Stable birational equivalence and geometric Chevalley-Warning
We propose a `geometric Chevalley-Warning' conjecture, that is a motivic extension of the Chevalley-Warning theorem in number theory. It is equivalent to a particular case of a recent conjecture of F. Brown and O. Schnetz. In this paper, we show the conjecture is true for linear hyperplane arrangments, quadratic and singular cubic hypersurfaces of any dimension, and cubic surfaces in P3. The last section is devoted to verifying the conjecture for certain special kinds of huypersurfaces of any dimension. As a by-product, we obtain information on the Grothendieck classes of the affine `Potts model' hypersurfaces considered in [AM].