### Une version du théorème d'Amer et Brumer pour les zéro-cycles, by Jean-Louis Colliot-Thélène and Marc Levine

M. Amer and A. Brumer have shown that, for two homogeneous quadratic
polynomials f and g in at least 3 variables over a field k of characteristic
different from 2, the locus f=g=0 has 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, and show that the
projective variety over k defined by f_{0}=...=f_{r}=0, where
the f_{i} are homogeneous polynomials over k of the same degree d ≥ 2
in n+1 variables (with n+1 ≥ r+2), has a 0-cycle of degree 1 over k if and only
if the generic hypersurface
f_{0}+t_{1}f_{1}+...+t_{r}f_{r}=0
has a 0-cycle of degree 1 over k(t_{1},...,t_{r}).

Jean-Louis Colliot-Thélène <jlct@math.u-psud.fr>

Marc Levine <marc.levine@uni-due.de>