Rec;u et accept6 le 1" mars 1999) RksumC. En s'inspirant des mkthodes dCveloppCes par Beauville et Donagi, on montre qu'une section hyperplane gCnCrique d'une hypersurface couverte par des k-plans vtrifie CH,(X) @ Q = Q pour i 5 k -1. Cela permet d'exhiber en tout degrk une famille d'hypersurfaces vkrifiant la conjecture de Bloch-Beilinson. 0 Acadkmie des Sciences/Elsevier. Paris Remarks on Chow groups of hypersurfaces of low degree Abstract. Following methods developed by Beauville and Donagi, we prove that a generic hyperplane section of a hypersurface covered by k-planes satisfies CH; (X)@Q = Q for i 5 k -1. It allows us to exhibit a family of hypersu$aces satisfying the Bloch-Beilinson conjectures. 0 AcadCmie des SciencesiElsevier, Paris Abridged English Version The Bloch-Beilinson conjectures in the case of hypersurfaces imply the following conjecture. CONJECTURE. -Let X C PE be a smooth hypersu&ce of degree d. Then CHk-1 (X),,, 8 Q = Ofor n =* kd, where CHk-l(X)hom -_ is the subgroup of C&-l(X) of the cycles homologically equivalent to 0. We prove the following special case of the conjecture. THEOREM 1. -Let Y c Pz+l be a smooth projective hypersu$ace covered by a family of linear projective varieties of dimension k in PE+l. Then for any smooth hyperplane section X C Y we have CHk-l(X)hom 63 Q = 0. If k(n -k) -(":") + 1 2 0, any smooth projective hypersurface of degree d in PF is covered by linear projective varieties of dimension k (see for instance [3]). Thus the theorem improves the bound given in [3]: Note prksentee par Jean-Pierre SERRE. 0764~4442/99/0329005 1 0 Acadkmie des Sciences/Elsevier. Paris >"'> c
Then Y is smooth and any smooth hyperplane section of Y satis$es the conjecture.
Corollary 2 provides examples of smooth hypersurfaces X of degree d in Pz with n = kd, without non-trivial automorphisms, satisfying CHk-r(X) horn @ Q = 0, as expected by the conjecture.