈ J such that A (( Pic Z j /k ) k ) is non-zero.Thus, the product of the polarizations θ j on the right-hand side of (3.32)induces on ( CH X/k ) k × A ( G k ) a polarization which is at the same time a principalpolarization and the product of two polarizations, and which is therefore theproduct of two principal polarizations; moreover, the first of these coincides with thecanonical principal polarization of Theorem 3.1 (vi). Proposition 3.5 is proved. (cid:3) Now that Proposition 3.5 is proved, we let J (resp. J , resp. J ) be the set ofindices j such that c j = 2 and the curve Z j is smooth over k (resp. such that c j = 2and Z j is not smooth over k , resp. such that c j = 3), and we proceed to show thatthe ( Z j ) j ∈ J do not contribute to ( CH X/k ) . Lemma 3.8.
The map Y j ∈ J Pic Z j /k → ( CH X/k ) induced by (3.12) vanishes.Proof. We fix j ∈ J . Let a : Pic Z j /k → ( CH X/k ) and b : Pic Z j /k → G denotethe maps induced by (3.12). Let us assume that a = 0 and derive a contradiction.When a = 0, we claim that Pic Z j /k = Ker( a ) × Ker( b ), that Ker( a ) is affine andthat Ker( b ) is a non-trivial abelian variety; Corollary 1.8 then provides the desiredcontradiction. It thus suffices to prove the claim. To this end, it is enough to checkthat ( Pic Z j /k ) k p = Ker( a k p ) × Ker( b k p ), that Ker( a k p ) is affine and that Ker( b k p )is a non-trivial abelian variety, as these three properties descend to k .By functoriality, the map b induces maps A ( b k p ) : A (( Pic Z j /k ) k p ) → A ( G k p )and L ( b k p ) : L (( Pic Z j /k ) k p ) → L ( G k p ). As ( CH X/k ) k p is an abelian variety (seeTheorem 3.1 (ii)), we have L ( a k p ) = 0 and the map A ( a k p ) can be viewed as amap A ( a k p ) : A (( Pic Z j /k ) k p ) → ( CH X/k ) k p through which a k p factors, so that ourassumption that a k p = 0 implies that A ( a k p ) = 0. On the other hand, the map L ( b k p ) is a closed immersion since a × b is one and L ( a k p ) = 0.Proposition 3.5 allows us to view A (( Pic Z j /k ) k p ) as a principally polarized directfactor of the product of principally polarized abelian varieties ( CH X/k ) k p × A ( G k p )over k p , through A ( a k p ) × A ( b k p ). As the decomposition of a principally polarizedabelian variety into its indecomposable factors is unique, as A (( Pic Z j /k ) k p ) is itselfindecomposable (see §1.4.2), and as A ( a k p ) = 0, necessarily A ( b k p ) = 0 and A ( a k p )is a closed immersion, so that Ker( a k p ) = L (( Pic Z j /k ) k p ) (see (1.1)). NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 33
All in all, the exact sequences (1.1) fit into a commutative diagram(3.33) 0 / / Ker( a k p ) / / (cid:127) _ (cid:15) (cid:15) ( Pic Z j /k ) k p / / b k p (cid:15) (cid:15) A (( Pic Z j /k ) k p ) (cid:15) (cid:15) / / / / L ( G k p ) / / G k p / / A ( G k p ) / / b k p ) ∼ −→ A (( Pic Z j /k ) k p ) and then all of the desired statementsnow result from this diagram, in view of the remark that the snake homomorphismis trivial since it goes from an abelian variety to an affine group. (cid:3) Proof of Theorem 3.1 (vii).
In view of Lemma 3.8, we may consider the quotient H of G by its subgroup scheme Q j ∈ J Pic Z j /k . Thanks to the exact sequences0 → Pic Z j /k → Pic Z j /k → Z Z j /k → j ∈ J , wededuce from (3.12) an isomorphism(3.34) CH X/k × H ∼ ←− Z × Y j ∈ J Pic Z j /k × Y j ∈ J ∪ J Z Z j /k .Let B be the disjoint union of P k , of the curves P π ( Z j /k ) for all j ∈ J ∪ J ,and of the curves Z j for all j ∈ J . It is a smooth projective curve over k ,and it has the property that Pic
B/k ≃ Z × Q j ∈ J Pic Z j /k × Q j ∈ J ∪ J Z Z j /k since Pic P π Zj/k ) /k ≃ Res π ( Z j /k ) /k ( Z ) ≃ Z Z j /k for j ∈ J ∪ J by Proposition 1.1. Theisomorphism Pic
B/k ∼ −→ CH X/k × H deduced from (3.34) realizes CH X/k as aprincipally polarized direct factor of
Pic
B/k by Proposition 3.5, as desired. (cid:3)
The proof of Theorem 3.1 is now complete.3.3.
Blow-ups.
The next proposition, which relies on arguments already used inthe proof of Theorem 3.1, allows one to compute CH X/k in concrete situations.
Proposition 3.9.
Let X be a smooth projective k -rational threefold over k , let i : Y → X be the inclusion of a smooth closed subvariety of pure codimension c ,and let p : X ′ → X be its blow-up. Then the formula (2.8) induces an isomorphism (3.35) CH X/k × Pic
Y/k ∼ −→ CH X ′ /k if c = 2 (resp. CH X/k × Z Y/k ∼ −→ CH X ′ /k if c = 3 ),respecting the principal polarizations furnished by Theorem 3.1 (applied to X andto X ′ ) and by §1.4.2 (applied to Y ).Proof. The isomorphism (2.8), Corollary 1.5 and Remark 1.2 (ii) yield canonicalisomorphisms K ,X/k × K c − ,Y/k ∼ −→ K ,X ′ /k , Pic
X/k × Z Y/k ∼ −→ Pic X ′ /k and Z X/k ∼ −→ Z X ′ /k . Identifying (rk , det) : K ,X ′ /k → Z X ′ /k × Pic X ′ /k in terms ofthese decompositions and using Proposition 2.4, we obtain an isomorphism(3.36) SK ,X/k, fppf × Pic
Y/k, fppf ∼ −→ SK ,X ′ /k, fppf if c = 2(resp. SK ,X/k, fppf × Z Y/k ∼ −→ SK ,X ′ /k, fppf if c = 3).If l/k is a finite extension and x ∈ ( X \ Y )( l ), one has p ∗ [ O x ] = [ O p − ( x ) ] ∈ K ( X ′ l ).It follows that p ∗ ν X (1) = ν X ′ (1) ∈ SK ,X ′ /k, fppf ( k ) ⊂ SK ,X ′ /k, fppf ( l ). We thusdeduce from (3.36) the required isomorphism (3.35) of k -group schemes. If c = 2, considering the commutative diagram(3.37) Ab ( X k p ) × Pic Y k p /k p ∼ / / ≀ (cid:15) (cid:15) Ab ( X ′ k p ) ≀ (cid:15) (cid:15) ( CH X k p /k p ) × Pic Y k p /k p ∼ / / ( CH X ′ k p /k p ) whose vertical arrows stem from Theorem 3.1 (vi), whose lower horizontal arrowis the above constructed isomorphism and whose upper horizontal arrow is that of[BW19, Lemma 2.10] concludes the proof, as the latter arrow respects the principalpolarizations by [BW19, Lemma 2.10]. If c = 3, one can argue in the same way,using a diagram similar to (3.37) in which Pic Y k p /k p does not appear. (cid:3) Obstructions to k -rationality. The most general obstruction to the k -ratio-nality of a smooth projective k -rational threefold obtained in this article is Theo-rem 3.1 (vii). In §3.4, we spell out concrete consequences of this theorem.We recall that a Γ k -module M is a permutation Γ k -module if it is free of finiterank as a Z -module and admits a Z -basis that is permuted by the action of Γ k ,and that it is stably of permutation if there exists a Γ k -equivariant isomorphism M ⊕ N ≃ N for some permutation Γ k -modules N and N .If X is a smooth projective k -rational threefold, we associate with any class α ∈ NS ( X k ) Γ k = ( CH X/k / ( CH X/k ) )( k ) (see Theorem 3.1 (v)) its inverseimage ( CH X/k ) α in CH X/k . It is an fppf torsor under ( CH X/k ) , hence an étaletorsor under ( CH X/k ) by [Mil80, III, Corollary 4.7, Remark 4.8 (a)]. We let[( CH X/k ) α ] ∈ H ( k, ( CH X/k ) ) be its Galois cohomology class. Theorem 3.10.
Let X be a smooth projective k -rational threefold over k . Then:(i) The Γ k -module NS ( X k ) is a direct factor of a permutation Γ k -module.(ii) There exists an isomorphism ( CH X/k ) ≃ Pic C/k of principally polarizedabelian varieties over k for some smooth projective curve C over k .(iii) For all smooth projective geometrically connected curves D of genus ≥ over k , all morphisms ψ : ( CH X/k ) → Pic D/k identifying
Pic D/k with aprincipally polarized direct factor of ( CH X/k ) and all α ∈ NS ( X k ) Γ k , thereexists d ∈ Z such that ψ ∗ [( CH X/k ) α ] = [ Pic dD/k ] ∈ H ( k, Pic D/k ) .(iv) For all elliptic curves E over k and all morphisms ψ : ( CH X/k ) → E identifying E with a principally polarized direct factor of ( CH X/k ) , thereexists a class η ∈ H ( k, E ) such that for all α ∈ NS ( X k ) Γ k , there exists d ∈ Z with ψ ∗ [( CH X/k ) α ] = dη ∈ H ( k, E ) .Proof. Theorem 3.1 (vii) shows the existence of a smooth projective curve B over k such that CH X/k is a principally polarized direct factor of
Pic
B/k . Wedenote by q : Pic
B/k → CH X/k the projection onto this direct factor and by r : CH X/k → Pic
B/k the inclusion of this direct factor.Passing to the groups of connected components shows that NS ( X k ) is a directfactor of Z B/k ( k ), which is a permutation Γ k -module, thus proving (i).Passing to the identity components shows that ( CH X/k ) is a principallypolarized direct factor of Pic B/k . In view of the uniqueness of the decomposition ofa principally polarized abelian variety as a product of indecomposable ones, and in
NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 35 view of the description of the indecomposable factors of
Pic B/k (see [BW19, §2.1]),there exists a union C of connected components of B such that ( CH X/k ) ≃ Pic C/k as principally polarized abelian varieties over k , thus proving (ii).Let us now fix D , ψ and α as in (iii). The composition p := ψ ◦ q : Pic B/k → Pic D/k realizes
Pic D/k as a principally polarized direct factor of
Pic B/k . Allindecomposable principally polarized direct factors of
Pic B/k are of the form
Pic B ′ /k for some connected component B ′ of B (see [BW19, §2.1]). Let B ′ bethe connected component of B corresponding to Pic D/k . Since D is geometricallyconnected of genus ≥
2, we see that
Pic D/k , hence also
Pic B ′ /k , is geometricallyindecomposable of dimension ≥
2, and it follows that B ′ is geometrically connectedof genus ≥
2. By the precise form of the Torelli theorem [Ser01, Théorèmes 1 et 2],after possibly replacing q with − q (and p with − p ) if D is not hyperelliptic, wecan identify D and B ′ in such a way that p is the pull-back by the inclusion i : D ≃ B ′ ֒ → B . Since q : Pic
B/k → CH X/k realizes CH X/k as a direct factorof
Pic
B/k , one can find β ∈ NS ( B k ) Γ k with q ( β ) = α ∈ NS ( X k ) Γ k . Letting d := i ∗ β ∈ NS ( D k ) ≃ Z , we obtain ψ ∗ [( CH X/k ) α ] = p ∗ [ Pic βB/k ] = [
Pic dD/k ] ∈ H ( k, Pic D/k ), which proves (iii).Fix E and ψ as in (iv). Arguing as above shows that p := ψ ◦ q : Pic B/k → E identifies E with Pic B ′ /k for some connected component i : B ′ ֒ → B of B which is geometrically connected of genus 1. The genus 1 curve B ′ has a naturalstructure of Pic B ′ /k -torsor, and we set η := [ B ′ ] ∈ H ( k, Pic B ′ /k ) = H ( k, E )be its class. Let α ∈ NS ( X k ) Γ k . Setting s := i ∗ ◦ r : CH X/k → Pic B ′ /k and d := s ∗ α ∈ NS ( B ′ k ) ≃ Z , we get ψ ∗ [( CH X/k ) α ] = s ∗ [( CH X/k ) α ] = [ Pic dB ′ /k ] = dη ∈ H ( k, Pic B ′ /k ) = H ( k, E ), which completes the proof of (iv). (cid:3) Remarks . (i) If X is a smooth projective k -rational variety and if k hascharacteristic 0, one can apply the weak factorization theorem [AKMW02, The-orem 0.3.1] to show that the Γ k -module NS ( X k ) is stably of permutation. Thisstatement is stronger than Theorem 3.10 (i). We do not know if it holds if k hascharacteristic p > X has dimension ≥
3. We do not know either whetherTheorem 3.10 (i) holds for smooth projective k -rational varieties of dimension ≥ ( X k ) of a smooth projective k -rationalvariety X is stably of permutation (see [Man66, Theorem 2.2] if X is a surface and[CTS87, Proposition 2.A.1] in general). Theorem 3.10 (i) and Remark 3.11 (i) maybe viewed as analogues of this classical statement for codimension 2 cycles.(iii) To obtain a variant of Theorem 3.10 (iii) in the case where D is connectedbut not geometrically connected, one can apply Theorem 3.10 (iii) to the l -rationalvariety X l over l , where l is the algebraic extension l := H ( D, O D ) of k . The sameremark applies to Theorem 3.10 (iv).4. Smooth complete intersections of two quadrics
In this last section, we apply the above results to k -varieties X that are three-dimensional smooth complete intersections of two quadrics, computing the variety CH X/k (in Theorem 4.5) and providing a necessary and sufficient criterion fortheir k -rationality (in Theorem 4.7). A more classical necessary and sufficient criterion for their k -unirationality (Theorem 4.8) allows us to give examples, for anyalgebraically closed field κ , of such varieties over κ (( t )) that are κ (( t ))-unirationalbut not κ (( t ))-rational (Theorem 4.11).Many of the geometric results that we need along the way are available in theliterature only in characteristic 0 or in characteristic = 2 [Rei72, Don80, Wan18],and we extend them to arbitrary characteristic.4.1. Lines in a complete intersection of two quadrics.
Let X ⊂ P k be athree-dimensional smooth complete intersection of two quadrics over k . We let F be the Hilbert scheme of lines in X (also called the Fano variety of lines of X ). Lemma 4.1.
The following assertions hold:(i) The variety X does not contain any plane.(ii) The normal bundle N L/X of a line L ⊂ X is isomorphic either to O ⊕ L or to O L (1) ⊕ O L ( − .(iii) The variety F is a non-empty geometrically connected smooth projective sur-face with trivial canonical bundle. Its tangent space at a k -point correspondingto a line L ⊂ X is naturally isomorphic to H ( X, N
L/X ) .Proof of (i). By the Lefschetz hyperplane theorem [SGA2, XII, Corollary 3.7],Pic( X ) is generated by O X (1). If X contained a plane P , we would have O X ( P ) ≃ O X ( l ) for some l ∈ Z , hence an equality of intersection numbers1 = O P (1) · O P (1) = O X ( P ) · O X (1) · O X (1) = 4 l ,which is a contradiction. Proof of (ii).
Since L and X are complete intersections in P k , the normal exactsequence 0 → N L/X → N L/ P k → N X/ P k | L → L ⊂ X ⊂ P k reads:(4.1) 0 → N L/X → O L (1) ⊕ → O L (2) ⊕ → . It follows that N L/X is a rank 2 vector bundle of degree 0 on L , hence is of theform O L ( l ) ⊕ O L ( − l ) for some l ≥ O L (1) ⊕ , one has l ∈ { , } . Proof of (iii).
The computation of the tangent space is [Kol96, Chapter I, Theo-rem 2.8.1]. To prove that F is smooth, geometrically connected and non-empty, wemay work over k . For all L ∈ F ( k ), one has h ( L, N
L/X k ) = 0 and h ( L, N
L/X k ) = 2by (ii). The variety F k is thus smooth of dimension 2 at L by [Kol96, Chap-ter I, Theorem 2.8.3]. That F k is non-empty and connected follows from [DM98,Théorème 2.1 b) c)]. To compute the canonical bundle of F , we let G be theGrassmannian of lines in P k and 0 → S → O ⊕ G → Q → G , where S and Q respectively have rank 4and rank 2. The variety of lines F is defined in G by the vanishing of a section of(Sym Q ) ⊕ . Since F is smooth of the expected dimension, the normal short exactsequence reads 0 → T F → T G | F → N F/G →
0, where N F/G ≃ (Sym Q ) ⊕ | F and T G | F ≃ ( S ∨ ⊗ Q ) | F . We deduce that K F ≃ (det((Sym Q ) ⊕ ) ⊗ det( S ⊗ Q ∨ )) | F ≃ det( Q ) | ⊗ F ⊗ det( Q ) | ⊗− F ≃ O F . (cid:3) NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 37
We will later show that F k is actually an abelian surface (see Theorem 4.5).Let Z ⊂ X × F be the universal line in X . If Λ ⊂ X is a line, the second projection Z ∩ (Λ × ( F \{ Λ } )) → F \{ Λ } is a closed immersion by [EGA43, Proposition 8.11.5].Its image is a subscheme W (Λ) ⊂ F \ { Λ } parametrizing the lines of X that aredistinct from Λ and that intersect Λ. If L ∈ W (Λ)( k ) and x = L ∩ Λ, then T L W (Λ) ⊂ T L F = H ( X, N
L/X ) is the subset of those σ ∈ H ( X, N
L/X ) such that σ x ∈ h T x L, T x Λ i /T x L (as can be seen using [Ser06, Remark 4.5.4 (ii)]). Lemma 4.2. If L and Λ are two distinct lines in X that intersect, the inclusionof tangent spaces T L W (Λ) ⊂ T L F is strict.Proof. In view of Lemma 4.1 (ii), we may distinguish two cases according to theisomorphism class of the normal bundle N L/X . Suppose first that N L/X ≃ O ⊕ L .Consider the point x = L ∩ Λ. Since N L/X is globally generated, we can choosea section σ ∈ H ( L, N
L/X ) such that σ x / ∈ h T x L, T x Λ i /T x L . Then, the tangentvector of F at L associated to σ by Lemma 4.1 (iii) is not tangent to W (Λ).Assume now that N L/X ≃ O L (1) ⊕ O L ( − X , . . . , X on P k such that L = { X = · · · = X = 0 } and use X , . . . , X toidentify N L/ P k with O L (1) ⊕ . After a coordinate change, we may assume that thecomposition O L (1) → O L (1) ⊕O L ( − → O L (1) ⊕ of the inclusion of the first factorand of the first arrow of (4.1) is the inclusion of the first factor. In such coordinates, { X + εX = X = X = X = 0 } and { X + εX = X = X = X = 0 } are two k [ ε ] / ( ε )-points of F . Assuming for contradiction that T L W (Λ) = T L F and using the characterization recalled above of T L W (Λ) viewed as a subspace of H ( X, N
L/X ), we see that Λ ⊂ P := { X = X = X = 0 } . Since X contains L ,the monomials X , X X and X do not appear in the equations of X . Since X alsocontains the above two infinitesimal deformations of L , neither do the monomials X X and X X . It follows that the intersection of X with the plane P is equaleither to P or to the double line { X = 0 } ⊂ P . Since Λ ⊂ X ∩ P but Λ = L , wededuce that P ⊂ X , which contradicts Lemma 4.1 (i). (cid:3) Projecting from a line.
We keep the notation of §4.1.Assume that X contains a line Λ ⊂ X , which we fix. We denote by µ : X ′ → X and e µ : ( P k ) ′ → P k the blow-ups of Λ in X and in P k , and by ν : X ′ → P k and e ν : ( P k ) ′ → P k the morphisms obtained by projecting away from Λ. Proposition 4.3.
There exists a smooth projective geometrically connected curve ∆ ⊂ P k of genus such that ν can be identified with the blow-up of ∆ in P k .Proof. Write Λ = { X = · · · = X = 0 } in appropriate homogeneous coordinates X , . . . , X of P k . The morphism e ν : ( P k ) ′ → P k realizes ( P k ) ′ as the projectiviza-tion (in Grothendieck’s sense) of the vector bundle E = O P k ⊕O P k ⊕O P k (1) on P k .In a natural way, we use homogeneous coordinates X , . . . , X on P k , and we let X , X (resp. X ) denote the global sections of E (resp. of E ( − P k correspond-ing to the direct sum decomposition of E . If X has equations { L X + L X + Q = 0 } and { L ′ X + L ′ X + Q ′ = 0 } in P k , with L , L , L ′ , L ′ (resp. Q, Q ′ ) linear (resp.quadratic) in X , . . . , X , then X ′ has equations { L X + L X + QX = 0 } and { L ′ X + L ′ X + Q ′ X = 0 } in ( P k ) ′ .The fibers of ν : X ′ → P k are defined by two linear equations in a 2-dimensionalprojective space, hence are isomorphic to P , to P or to P . The determinantal subscheme ∆ ⊂ P k defined by the vanishing of the maximal minors of the matrix(4.2) (cid:18) L L QL ′ L ′ Q ′ (cid:19) endows the subset of P k over which the fibers of ν are positive-dimensional witha schematic structure. We claim that ν | ν − (∆) : ν − (∆) → ∆ is a flat family oflines in the projective bundle e ν | e ν − (∆) : e ν − (∆) → ∆. To see it, we work on anaffine open subset of ∆ with coordinate ring R . One has to show that the cokernel M of the linear map R → R given by the transpose of the matrix (4.2) is freeof rank 2. This follows from [Eis95, Proposition 20.8] since Fitt ( M ) = 0 by thedefinition of ∆ and Fitt ( M ) = R by Lemma 4.1 (i).Let us show that ∆ is smooth of the expected dimension (equal to 1). To thisend, we fix x ∈ ∆( k ) such that T x ∆ k has dimension ≥ ν − ( x ) is the line { X = 0 } ⊂ e ν − ( x ). Then the linearforms L , L , L ′ and L ′ vanish at x . As T x ∆ k has dimension ≥
2, the differentialsat x of the cubic forms L Q ′ − L ′ Q and L Q ′ − L ′ Q are linearly dependent. Afterreplacing ( L , L ′ ) and ( L , L ′ ) with suitable k -linear combinations of ( L , L ′ ) and( L , L ′ ) (which is possible by a change of coordinates), we may therefore assumethat the cubic { L Q ′ − L ′ Q = 0 } is singular at x . Since Q and Q ′ do not both vanishat x (otherwise X k would contain the plane e µ ( e ν − ( x )), contradicting Lemma 4.1 (i)over k ), and since L and L ′ vanish at x , we deduce that L and L ′ are linearlydependent. Consequently, an appropriate k -linear combination of the degree 2equations defining X k is of the form L ′′ X + Q ′′ , where L ′′ and Q ′′ are respectivelylinear and quadratic in X , . . . , X . Since V := { L ′′ X + Q ′′ = 0 } is singular at p := [0 : 0 : 0 : 0 : 1 : 0] and X k ⊂ V is a Cartier divisor containing p , we see that X k is also singular at p , which is absurd.Thus the line ν − ( x ) is not equal to { X = 0 } ⊂ e ν − ( x ). In other words, itsimage by µ is a line L ⊂ X distinct from Λ. We note that the open subset of therelative Hilbert scheme of lines of e ν : ( P k ) ′ → P k consisting of those lines that arenot defined by the equation { X = 0 } in a fiber of e ν is naturally isomorphic tothe scheme parametrizing the lines in P k that are distinct from Λ but intersect Λ.Since moreover ν | ν − (∆) : ν − (∆) → ∆ is a family of lines over ∆, two independenttangent vectors of ∆ at x give rise to two independent tangent vectors of W (Λ)at L . As F is smooth of dimension 2 at L by Lemma 4.1 (iii), this contradictsLemma 4.2 and finishes the proof that ∆ is a smooth curve.It then follows from [Eis05, Theorem A2.60, Example A2.67] that O ∆ is resolvedby the Eagon–Northcott complex0 → O P k ( − ⊕ → O P k ( − ⊕ O P k ( − ⊕ → O P k → O ∆ → h (∆ , O ∆ ) = 1 and h (∆ , O ∆ ) = 2,hence that the smooth projective curve ∆ is geometrically connected of genus 2.To conclude, we denote by λ : Y → P k the blow-up of ∆ in P k . Since ν | ν − (∆) : ν − (∆) → ∆ is a flat family of lines, hence a smooth morphism, thesubscheme ν − (∆) ⊂ X ′ is a smooth divisor. Since X ′ is smooth, ν − (∆) is aCartier divisor in X ′ , and the universal property of a blow-up yields a morphism φ : X ′ → Y such that λ ◦ φ = ν . Both λ and ν are birational (the latter becausethe generic fiber of ν is a 0-dimensional projective space), hence so is φ . We deduce NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 39 that φ is an isomorphism, as is any birational morphism between smooth projectivevarieties with the same Picard number. (cid:3) Remark . By Lemma 4.1 (iii), a smooth complete intersection of two quadricsin P k contains a line over k , hence is k -rational by Proposition 4.3.4.3. The intermediate Jacobian.
In Theorem 4.5, we compute CH X/k forthreefolds X that are smooth complete intersections of two quadrics. Note thatsuch varieties are k -rational by Remark 4.4, so that Theorem 3.1 applies to them.Theorem 4.5 (iv) will be used in a crucial manner in the proof of Theorem 4.7.In characteristic = 2, it goes back to the work of Wang [Wan18].In the statement of Theorem 4.5 (ii), we denote by Alb V/k (resp. Alb V/k )the Albanese variety (resp. torsor) of a smooth proper geometrically connectedvariety V over k . This is the abelian variety over k (resp. torsor under Alb V/k )which underlies the solution of the universal problem of morphisms from V totorsors under abelian varieties over k . We recall that Alb V/k is canonically dualto the abelian variety (
Pic V/k ) red and that the formation of Alb V/k is compatiblewith arbitrary extensions of scalars (see [FGA, Exp. 236, Théorème 3.3 (iii)], inwhich the geometric fibers of X → S should be assumed to be connected). We alsorecall that Alb V/k = Pic V/k and Alb V/k = Pic V/k if in addition V is a curve.By a conic on X we mean a 1-dimensional closed subscheme of X which, whenviewed as a subscheme of P k , is the intersection of a quadric and a plane. Theorem 4.5.
Let X ⊂ P k be a smooth complete intersection of two quadrics,let F be its variety of lines, and let Z ⊂ X × F be the universal line. Then:(i) The degree map deg : CH ( X k ) → Z induces, via (3.1), a short exact sequence → ( CH X/k ) → CH X/k δ −→ Z → of k -group schemes.(ii) The class [ O Z ] ∈ K ( X F ) induces isomorphisms F ∼ −→ ( CH X/k ) := δ − (1) and Alb F/k ∼ −→ ( CH X/k ) .(iii) There exists a unique reduced closed subscheme D ⊂ ( CH X/k ) := δ − (2) such that D ( k ) coincides, via (3.1), with the subset of CH ( X k ) consistingof those classes represented by a conic on X k . The scheme D is a smoothprojective geometrically connected curve of genus over k .(iv) Via the identifications Pic D/k = Alb D/k and
Pic D/k = Alb D/k , the inclusion D ⊂ ( CH X/k ) induces isomorphisms of principally polarized abelian varieties Pic D/k ∼ −→ ( CH X/k ) and of torsors Pic D/k ∼ −→ ( CH X/k ) .Proof. The sheaf O Z induces a class [ O Z ] ∈ F G ( X F ) = F K ( X F ) = SK ( X F )by §2.1.5 (as F is smooth by Lemma 4.1 (iii)). It therefore induces a morphism F → CH X/k . This morphism factors through ( CH X/k ) because it sends a point x ∈ F ( k ) to the class in CH X/k ( k ) = CH ( X k ) of the line in X k associated with x (as Z is flat over F ), which has degree 1.A morphism a : F → ( CH X/k ) having been constructed, we are now free toextend the scalars from k to any finite Galois extension of k : indeed, the existence,unicity and smoothness of D can be tested over such an extension, and all otherconclusions of the theorem can even be tested over k . As F is smooth, we maytherefore, and will, assume that F ( k ) = ∅ (see [BLR90, 2.2/13]). Let us fix a line Λ ⊂ X defined over k . Proposition 4.3 yields a diagram X µ ←− X ′ ν −→ P k , where µ is the blow-up of Λ and ν is the blow-up of a smoothprojective geometrically connected curve ∆ ⊂ P k of genus 2. Our knowledge ofthe Chow groups of a blow-up [Ful98, Proposition 6.7 (e)] shows the existence ofa Γ k -equivariant short exact sequence 0 → CH ( X k ) alg → CH ( X k ) deg −−→ Z → f : Alb F/k → Alb CH X/k ) /k = ( CH X/k ) denote the morphism betweenAlbanese varieties induced by a : F → ( CH X/k ) . The morphism b : ∆ → F associating with x ∈ ∆ the line µ ( ν − ( x )) induces a morphism g : Pic /k → Alb F/k between Albanese varieties.The composition f ◦ g : Pic /k → ( CH X/k ) is an isomorphism of principallypolarized abelian varieties. Indeed, it coincides at the level of k -points with theprincipally polarized isomorphism Pic /k ∼ −→ ( CH X/k ) obtained by applyingProposition 3.9 to µ and ν , hence is equal to it. It follows that the kernel of g istrivial and that Alb F/k has dimension ≥
2, hence that the first Betti number of F k is ≥
4. Lemma 4.1 (iii) and the classification of surfaces with Kodaira dimension 0(see [BM77, Table p.25 and Theorem 6]) now show that F k is an abelian surface.We deduce that g is an isomorphism as its kernel is trivial and as Pic /k andAlb F/k are both abelian surfaces. It follows that f is an isomorphism.The variety F is isomorphic to an abelian variety since so is F k and since F ( k ) = ∅ . Thus a and f can be identified; hence a is an isomorphism as well,and (ii) is proved.Let c : ∆ → ( CH X/k ) be defined by c ( x ) = a ( b ( x )) + a (Λ), where Λ denotesthe rational point of F corresponding to Λ. Passing to Albanese torsors yieldsa morphism Pic /k → ( CH X/k ) which is an isomorphism since the underlyingmorphism of abelian varieties is the (principally polarized) isomorphism f ◦ g . Itfollows, in particular, that c is a closed immersion.Let us set D = c (∆) and check that D ( k ) = ψ X (Ξ), where ψ X is as in (3.1) andΞ ⊂ CH ( X k ) is the subset appearing in (iii). The theorem will then be proved.We take up the notation e µ , e ν of §4.2 and the notation introduced in the proofof Proposition 4.3. The subvariety e µ − ( X ) ⊂ ( P k ) ′ is given by the system ofequations { ( L X + L X + QX ) X = 0 } and { ( L ′ X + L ′ X + Q ′ X ) X = 0 } . If X , . . . , X are the homogeneous coordinates of x ∈ ∆( k ), this system of equationsdefines a conic in the plane e ν − ( x ) since the matrix (4.2) has rank ≤
1. Thus, forany x ∈ ∆( k ), we have exhibited a (singular) conic on X k whose class in CH ( X k ) isvisibly equal to [ µ ( ν − ( x ))] + [Λ k ], that is, to ( ψ X ) − ( c ( x )). Hence D ( k ) ⊂ ψ X (Ξ).Let us now fix a conic C on X k and prove that ψ X ([ C ]) ∈ D ( k ). Thereexist a (unique) plane P ⊂ P k such that C = X k ∩ P and a (unique) quadric Y ⊂ P k containing X k ∪ P . As X is smooth, the singular locus of Y is disjointfrom X k and has dimension ≤
0. Lemma 4.6 below provides a plane P ′ ⊂ Y containing Λ k such that [ P ] = [ P ′ ] ∈ CH ( Y ). As X k is an effective Cartierdivisor on Y and as X k contains neither P nor P ′ (see Lemma 4.1 (i)), it followsthat [ C ] = [ X k ∩ P ] = [ X k ∩ P ′ ] ∈ CH ( X k ) [Ful98, Proposition 2.6(a)]. Now P ′ = e ν − ( x ) for a unique x ∈ P ( k ); as X k ∩ P ′ is a conic in P ′ , a glance at theequations of e µ − ( X ) ∩ e ν − ( x ) shows that (4.2) has rank ≤
1, hence x ∈ ∆( k ) and[ X k ∩ P ′ ] = [ µ ( ν − ( x ))] + [Λ k ], so that ψ X ([ C ]) = c ( x ) ∈ D ( k ), as desired. (cid:3) NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 41
Lemma 4.6.
Let Y ⊂ P k be a quadric whose singular locus has dimension ≤ .Let P ⊂ Y be a plane. Let Λ ⊂ Y be a line along which Y is smooth. There existsa (unique) plane P ′ ⊂ Y rationally equivalent to P on Y such that Λ ⊂ P ′ .Proof. Let X , . . . , X denote the homogeneous coordinates of P k . After a linearchange of coordinates, we may assume that Λ is the line X = X = X = X = 0and that Y is defined by the equation X X + X X + X X = 0 (if Y is smooth)or by the equation X X + X X + X = 0 (otherwise). Indeed, letting H ⊂ P k be a hyperplane containing Λ and avoiding the singular locus of Y , if Y is singular(so that Y is in this case a cone over the quadric Y ∩ H , which is smooth), andsetting H = P k otherwise, one can use Λ to split off two hyperbolic planes froma quadratic form defining the smooth quadric Y ∩ H [EKM08, Proposition 7.13,Lemma 7.12]; the remaining regular quadratic form of dimension 1 or 2 has thedesired shape since the ground field is algebraically closed.If Y is smooth, the intersection of Y with the linear subspace X = X = 0 isthe union of two planes containing Λ. According to op. cit. , Proposition 68.2, oneof them is rationally equivalent to P on Y .If Y is singular, let P ′ be the plane that contains Λ and the singular point of Y .The smooth quadric Y ∩ H cannot contain a plane ( op. cit. , Lemma 8.10), therefore P ∩ H is a line. The two lines P ∩ H and Λ are rationally equivalent on Y ∩ H ( op.cit. , Proposition 68.2); hence P and P ′ , being the cones over P ∩ H and over Λinside Y , are rationally equivalent on Y [Ful98, Example 2.6.2]. (cid:3) Rationality.
We can now prove a necessary and sufficient criterion for the k -rationality of three-dimensional smooth complete intersections of two quadrics. Theorem 4.7.
Let X ⊂ P k be a smooth complete intersection of two quadrics.Then X is k -rational if and only if it contains a line defined over k .Proof. If X contains a line Λ, then projecting from Λ induces a birational map X P k (see the more precise Proposition 4.3).Assume, conversely, that X is k -rational. Let D be as in Theorem 4.5 (iii). Let ψ : ( CH X/k ) ∼ −→ Pic D/k be the inverse of the isomorphism of Theorem 4.5 (iv).Using Theorem 4.5 (ii), we identify the variety of lines F of X with the torsor( CH X/k ) under ( CH X/k ) . Theorem 3.10 (iii) shows the existence of d ∈ Z suchthat ψ ∗ [ F ] = [ Pic dD/k ] ∈ H ( k, Pic D/k ) and Theorem 4.5 (iv) yields the identity[
Pic D/k ] = 2 ψ ∗ [ F ] ∈ H ( k, Pic D/k ). Combining these two equalities shows that(4.3) ψ ∗ [ F ] = [ Pic dD/k ] = [
Pic − dD/k ] ∈ H ( k, Pic D/k ) . Noting that K D ∈ Pic D/k ( k ) since D has genus 2, we see that the Pic D/k -torsor
Pic D/k is trivial. As one of d and 1 − d is even, it follows from (4.3) that ψ ∗ [ F ] = 0 ∈ H ( k, Pic D/k ). Consequently, F ( k ) = ∅ and X contains a linedefined over k . (cid:3) Unirationality.
In §4.5, we study the k -unirationality of smooth completeintersections of two quadrics. Over infinite perfect fields of characteristic not 2,Theorem 4.8 can be found in [CTSSD87, Remark 3.28.3]. The analogue of Theorem4.8 for cubic hypersurfaces is due to Kollár [Kol02, Theorem 1]. Theorem 4.8.
Fix N ≥ , and let X ⊂ P Nk be a smooth complete intersection oftwo quadrics. Then X is k -unirational if and only if X ( k ) = ∅ . Proof.
Since projective k -unirational varieties always have k -points, we only needto prove the converse implication. We argue by induction on N . If N = 4and k has cardinality ≥
23, the assertion is due to Manin [Man86, Theorems 29.4and 30.1]. (The hypothesis that k is perfect is not used in the proof of [Man86,Theorem 30.1 (i)] for degree 4 del Pezzo surfaces. One only needs to notice that thevariety of lines in a degree 4 del Pezzo surface over k is étale over k .) If N = 4 and k is an arbitrary finite field, the assertion is due to Knecht [Kne15, Theorem 2.1].If N ≥
5, choose x ∈ X ( k ), and consider the space B of hyperplanes in P Nk thatcontain x . Let Z = { ( w, b ) ∈ X × B ; w ∈ b } and let p : Z → B and q : Z → X bethe natural projections. The generic fiber of p is a smooth complete intersection oftwo quadrics in P N − k ( B ) by Lemma 4.9, and has a k ( B )-point induced by x , hence is k ( B )-unirational by the induction hypothesis. Since B is a projective space, Z is k -unirational, and the dominant map q shows that so is X . (cid:3) Lemma 4.9.
Fix N ≥ , let X ⊂ P Nk be a smooth complete intersection of twoquadrics, and let x ∈ X ( k ) . Then, if H ⊂ P Nk is a general hyperplane containing x ,the variety X ∩ H is smooth of dimension N − .Proof. Suppose for contradiction that the conclusion of the lemma does not hold.It follows that for every hyperplane H ⊂ P Nk containing x , there exists a quadric Q ⊂ P Nk in the pencil defining X such that Q ∩ H is not smooth of dimension N − X . Consequently, the variety W parametrizing suchpairs ( Q, H ) is at least ( N − Q ⊂ P Nk be a singular quadric in the pencil defining X . (Such a Q exists since the locus of quadrics with a unique singular point has codimension 1in the projective space of quadrics in P Nk , so that its closure contains an ampledivisor.) Note that x ∈ X ⊂ Q . As X is smooth, the singular locus of Q iszero-dimensional, hence Q is a cone over a smooth quadric Q ′ , and X does notcontain the vertex of this cone. The projective dual ( Q ) ∨ of Q can be naturallyidentified with the projective dual of Q ′ . By Lemma 4.10 below, the variety ( Q ) ∨ has dimension N − Q in its vertex and nowhere else. In both cases, the subset of ( Q ) ∨ consisting of the hyperplanes that contain x is a non-trivial hyperplane sectionof ( Q ) ∨ , hence is ( N − H ⊂ P Nk with ( Q , H ) ∈ W is at most ( N − Q ⊂ P Nk in thepencil defining X such that ( Q, H ) ∈ W for all hyperplanes H ⊂ P Nk containing x .Consequently, the projective dual of Q is the hyperplane dual to the point x . Thiscontradicts Lemma 4.10 below since x ∈ X ⊂ Q . (cid:3) Lemma 4.10.
Fix N ≥ and let Q ⊂ P Nk be a smooth quadric. Then the projectivedual of Q is a smooth quadric if k has characteristic = 2 or if N is odd; otherwiseit is a hyperplane whose dual is a point not contained in Q .Proof. In appropriate homogeneous coordinates, the quadric Q is defined by theequation { X X + · · · + X N − X N = 0 } if N is odd, and by the equation { X + X X + · · · + X N − X N = 0 } if N is even (see [SGA72, XII, Proposition 1.2]).The lemma follows by a direct computation. (cid:3) NTERMEDIATE JACOBIANS AND RATIONALITY OVER ARBITRARY FIELDS 43
Examples over fields of Laurent series.
Here are examples of smoothcomplete intersections of two quadrics in P κ (( t )) which are not κ (( t ))-rational. Theequations we use in characteristic 2 are borrowed from Bhosle [Bho90]. Theorem 4.11.
Let κ be an algebraically closed field. If the characteristic of κ is = 2 , let a , . . . , a ∈ κ be pairwise distinct elements, and consider the smoothprojective variety X ⊂ P κ (( t )) with equations ( tX + tX + X + · · · + X = 0 ta X + ta X + a X + · · · + a X = 0 . If κ has characteristic , let a, b, c ∈ κ be pairwise distinct elements and considerthe smooth projective variety X ⊂ P κ (( t )) with equations ( tX X + X X + X X = 0 t ( X + aX X + X ) + ( X + bX X + X ) + ( X + cX X + X ) = 0 . Then X is κ (( t )) -unirational, κ (( t )) -rational, but not κ (( t )) -rational.Proof. In view of Theorem 4.7 and Theorem 4.8, the conclusion of the theorem isequivalent to the assertion that X contains a point over κ (( t )), a line over κ (( t )),but no line over κ (( t )), and this is what we shall now prove.Let Y ⊂ P κ be the subvariety with equations { P i X i = P i a i X i = 0 } if κ hascharacteristic = 2, with equations ( X X + X X + X X = 0( X + aX X + X ) + ( X + bX X + X ) + ( X + cX X + X ) = 0if κ has characteristic 2. Let κ (( t )) ⊂ κ (( u )) be the quadratic extension with u = t and φ : X κ (( u )) ∼ −→ Y κ (( u )) the isomorphism ( X , . . . , X ) ( uX , uX , X , . . . , X ).As κ is algebraically closed, the variety X has a κ (( t ))-point in the subspace { X = X = 0 } . The variety Y contains a line (by Lemma 4.1 (iii)), so that X κ (( u )) contains a line as well. Let us now assume that X itself contains a line L ⊂ X , andderive a contradiction.We denote by τ : µ × P κ → P κ the action of the κ -group scheme µ on P κ , viaits non-trivial character on X and X and trivially on X , X , X and X . Thesubvariety Y ⊂ P κ is τ -invariant. Let σ : µ × Spec( κ (( u ))) → Spec( κ (( u ))) bethe µ -action endowing Spec( κ (( u ))) with its natural structure of µ -torsor overSpec( κ (( t ))). It extends to an action of µ on Spec( κ [[ u ]]) for which the closedpoint is an invariant subscheme.Let us regard the line L ′ := φ ( L κ (( u )) ) ⊂ Y κ (( u )) as a κ (( u ))-point of the variety oflines F of Y . In view of the equation defining φ and since L is defined over κ (( t )),the morphisms µ × Spec( κ (( u ))) → F given by the orbits of L ′ with respectto the actions of µ on F κ (( u )) induced by τ and by σ coincide. It follows thatthe specialization L ′′ ⊂ Y of L ′ with respect to the u -adic valuation of κ (( u ))is τ -invariant. Since Y does not meet the projectivization of the subspace of κ where µ acts via its non-trivial character, the line L ′′ must be contained in theprojectivization { X = X = 0 } of the subspace where µ acts trivially. But theintersection of Y with { X = X = 0 } is an elliptic curve, which contains no line.This is the required contradiction, and the proposition is proved. (cid:3) Remarks . (i) We do not know whether the variety X appearing in Proposi-tion 4.11 is stably rational over κ (( t )), even when κ = C .(ii) If κ has characteristic 2, the variety X considered in Proposition 4.11is not κ (( t ))-rational, but it becomes rational over the perfect closure κ (( t )) p of κ (( t )). It follows that one cannot prove Proposition 4.11 by applying Theorem 4.7over κ (( t )) p . A theory of intermediate Jacobians over imperfect fields is thereforecrucial for our proof of Theorem 4.11.(iii) The variety X appearing in Proposition 4.11 when κ has characteristic 2 isthe first example of a smooth projective variety over a field k which is k p -rational,which has a k -point, but which is not k -rational. There are no such examples indimension ≤ k which is k -rational but not k -rational.We include the following proposition, which generalises [Coo88, Theorem 1], tojustify Remark 4.12 (iii). Proposition 4.13 (Segre, Manin, Iskovskikh) . Let X be a smooth projectivesurface over k . The following assertions are equivalent:(i) X is k -rational,(ii) X is k p -rational and X ( k ) = ∅ .If X is minimal, they are also equivalent to(iii) K X ≥ and X ( k ) = ∅ .Proof. To prove the proposition, we may assume that X is minimal, from which itfollows that X k p is minimal (see [Poo17, Corollary 9.3.7]). That (i) implies (ii) isobvious. That (ii) and the minimality of X k p imply (iii) results from the birationalclassification of geometrically rational surfaces over perfect fields, due to Segre,Manin and Iskovskikh (see [Isk96, p. 642]). It remains to prove that (iii) implies (i).If X is a del Pezzo surface, this is [VA13, Theorem 2.1]. Suppose now that X isnot a del Pezzo surface. Then X belongs to the family II described in [Isk79,Theorem 1]. Since 5 ≤ K X ≤ K X = 8by [Isk79, Theorem 5]. It then follows from [Isk79, Theorem 3 (2)] that X is eithera product of curves of genus 0, or a projective bundle over a curve of genus 0. As X ( k ) = ∅ , these curves of genus 0 are isomorphic to P k , and X is k -rational. (cid:3) References [ABB14] A. Auel, M. Bernardara and M. Bolognesi,
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