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 f0=...=fr=0, where the fi 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 f0+t1f1+...+trfr=0 has a 0-cycle of degree 1 over k(t1,...,tr).

Jean-Louis Colliot-Thélène <jlct@math.u-psud.fr>
Marc Levine <marc.levine@uni-due.de>