Stable Birational Equivalence and Geometric Chevalley-Warning

TL;DR

This paper introduces a geometric extension of the Chevalley-Warning theorem, proving it for various hypersurfaces and exploring its implications for motivic invariants and Potts model hypersurfaces.

Contribution

It formulates and verifies a geometric Chevalley-Warning conjecture for specific classes of hypersurfaces, linking it to motivic invariants and recent conjectures.

Findings

Proves the conjecture for linear hyperplane arrangements

Establishes the conjecture for quadratic and singular cubic hypersurfaces

Provides insights into Grothendieck classes of Potts model hypersurfaces

Abstract

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 arrangements, quadratic and singular cubic hypersurfaces of any dimension, and cubic surfaces in $\Pbb^3$. 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 in \cite{aluffimarcolli1}.