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

Abstract

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 ).