Stable birational equivalence and geometric Chevalley-Warning

Abstract

We propose a “geometric Chevalley-Warning” conjecture, that is, a motivic extension of the Chevalley-Warning theorem in number theory. Its statement is equivalent to a recent question raised by F. Brown and O. Schnetz. In this paper we show that the conjecture is true for linear hyperplane arrangements, quadratic and singular cubic hypersurfaces of any dimension, and cubic surfaces in P 3 \mathbb {P}^3 . The last section is devoted to verifying the conjecture for certain special kinds of hypersurfaces of any dimension. As a by-product, we obtain information on the Grothendieck classes of the affine “Potts model” hypersurfaces considered by Aluffi and Marcolli.