Hypersurfaces cubiques : équivalence rationnelle, R-équivalence et approximation faible

Abstract : This thesis presents some results concerning the arithmetic of rationally connected varieties and, more specifically, cubic hypersurfaces, in three main directions: rational equivalence, R-equivalence, and weak approximation. In the first part, we describe explicitly the specialization of R-equivalence. The second part deals with the vanishing of the Chow group of 0-cycles of degree 0 on a cubic hypersurface having good reduction over the p-adics. The third shows a result of weak approximation at places of good reduction for cubic surfaces over function fields. The fourth shows the R-triviality of cubic hypersurfaces of large dimension over the p-adics. The fifth part shows, by an explicit computation, the non-vanishing of the Chow group of 0-cycles of degree 0 of a certain cubic hypersurface of dimension 3 over a field of dimension 2. Finally, we study very free R-equivalence on toric varieties.
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David Madore. Hypersurfaces cubiques : équivalence rationnelle, R-équivalence et approximation faible. Mathématiques [math]. Université Paris Sud - Paris XI, 2005. Français. ⟨tel-00009887v2⟩

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