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=...=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 f₀+t₁f₁+...+t_rf_r=0 has a 0-cycle of degree 1 over k(t₁,...,t_r).

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