Équivalence rationnelle sur les hypersurfaces cubiques de mauvaise réduction

Abstract

This article consists of two independent, but related, parts. The first one proves the vanishing of the Chow group of classes of zero-cycles of degree zero modulo rational equivalence for a cubic hypersurface of dimension ⩾10 on a p-adic or C 2 field (and, in fact, the R-triviality of such a hypersurface). This is done without the assumption of good reduction (or even smoothness). The second part goes in the other direction and gives an explicit example of a smooth cubic hypersurface of dimension 3 (necessarily of bad reduction) on a field such as C ( ( ν , t ) ) (or C ( ( ν ) ) ( ( t ) ) ) whose Chow group of classes of zero-cycles of degree zero modulo rational equivalence does not vanish.