Graph Hypersurfaces and a Dichotomy in the Grothendieck Ring

Abstract

The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of $${\mathbb{Z}}$$ generated by the class of a point. This clarifies the extent to which the graph hypersurfaces ‘generate the Grothendieck ring of varieties’: while it is known that graph hypersurfaces generate the Grothendieck ring over a localization of $${\mathbb{Z}[\mathbb{L}]}$$ in which $${\mathbb{L}}$$ becomes invertible, the span of the graph hypersurfaces in the Grothendieck ring itself is nearly killed by setting the Lefschetz motive $${\mathbb{L}}$$ to zero. In particular, this shows that the graph hypersurfaces do not generate the Grothendieck ring prior to localization. The same result yields some information on the mixed Hodge structures of graph hypersurfaces, in the form of a constraint on the terms in their Deligne–Hodge polynomials. These observations are certainly not surprising for the expert reader, but are somewhat hidden in the literature. The treatment in this note is straightforward and self-contained.