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 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 in [AM].

Let X be a nonsingular variety defined over an algebraically closed field of characteristic 0, and D be a free divisor. We study the motivic Chern class of D in the Grothendieck group of coherent sheaves G0(X), and another class defined by the sheaf of logarithmic differentials along D. We give explicit calculations of the difference of these two classes when: D is a divisor on a nonsingular surface; D is a hyperplane arrangement whose affine cone is free.

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Let X be a nonsingular variety defined over an algebraically closed field of characteristic 0, and D be a free divisor with Jacobian ideal of linear type. We compute the Chern class of the sheaf of logarithmic derivations along D and compare it with the Chern-Schwartz-MacPherson class of the hypersurface complement. Our result establishes a conjecture by Aluffi raised in [Alu12b].