. Proof, The proof works as in [90, Lemma 5.1]. Indeed, by [8], the monodromy group of a smooth complete intersection of dimension 3 is Zariski dense in the symplectic group of H 3

. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 1966.
DOI : 10.1007/978-3-662-03580-1

A. B. Altman and S. L. Kleiman, Foundation of the theory of the Fano schemes, Compositio Mathematica, vol.34, issue.37, pp.3-47, 1977.

M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3), p.7595, 1972.

M. Atiyah and F. Hirzebruch, Analytic cycles on complex manifolds, Topology, vol.1, issue.1, pp.25-45, 1962.
DOI : 10.1016/0040-9383(62)90094-0
URL : https://doi.org/10.1016/0040-9383(62)90094-0

A. Auel, J. Colliot-thélène, and R. Parimala, Universal Unramified Cohomology of Cubic Fourfolds Containing a Plane, Progress in Mathematics, vol.320, issue.53, pp.29-56, 2017.
DOI : 10.1007/978-3-319-46852-5_4

W. Barth and A. , Fano-Varieties of lines on hypersurfaces, Archiv der Mathematik, vol.108, issue.1, pp.96-104, 1978.
DOI : 10.1007/BF01226420

A. Beauville, Vari??t??s de Prym et jacobiennes interm??diaires, Annales scientifiques de l'??cole normale sup??rieure, vol.10, issue.3, pp.309-391, 1977.
DOI : 10.24033/asens.1329

A. Beauville, Le groupe de monodromie des familles universelles d'hypersurfaces et d'intersections complètes, Complex analysis and Algebraic Geometry, p.78, 1986.
DOI : 10.1007/bfb0076991

A. Beauville, J. Colliot-thélène, J. Sansuc, and P. Swinnerton-dyer, Varietes Stablement Rationnelles Non Rationnelles, Variétés stablement rationnelles non rationnelles, pp.283-318, 1985.
DOI : 10.2307/1971174

A. Beauville and R. Donagi, La variété des droites d'une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sér. I Math, vol.301, issue.14, pp.703-706, 1985.

S. Bloch, Torsion algebraic cycles and a theorem of Roitman, Comp. Math, vol.39, pp.107-127, 1979.

S. Bloch and A. , Ogus Gersten's conjecture and the homology of schemes, Ann. Sci. École Norm. Sup., Sér, vol.4, issue.7, pp.181-201, 1974.

S. Bloch and V. Srinivas, Remarks on Correspondences and Algebraic Cycles, American Journal of Mathematics, vol.105, issue.5, pp.1235-1253, 1983.
DOI : 10.2307/2374341

H. Clemens, Homological equivalence modulo algebraic equivalence is not nitely generated
DOI : 10.1007/bf02953771

H. Clemens and P. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math, pp.281-356

J. Colliot-thélène, Birational invariants, purity and the Gersten conjecture, in K-Theory and Algebraic Geometry : Connections with Quadratic Forms and Division Algebras, Proceedings of Symposia in Pure Mathematics 58, pp.1-64, 1992.

J. Colliot-thélène and D. Coray, L'équivalence rationnelle sur les points fermés des surfaces rationnelles brées en coniques, Compositio Mathematica, vol.39, issue.3, pp.301-332, 1979.

J. Colliot-thélène and M. , Vari??t??s unirationnelles non rationnelles: au-del?? de l'exemple d'Artin et Mumford, Inventiones Mathematicae, vol.55, issue.1, pp.141-158, 1989.
DOI : 10.1007/BF01850658

J. Colliot-thélène and A. Pirutka, Hypersurfaces quartiques de dimension 3 : non-rationalit?? stable, Annales scientifiques de l'??cole normale sup??rieure, vol.49, issue.2, pp.49-373, 2016.
DOI : 10.24033/asens.2285

J. Colliot-thélène, J. Sansuc, and C. Soulé, Torsion dans le groupe de Chow de codimension deux, Duke Math, J, vol.50, pp.763-801, 1983.

J. Colliot-thélène, T. Szamuely, R. Akhtar, P. Brosnan, and R. Joshua, Autour de la conjecture de Tate à coecients Z pour les variétés sur les corps nis, in The Geometry of Algebraic Cycles, pp.83-98, 2010.

J. Colliot-thélène and C. Voisin, Cohomologie non ramiée et conjecture de Hodge entière, Duke Math, J, vol.161, issue.23, pp.735-801

V. Cossart and O. Piltant, Resolution of singularities of threefolds in positive characteristic. I., Journal of Algebra, vol.320, issue.3, pp.1051-1082, 2008.
DOI : 10.1016/j.jalgebra.2008.03.032
URL : https://hal.archives-ouvertes.fr/hal-00139445

V. Cossart and O. Piltant, Resolution of singularities of threefolds in positive characteristic. I., Journal of Algebra, vol.320, issue.3, pp.1836-1976, 2009.
DOI : 10.1016/j.jalgebra.2008.03.032
URL : https://hal.archives-ouvertes.fr/hal-00139445

O. Debarre and L. , Sur la vari??t?? des espaces lin??aires contenus dans une intersection compl??te, Mathematische Annalen, vol.312, issue.3, pp.312-549, 1998.
DOI : 10.1007/s002080050235

P. Deligne, Variétés unirationnelles non rationnelles Lectures Notes in Math, Séminaire Bourbaki, vol.402, issue.317, pp.45-57, 1971.

H. Esnault, M. Levine, and E. Viehweg, Chow groups of projective varieties of very small degree, Duke Math, J, vol.87, issue.30, pp.29-58, 1997.

W. Fulton, Intersection Theory Ergebnisse der Mathematik und ihrer Grenzgebi ete. 3. Folge. A Series of Modern Surveys in Mathematics, Mathematics an d Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], pp.12-44, 1998.

P. Griffiths, Periods of Integrals on Algebraic Manifolds, I. (Construction and Properties of the Modular Varieties), American Journal of Mathematics, vol.90, issue.2, pp.568-626805, 1968.
DOI : 10.2307/2373545

P. Griffiths, On the periods of certain rational integrals I, II, Ann. of Math, pp.90-460, 1969.

A. Grothendieck, Standard conjectures on algebraic cycles, Algebraic Geometry (Internat. Colloq . Tata Inst. Fund. Res., Bombay, pp.193-199, 1968.

J. Harris and I. Morrison, Moduli of curves, Graduate Texts in Mathematics, vol.187, 1998.

J. Harris, M. Roth, and J. Starr, Abel-Jacobi maps associated to smooth cubic three-folds, arXiv preprint math/0202080

R. Hartshorne, Complete Intersections and Connectedness, American Journal of Mathematics, vol.84, issue.3, pp.497-508, 1962.
DOI : 10.2307/2372986

R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, 1977.

R. Hartshorne, Stable reflexive sheaves, Mathematische Annalen, vol.47, issue.2, pp.121-176, 1980.
DOI : 10.1017/S0027763000014896

B. Hassett, A. Pirutka, and Y. , Stable rationality of quadric surface bundles over surfaces

H. Hironaka, Resolution of singularities of an algebraic variety over a eld of characteristic zero, I Ann, of Math, vol.79, issue.21, pp.109-203

H. Hironaka, Smoothing of Algebraic Cycles of Small Dimensions, American Journal of Mathematics, vol.90, issue.1, pp.1-54, 1968.
DOI : 10.2307/2373425

K. Hulek, Projective geometry of elliptic curves, Astérisque, vol.27, pp.65-67, 1986.
DOI : 10.1007/BF01452062

A. Iliev and D. Markushevich, The Abel-Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14, Documenta Mathematica, vol.5, issue.55, pp.23-47, 2000.

V. Iskovskikh and Y. Manin, THREE-DIMENSIONAL QUARTICS AND COUNTEREXAMPLES TO THE L??ROTH PROBLEM, Mathematics of the USSR-Sbornik, vol.15, issue.1, pp.141-166, 1971.
DOI : 10.1070/SM1971v015n01ABEH001536

B. Kahn and R. Sujatha, Unramied cohomology of quadrics, II, Duke Math, J, vol.106, issue.38, pp.449-484, 2001.

J. Kollár and L. P. , 134 in Classication of irregular varieties, Lecture Notes in Math, vol.1515, p.17, 1990.

J. Kollár, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in M athematics, Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in M athematics, pp.21-37, 1996.

S. Lang, On Quasi Algebraic Closure, The Annals of Mathematics, vol.55, issue.2, pp.373-390, 1952.
DOI : 10.2307/1969785

M. Larsen and V. A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J, vol.3, issue.1, pp.85-95, 2002.

C. Liedtke, Supersingular K3 surfaces are unirational, Inventiones mathematicae, vol.230, issue.84, pp.979-1014, 2015.
DOI : 10.1017/CBO9780511569197.010
URL : http://arxiv.org/pdf/1304.5623.pdf

J. Lüroth, Beweis eines Satzes ???ber rationale Curven, Mathematische Annalen, vol.9, issue.2, pp.163-165, 1876.
DOI : 10.1007/BF01443371

D. Markushevich and A. Tikhomirov, The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold, J. Algebraic Geometry, vol.10, issue.55, pp.37-62, 2001.

R. Mboro, On the universal $$\mathrm {CH}_0$$ CH 0 group of cubic threefolds in positive characteristic, manuscripta mathematica, vol.22, issue.1, pp.147-168
DOI : 10.1090/S1056-3911-2012-00597-9

R. Mboro, Remarks on the CH 2 of cubic hypersurfaces
URL : https://hal.archives-ouvertes.fr/hal-01478975

R. Mboro, Remarks on approximate decompositions of the diagonal
URL : https://hal.archives-ouvertes.fr/hal-01587517

J. S. Milne, Lectures on étale cohomology

J. S. Milne, Abelian Varieties, Arithmetic geometry, pp.103-150, 1986.
DOI : 10.1007/978-1-4613-8655-1_5

B. Moonen and A. Polishchuk, Divided powers in Chow rings and integral Fourier transforms, Advances in Mathematics, vol.224, issue.5, pp.2216-2236, 2010.
DOI : 10.1016/j.aim.2009.12.025

D. Mumford, Rational equivalence of 0-cycles on surfaces, Journal of Mathematics of Kyoto University, vol.9, issue.2, pp.195-204, 1968.
DOI : 10.1215/kjm/1250523940

D. Mumford, Prym varieties I, in Contributions to analysis, 1974.

J. P. Murre, Algebraic equivalence modulo rational equivalence on a cubic threefold, Compositio Mathematica, vol.25, pp.161-206, 1972.

J. P. Murre, Some results on cubic threefolds, in Classication of algebraic varieties and compact complex manifolds, Lecture Notes in Math, pp.140-164, 1974.

J. P. Murre, Applications of algebraic K-theory to the theory of algebraic cycles, Proc. Conf. Algebraic Geometry, pp.216-261, 1983.
DOI : 10.1017/S0027763000024855

J. P. Murre-albano and F. Bardelli, Algebraic Cycles and Algebraic Aspects of Cohomology and K-Theory, Lecture Notes in Math, vol.1594, issue.76, pp.93-152, 1994.
DOI : 10.1007/978-3-540-49046-3_2

A. Otwinowska, Remarques sur les groupes de Chow des hypersurfaces de petit degr??, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.329, issue.1
DOI : 10.1016/S0764-4442(99)80460-1

X. Pan, 2-Cycles on higher Fano hypersurfaces, Mathematische Annalen, vol.2, issue.1, pp.1-28, 2016.
DOI : 10.1007/s40304-014-0029-7
URL : http://arxiv.org/pdf/1512.01721

A. A. Roitman, The Torsion of the Group of 0-Cycles Modulo Rational Equivalence, The Annals of Mathematics, vol.111, issue.3, pp.553-569, 1980.
DOI : 10.2307/1971109

C. Schnell, Hypersurfaces containing a given subvariety, online lecture notes. (Cité à la page 49

C. Schoen, An integral analog of the Tate conjecture for one-dimensional cycles on varieties over nite eld, Math. Ann, vol.53, issue.3, pp.311-493, 1998.

J. Serre, On the fundamental group of a unirational variety, J. London Math. Soc, vol.34, pp.481-484, 1959.

M. Shen, On relations among $1$-cycles on cubic hypersurfaces, Journal of Algebraic Geometry, vol.23, issue.3, pp.539-569, 2014.
DOI : 10.1090/S1056-3911-2014-00631-7

M. Shen, Rationality, universal generation and the integral Hodge conjecture, pp.22-31

I. Shimada, On the cylinder homomorphism for a family of algebraic cycles, Duke Math, J, vol.64, pp.201-205, 1991.

J. Starr, Brauer groups and Galois cohomology of function elds of varieties, Lecture notes, 2008.

Z. Tian and R. Zong, Abstract, Compositio Mathematica, vol.7, issue.03, pp.396-408, 2014.
DOI : 10.1090/pspum/062.2/1492534

B. Totaro, Torsion algebraic cycles and complex cobordism, Journal of the American Mathematical Society, vol.10, issue.02, pp.467-493, 1997.
DOI : 10.1090/S0894-0347-97-00232-4

B. Totaro, THE INTEGRAL COHOMOLOGY OF THE HILBERT SCHEME OF TWO POINTS, Forum of Mathematics, Sigma, vol.49, pp.49-51
DOI : 10.1007/s00222-014-0551-y

C. Tsen, Quasi-algebraische-abgeschlossene Funktionenkörper, J. Chinese Math, vol.1, pp.81-92, 1936.

V. Voevodsky, A nilpotence theorem for cycles algebraically equivalent to zero, Internat. Math. Res. Notices, vol.75, issue.4, pp.187-198, 1995.

C. Voisin, Remarks on zero-cycles of self-products of varieties, Moduli of vector bundles (Proceedings of the Taniguchi congress on vector bundles, pp.265-285, 1994.

C. Voisin, Hodge theory and complex algebraic geometry I, Cambridge Studies in Advanced Mathematics 77, 2003.

C. Voisin, Hodge theory and complex algebraic geometry II, Cambridge Studies in Advanced Mathematics, vol.77, 2003.
DOI : 10.1017/CBO9780511615177

C. Voisin and H. Loci, Farkas and I. Morrison) Advanced Lectures in, Mathematics, vol.25, pp.507-546

C. Voisin, On integral Hodge classes on uniruled and Calabi-Yau threefolds, Moduli Spaces and Arithmetic Geometry, Adv. Stud. Pure Math. Math. Soc. Japan, vol.45, issue.18, pp.43-73, 2006.

C. Voisin, Some aspects of the Hodge conjecture, Japanese Journal of Mathematics, vol.34, issue.2, pp.261-296, 2007.
DOI : 10.1112/S0010437X07002837

C. Voisin, Degree 4 unramied cohomology with nite coecients and torsion codimension 3 cycles, Geometry and Arithmetic Series of Congress Reports, pp.347-368
DOI : 10.4171/119-1/20
URL : http://www.math.polytechnique.fr/~voisin/Articlesweb/hquatrenr.pdf

C. Voisin, Remarks on curve classes on rationally connected varieties, in: A Celebration of Algebraic Geometry, Clay Math, Proc. 18, pp.591-599

C. Voisin, Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal, Journal of Algebraic Geometry, vol.22, issue.1, pp.141-174, 2013.
DOI : 10.1090/S1056-3911-2012-00597-9
URL : https://hal.archives-ouvertes.fr/hal-00473977

C. Voisin, On the universal CH$_ 0$ group of cubic hypersurfaces, Journal of the European Mathematical Society, vol.19, issue.6, pp.1619-1653, 2017.
DOI : 10.4171/JEMS/702
URL : https://hal.archives-ouvertes.fr/hal-01079297

C. Voisin, Unirational threefolds with no universal codimension $$2$$ 2 cycle, Inventiones mathematicae, vol.351, issue.1???2, pp.207-237, 2015.
DOI : 10.1016/j.crma.2012.12.002
URL : https://hal.archives-ouvertes.fr/hal-01002966

C. Voisin, Stable birational invariants and the Lüroth problem, Surveys in Dierential Geometry XXI (2016), p.16

G. E. Welters, Abel-Jacobi isogenies for certain types of Fano threefolds, Mathematical Centre Tracts Math. Centrum, vol.141, issue.68, p.67, 1981.

O. Zariski, On Castelnuovo's criterion of rationality p a = P 2 = 0 of an algebraic surface, Illinois Journal of Mathematics, vol.2, pp.303-315