Une version du th\'eor\`eme d'Amer et Brumer pour les z\'ero-cycles

TL;DR

This paper extends Amer and Brumer's theorem, establishing conditions under which certain projective varieties over a field have 0-cycles of degree 1, linking solutions over the base field to solutions over a rational function field.

Contribution

It generalizes the theorem to higher-dimensional varieties defined by multiple homogeneous polynomials, connecting the existence of 0-cycles over the base field to those over a rational function field.

Findings

Characterization of 0-cycles of degree 1 on projective varieties

Equivalence between solutions over k and over k(t_1,...,t_r)

Extension of Amer and Brumer's theorem to multiple polynomials

Abstract

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\ge2 in n+1 variables (with n+1\ge r+2) , has a 0-cycle of degree 1 over k if and only if the generic hypersurface f_0+t_1f_1+...+t_rf_r=0 has a 0-cycle of degree 1 over k(t_1,...,t_r).