NOTE ON TORSION CONJECTURE. par. Anna Cadoret and Akio Tamagawa

Transcription

1 NOTE ON TORION CONJECTURE par Anna Cadoret and Aio Tamagawa Résumé. In this note, we give an elementary and effective proof of the fact that the torsion conjecture for jacobian varieties implies the torsion conjecture for abelian varieties. 010 Mathematics ubject Classification. Primary: 14K15; econdary: 14H Introduction The classical torsion conjecture for abelian varieties over number fields can be stated as follows. Conjecture 1.1. Let d 0 be an integer then: - (Wea form): Given a number field, there exists an integer N := N(, d) 0 such that for any d-dimensional abelian variety A over, one has: A() tors A[N]. - (trong form): Given an integer δ 1, there exists an integer N := N(δ, d) 0 such that for any number field of degree δ and any d-dimensional abelian variety A over, one has: A() tors A[N]. Completing a body of wors initiated by B. Mazur in the mid-1970 s [Ma77], L. Merel achieved a proof of the d = 1 case of the strong torsion conjecture in the mid-1990 s [Me96]. But the d > 1 case remains widely open though recent results of the authors show that the strong torsion conjecture for the p-primary part of the torsion holds for d-dimensional abelian varieties parametrized by curves [CT09]. The aim of this note is to give a proof of the following statement, which, in particular, shows that the torsion conjecture for abelian varieties is equivalent to the torsion conjecture for jacobian varieties. Theorem 1.. Let d > 0 be an integer. Then there exists an integer g(d) > 0 satisfying the following property: For any infinite field and any d-dimensional principally polarized abelian variety A over there exists a smooth, geometrically connected curve C A of genus g C g(d) that induces a smooth surjective homomorphism with connected ernel J C A and a closed immersion A J C of abelian varieties. More precisely, one may tae C A of genus g C = g(d) with: g(d) = d (d 1)! d(d 1).

2 ANNA CADORET AND AKIO TAMAGAWA Note that if A is an arbitrary (i.e., a not necessarily principally polarized) d-dimensional abelian variety over then, by Zarhin s tric, (A A ) 4 is an 8d-dimensional principally polarized abelian variety over. In particular, to prove the torsion conjecture for d-dimensional abelian varieties, it is enough to prove it for g(8d)-dimensional jacobian varieties. Roughly speaing, the curve C in the statement of theorem 1. is constructed by cutting d 1 times A by nice hyperplanes. For the proof of the crucial fact that C can be chosen to have genus bounded only in terms of d, we give two different approaches in section and section 3 respectively. The first approach given in section is very elementary and relies on a certain explicit genus computation. More precisely, in subsection.1, we show that given a smooth, geometrically connected projective variety X over an infinite field and a fixed embedding X P n, the curves obtained by cutting X by nice hyperplanes all have the same Hilbert polynomial, which depends only on the Hilbert polynomial of X. We then compute in subsection. effectively the Hilbert polynomial of the resulting curves. The second approach given in section 3 is more conceptual and relies on the theory of Hilbert schemes. More precisely, in subsection 3.1 we show that given a scheme and a closed subscheme X P n smooth, geometrically connected and (purely) of relative dimension d > 0 over there exists a surjective smooth morphism π : U of finite type and a universal curve q : C X U U such that any curve constructed in X s by cutting d 1 times X s by nice hyperplanes arises as the fiber of q at some h U s (see proposition 3.1 for the precise statement). In subsection 3., we discuss the representability of the Hilbert functor. In subsection 3.3, we combine these results to recover the desired boundedness of the genus. Eventually, in section 4, we conclude the proof of theorem 1. by resorting to a wea version of Lefschetz theorem and a duality argument. Remar If is a finite field, one may still show that for any (positive-dimensional) abelian variety A over there exists a smooth, geometrically connected curve C A for which the assertions of theorem 1. hold. Indeed, one has only to replace the classical hyperplane Bertini theorem by more recent hypersurface Bertini theorems due to O. Gabber [G01, Cor. 1.6] and B. Poonen [P04, Thm. 1.1] and note that lemma 4.1 also wors for hypersurfaces. However, the genus of the curve constructed by this method is uncontrolled in Gabber s method and depends on the poles of the zeta functions of the successive sections in Poonen s method. o it is difficult to obtain a bound of the genus independent of the finite base field as in theorem 1... If the characteristic of is 0, our proof of theorem 1. is entirely elementary and classical. On the other hand, if the characteristic of is positive, this argument only shows that the morphism J C A is surjective with connected ernel and the morphism A J C is finite with ernel having connected Cartier dual. To get the full statement in positive characteristic, we resort to [G01, Prop..4], which may be less elementary. 3. The problem of how to realize abelian varieties as quotients of jacobian varieties is classical and the first proof of the fact that this can always be done (over an algebraically closed field) seems to go bac to [M5]. Other references include [Mi86] and the already mentioned [G01]. We end this section by the following lemma, which is used in both of the two approaches. Lemma 1.4. Let be a field and A a d-dimensional abelian variety over equipped with a polarization λ : A A of degree δ (δ > 0). Let L denote the invertible sheaf (id A, λ) (P A ) on A, where P A is the normalized Poincaré sheaf on A A, so that φ L = λ [MF8, Chap.6,, Prop. 6.10], and that O A (1) := L 3 is very ample relatively to A [Mu70, III, 17] and induces a closed immersion A P n. Then the Hilbert polynomial P (T ) of A with respect to this embedding is given by: In particular, P (T ) depends only on (d, δ). P (T ) = 6 d δt d.

3 NOTE ON TORION CONJECTURE 3 Proof. This is stated in [MF8, Chap.7, ] with a hint of proof. More explicitly, from the Riemann- Roch theorem [Mu70, III, 16], one has: χ(o A (n)) = deg(φ L 3n) = deg(3nφ L ) = deg(6nλ) = (6n) d deg(λ), whence χ(o A (n)) = 6 d δn d and P (T ) = 6 d δt d.. First approach Genus computation.1. General case. Let be an infinite field and let X 0 P n be a smooth, projective and geometrically connected variety of dimension d > 0 over. If d 1 > 0, it follows from Bertini s theorem [J83, I, Th and Th. 7.1] that there exists a hyperplane H 1 P n such that X 1 := X 0 P n H 1 is a smooth, geometrically connected -variety of dimension d 1. Iterating the process, one can construct a smooth geometrically connected -variety X i of dimension d i inductively for 0 i d 1. More precisely, one set X i := X i 1 P n H i = X 0 P n H 1 P n P n H i. Also, let O Xi (1) denote the very ample invertible sheaf relatively to induced by the projective embedding X i P n. Then, one has the short exact sequences of O Xi -modules: Hence, tensoring by O Xi (n): 0 O Xi ( 1) O Xi O Xi O Xi (n 1) O Xi (n) O Xi+1 (n) 0, from which it follows that P i+1 (T ) = P i (T ) P i (T 1), where P i (T ) denotes the Hilbert polynomial of X i. A straightforward inductive computation then yields: P i (T ) = ( ) i ( 1) P 0 (T ). 0 i And, in particular, one can compute the genus g(x d 1 ) of X d 1 : g(x d 1 ) = dim (H 1 (X d 1, O Xd 1 )) = 1 ( ) d 1 ( 1) P 0 ( ). 0 d 1 Remar.1. (Comparison with Castelnuovo s bound) By construction, the curves X d 1 obtained by cutting out X by d 1 hyperplanes all have the same degree as X - say a. From Castelnuovo s bound [ACGH85, p. 116], this implies that the genus of X d 1 is bounded from above by a constant q(q 1) π(a, n) = (n 1) + qr, where q and r denote the quotient and remainder of the euclidean division of a 1 by n 1 respectively... The case of polarized abelian varieties. We would lie to apply the preceding computation to a d-dimensional abelian variety X 0 = A over equipped with a degree δ polarization λ : A A (δ > 0). o, as in lemma 1.4, let L be the invertible sheaf on A such that φ L = λ, hence O A (1) := L 3 is very ample relatively to A and induces a closed immersion A P n. Now, by lemma 1.4, the Hilbert polynomial P 0 (T ) with respect to this embedding is given by: P 0 (T ) = 6 d δt d. As a result: It remains to compute: g(x d 1 ) = d δ( 1) d 1 u(d) := 0 d 1 0 d 1 ( d 1 ( ) d 1 ( 1) d. ) ( 1) d.

4 4 ANNA CADORET AND AKIO TAMAGAWA For this, set u 0 (x) = (1 x) d 1 and u i (x) = xu i 1 (x), i 1. computations show that, on the one hand: u i (x) = ( d 1 and that, on the other hand: with a 1,1 = 1 and 1 d 1 u i (x) = 1 i ) ( 1) i x, i 1, a i, x u () 0 (x), i 1, a i 1,i 1, = i, a i, = a i 1, + a i 1, 1, i 1, a i 1,1, = 1. In particular, a i,i = a i 1,i 1 = = a 1,1 = 1, a i,1 = a i 1,1 = = a 1,1 = 1, and a i,i 1 = (i 1) + a i 1,i = = (i 1) + (i ) a,1 = o, as u () 0 (x) = ( 1) (d 1)! (d 1)! (1 x)d 1, one eventually gets: and: u(d) = u d (1) = a d,d 1 u (d 1) 0 (1) = ( 1) d 1 d(d 1) (d 1)! g(x d 1 ) = d δ(d 1)! d(d 1). Then straightforward inductive i(i 1). Remar.. (Comparison with Castelnuovo s bound - continued) In our situation, X 0 = A is embedded with degree d!(6 d δ) into P n, where n = 6 d δ 1. As X d 1 is obtained by cutting X 0 out by generic hyperplane sections, the degree of X d 1 is, again, a = d!(6 d δ). In particular, for d +, Castelnuovo s bound is asymptotically equivalent to 1 (d!) 6 d δ whereas our bound is asymptotically equivalent to 1 d!d6d δ. (The authors are grateful to the referee for mentioning the connection of their wor with Castelnuovo s bound.) 3. econd approach Universal curve 3.1. Existence. Let be a scheme and let X P n be a closed subscheme smooth, geometrically connected and (purely) of relative dimension d > 0 over. Let G (n) P n denote the Grassmannian of projective linear subschemes of codimension 1 in P n and H Pn G (n) the universal hyperplane. et π (n) := G (n) G (n), }{{} d 1 and write p i : (n) G (n) for the natural ith projection (i = 1,..., d 1). Proposition 3.1. (Universal curve) There exist an open subscheme U (n) such that π : U remains surjective and a smooth, geometrically connected projective curve C X U P n U over U with the following universal property : for any h = (H 1,..., H d 1 ) (n), the scheme I h := (X s ) (h) P n (h) H 1 P n (h) P n (h) H d 1, where s := π(h), is a smooth, geometrically connected curve over (h) if and only if h U. Moreover, one has I h = C h P n (h) for any h U.

5 NOTE ON TORION CONJECTURE 5 Proof. ince the problem is local on, one may reduce to the case where is affine. Then, since X is of finite presentation over, it follows from the standard argument (cf. [EGA4-3, Prop. (8.9.1)(i)]) that one may reduce to the case where is noetherian. Consider the closed subscheme C 0 := π X P n (n) p 1H P n (n) P n (n) p d 1 H Pn (n) obtained as the (scheme-theoretic) intersection of π X = X (n) and d 1 hyperplanes p 1 H,..., p d 1 H in Pn. Note that one may also write: (n) P n (n) = Pn G (n) P n P n Pn G (n) }{{} d 1 and correspondingly: hence one gets: p i H = P n G (n) P n P n H i P n P n Pn G (n), C 0 = X P n H P n P n H }{{} d 1 (that is, for any (x, h = (H 1,..., H d 1 )) X (n), (x, h) C 0 if and only if x H 1 P n (h) P n (h) H d 1 ). We will use the following notation: C 0 pr 1 X q 0 X (n) pr (n) By construction, q 0 : C 0 (n) is projective (as a closed immersion composed with the base change of a projective morphism) with geometrically connected fibers of dimension 1 ([J83, I, Th. 7.1]). From the semicontinuity of dimension, the subset: U 1 := {h (n) dim(q0 1 (h)) = 1} is open in (n). Write q 1 : C 1 U 1 for the base change of q 0 : C 0 (n) via the open immersion U 1 (n). Then q 1 : C 1 U 1 remains projective over U 1 hence closed. Define the following loci: C1 smooth := {c C 1 q 1 : C 1 U 1 is smooth at c} C reg 1 := {c C 1 C 1,q1 (c) spec((q 1 (c))) is smooth (that is, C 1,q1 is regular)} (c) and set U := U 1 \ q 1 (C 1 \ C1 smooth ), U := U 1 \ q 1 (C 1 \ C reg 1 ). Then - U is open in U 1 (since C1 smooth is open in C 1 and q 1 : C 1 U 1 is closed); - U U ; - π(u ) = ([J83, I, Th. 6.10] ). Write q : C U for the base change of q 1 : C 1 U 1 via the open immersion U U 1. It only remains to prove that π : U is surjective and that C P n U satisfies the announced universal property. This will follow from the claim below which, in particular, shows that U = U : Claim. C smooth 1 = C reg 1. Indeed, this follows from the following claim: Claim. q 1 : C 1 U 1 is flat. π

6 6 ANNA CADORET AND AKIO TAMAGAWA To prove the latter claim, let x C 1 and set h = q 1 (x). Let A (resp. B, resp. B) be the local ring of U 1 at h (resp. of X U1 at x, resp. of C 1 at x), and the residue field of A (i.e., = (h)). It follows from the reduction step at the beginning of the proof that A, B and B are noetherian. Let f i B be the image in B of a local defining equation of the hyperplane (p i H) U 1 P n U 1. Then one has B = B/(f 1,..., f d 1 ). Now, one has to prove that B is flat as an A-module. For this, it suffices to show that g 1,..., g d 1 is a regular sequence of B A, where g i is the image of f i in B A, by [EGA4-1, Chap. 0 IV, Prop. ( ), c) b)]. But this latter fact follows from [EGA4-1, Chap. 0 IV, Cor. (16.5.6), b) a)]. (The second author would lie to than. Yasuda very much for the useful discussion on this proof.) 3.. Representability of the Hilbert functor. Let ets and ch op denote the category of sets and the opposite category of locally noetherian schemes, respectively. From [FGA] that for any N 0 and P Q[T ] the Hilbert functor Hilb N,P : ch op ets defined by Hilb N,P (X) = { Y P N X closed subscheme Y X is flat and has Hilbert polynomial P } is representable by a projective Z-scheme Hilb N,P Z; we will denote by Y N,P P N Hilb N,P the universal object over it. Now, set P d,δ (T ) := 6 d δt d Q[T ] and m := 6 d δ 1. Define the functor A d,δ : ch op ets by A d,δ (X) = (A π X, λ, φ) A π X is an abelian scheme of dimension d; λ : A A is a polarization of degree δ ; φ : P(π (L (λ) 3 )) P m X is a linearization. Here L (λ) stands for the invertible sheaf (id A, λ) (P A ) on A, where P A is the normalized Poincaré sheaf on A X A. et Y (1) pr := Y m,pd,δ Hilbm,Pd,δ Y m,pd,δ Ym,Pd,δ, which is naturally equipped with the diagonal section ɛ : Y m,pd,δ Y (1). Then (Y (1) P m Y m,pd,δ, ɛ) Y (1) is the universal family for closed subschemes of P m X flat over X with Hilbert polynomial P d,δ and 1 section. Thus, by definition of A d,δ : ch op ets, one has a natural functor morphism Φ : A d,δ Hom(, Y (1) ). Then, applying steps (I), (II), (V) and (VI) of the proof of [MF8, Prop. 7.3] (where level structures are also considered), one can show: Claim. There exists a locally closed subscheme A d,δ Y (1) such that Φ induces a functor isomorphism A d,δ Hom(, A d,δ ) Hom(, Y (1) ). In other words, A d,δ represents A d,δ Boundedness of the genus. We combine the results of subsections 3.1 and 3. to prove that there exists an integer g(d) > 0 such that for any d-dimensional principally polarized abelian variety A over an infinite field, any curve C A constructed as in subsection. has genus g C g(d). As Y (1) m,p d,1 is projective over Z, it is noetherian and, consequently, A d,1 is as well. Let (A π A d,1, λ, φ) denote the universal family over A d,1 and A P(π (L (λ) 3 )) φ P m A d,1 the closed immersion induced by λ and φ. Now, proposition 3.1 applied to A P m A d,1 yields an open subscheme U A d,1 (m) and a universal (projective, smooth, geometrically connected) curve q : C U. As U is noetherian, it has only finitely many connected components. Thus, only finitely many values appear as the genus of a fiber of q : C U. Let g(d) be the maximum of such values. Any d-dimensional abelian variety A π spec() over a field with a principal polarization λ : A A can be equipped with a linearization φ : P(π (L (λ) 3 )) P m defined over, which

7 NOTE ON TORION CONJECTURE 7 corresponds to a -rational point a of A d,1. Further, a choice of d 1 nice hyperplanes H 1,..., H d 1 corresponds to a -rational point b of the fiber U a (which is a nonempty open subscheme of (m)) and the resulting curve C A is identified with the fiber C b at b of q : C U. Thus, g C g(d). 4. End of the proof Recall first the following classical result [GA1, Chap. X, Lem..10]: Lemma 4.1. Let be an algebraically closed field, and X a normal, irreducible scheme proper over. Let g : X P n be a morphism such that g(x) has dimension. Then, for any hyperplane H P n, the scheme X P n H is connected and the natural homomorphism π 1 (X P n H) π 1 (X) of étale fundamental groups induced by the first projection X P n H X is surjective. From now on, fix an integer d (as the d = 1 case is trivial), an infinite field and a d-dimensional abelian variety A over equipped with a principal polarization λ : A A. As in subsection. let L be the invertible sheaf on A defining λ and A P n the closed immersion induced by L 3. Then, for any hyperplane H 1 P n such that A H 1 is a smooth, geometrically connected - variety of dimension d 1 it follows from lemma 4.1, that the closed immersion A H 1 A induces a surjective homomorphism π 1 (A H 1 ) π 1 (A) over Γ. Iterating d 1 times the process, one can construct a smooth geometrically connected -curve C := A P n H 1 P n P n H d 1 such that the closed immersion C A induces a surjective homomorphism of fundamental groups π 1 (C) π 1 (A) over the absolute Galois group Γ of or, in other words, a surjective homomorphism of short exact sequences: (1) 1 π 1 (C ) π 1 (C) Γ 1 1 π 1 (A ) π 1 (A) Γ 1 ince π 1 (A ) = T (A) (the full Tate module) is abelian, the homomorphism π 1 (C ) π 1 (A ) ills the commutator subgroup π 1 (C ) π 1 (C ) and one gets: () 1 π 1 (C ) ab π 1 (C) (ab) Γ 1 1 π 1 (A ) π 1 (A) Γ 1, where π 1 (C) (ab) denotes the quotient π 1 (C)/π 1 (C ). Further, the homomorphism π 1 (C ) ab π 1 (A ) is identified with the natural homomorphism T (J) T (A) of full Tate modules induced by the albanese morphism J A associated with C A, where J = J C denotes the jacobian variety of C over. ince the morphism J A induces a surjection T (J) T (A) of full Tate modules, it is surjective with connected ernel. Indeed, let K, I and C denote the ernel, the image and the coernel of J A, respectively. Note that I and C are abelian varieties, while K may not in general. The exact sequence of abelian varieties induces an exact sequence 0 I A C 0 0 T (I) T (A) T (C) 0 of full Tate modules. ince the image of T (J) in T (A) is contained in T (I) T (A), one must have T (C) = 0, hence C = 0, i.e., J A is surjective. Next, let K 0 denote the connected component at 0 of K, and K 0,red the reduced closed subscheme associated with K 0. Then K 0,red K 0 K J are

8 8 ANNA CADORET AND AKIO TAMAGAWA closed subgroup schemes; K et := K/K 0 is finite and étale; K loc := K 0 /K 0,red is finite and connected; and K 0,red is an abelian variety. Now, by using the snae lemma, one gets the exact sequence 0 T (K 0,red ) T (J) T (A) K et () 0, from which one gets K et () = 0, hence K et = 0, and K = K 0 is connected. Next, the dual morphism A (= A) J (= J) is finite with ernel having connected Cartier dual. Indeed, set à := J/K0,red. Then one gets the following two exact sequences: 0 K 0,red J à 0, 0 K loc à A 0. The dual morphism A J factors as A à J. Here, the ernel of the dual isogeny A à is identified with the Cartier dual of the ernel K loc of the morphism à A, and the dual morphism à J is a closed immersion. Indeed, for the former fact, see, e.g., [Mu70, III, 15, Th. 1]. For the latter fact, write K 1 and I 1 for the ernel and the image of the morphism à J, respectively. From Poincaré s complete reducibility theorem, there exists an abelian subvariety A J such that the composite of A J à is an isogeny. Thus, the ernel K 1 is contained in the ernel of the dual isogeny à (A ), hence is finite. ince the dual morphism à J factors as à I 1 J, the original morphism J à factors as J(= J ) (I 1 ) Ã(= à ). Here, the morphism (I 1 ) à is an isogeny whose ernel is identified with the Cartier dual (K 1) D of K 1. This, together with the surjectivity of J Ã, implies that the morphism J (I 1) must be also surjective. Now, the surjections J (I 1 ) à yield an epimorphism from the ernel K0,red of J à to the ernel (K 1 ) D of (I 1 ) Ã. As K0,red is an abelian variety (hence divisible) and (K 1 ) D is finite, this implies that (K 1 ) D = 0, hence K 1 = 0, as desired. Combining these facts, one sees that the ernel of A J coincides with the Cartier dual of the (finite, connected) group scheme K loc. Now, if the characteristic of is 0, the surjectivity of the morphism J A automatically implies the smoothness (cf. [Co0, Lem..1]), and the connectedness of the Cartier dual of the ernel of the morphism A J automatically implies the triviality. Thus, the proof is completed in characteristic 0. On the other hand, if the characteristic of is positive, the assertion that the morphism J A is smooth with connected ernel follows from [G01, Prop..4(iii)]. Namely, notations being as above, one has K = K 0,red and A = Ã. Thus, it follows from the above argument that the morphism A J is a closed immersion. (ee also [G01, Cor..5].) Références [ACGH85] E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris, Geometry of algebraic curves. Vol. 1, G.M.W. 67, pringer-verlag, [CT09] A. Cadoret and A. Tamagawa, A uniform open image theorem for l-adic representations II, preprint, 009. [Co0] B. Conrad, A modern proof of Chevalley s theorem on algebraic groups, J. Ramanujan Math. oc. 17 (00), [G01] O. Gabber, On space filling curves and Albanese varieties, Geom. Funct. Anal. 11 (001), [EGA4-1] A. Grothendiec, Éléments de géometrie algébrique IV. Étude locale des schémas et des morphismes de schémas (Première Partie), Inst. Hautes Etudes ci. Publ. Math. 0 (1964). [EGA4-3] A. Grothendiec, Éléments de géometrie algébrique IV. Étude locale des schémas et des morphismes de schémas (Troisième Partie), Inst. Hautes Etudes ci. Publ. Math. 8 (1966). [FGA] A. Grothendiec, Fondements de la géométrie algébrique. Extraits du éminaire Bourbai, , ecrétariat mathématique, Paris, 196. [GA1] A. Grothendiec et al., Revêtements étales et groupe fondamental, éminaire de Géometrie Algébrique du Bois Marie (GA 1), Lecture Notes in Mathematics 4, pringer-verlag, [H77] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 5, pringer-verlag, 1977.

9 NOTE ON TORION CONJECTURE 9 [J83] J.-P. Jouanolou, Théorèmes de Bertini et applications, Progress in Mathematics 4, Birhauser, [M5] T. Matsusaa, On a generating curve on an Abelian variety, Nat. ci. Rep. Ochanomizu Univ. 3 (195), 1 4. [Ma77] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Etudes ci. Publ. Math. 47 (1977), (1978). [Me96] L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 14 (1996), [Mi80] J.. Milne, Étale cohomology, Princeton Mathematical eries 33, Princeton University Press, [Mi86] J.. Milne, Jacobian varieties, in Arithmetic Geometry, G.Cornell and J.H. ilverman ed., pringer Verlag, [Mu70] D. Mumford, Abelian varieties, Tata Institute of Fundamental Research tudies in Mathematics 5, Oxford University Press, [MF8] D. Mumford and J. Fogarty, Geometric invariant theory, econd edition, Ergebnisse der Mathemati und ihrer Grenzgebiete 34, pringer-verlag, 198. [P04] B. Poonen, Bertini theorems over finite fields, Annals of Math. () 160 (004), anna.cadoret@math.u-bordeaux1.fr Institut de Mathématiques de Bordeaux - Université Bordeaux 1, 351 Cours de la Libération, F33405 TALENCE cedex, FRANCE. tamagawa@urims.yoto-u.ac.jp Research Institute for Mathematical ciences - Kyoto University, KYOTO , JAPAN. Anna Cadoret and Aio Tamagawa