Une version du théorème d’Amer et Brumer pour les zéro-cycles

Résumé (anglais).

M. Amer and A. Brumer have shown that, for two homogeneous quadratic forms f and g over a field k, the locus $f = g = 0$ has a non-trivial solution over k if and only if, for a variable t, the equation $f + tg = 0$ has a non-trivial solution over k(t). We consider a modified version of this result. We show that the projective variety over k defined by ${f}_{0}=\cdots={f}_{r}= 0$, where the f _i are homogeneous forms over k of the same degree d≥2 in n+1 variables (with $n + 1 \geq r + 2$), has a 0-cycle of degree 1 over k if and only if the generic hypersurface ${f}_{0} + {t}_{1}{f}_{1} + \cdots + {t}_{r}{f}_{r} = 0$ has a 0-cycle of degree 1 over k(t ₁,…,t _r ).

À Parimala

Classification thématique AMS. 14C25