Graph hypersurfaces and a dichotomy in the Grothendieck ring

TL;DR

This paper investigates the algebraic structure of graph hypersurfaces within the Grothendieck ring, revealing a dichotomy that impacts their generating capacity and provides insights into their mixed Hodge structures.

Contribution

It demonstrates that graph hypersurfaces generate a trivial subring in the Grothendieck ring unless localized, highlighting a fundamental dichotomy in their algebraic behavior.

Findings

The image of graph hypersurfaces in the Grothendieck ring is generated by a point.

Graph hypersurfaces do not generate the Grothendieck ring without localization.

Constraints on Deligne-Hodge polynomials of graph hypersurfaces are established.

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 Z generated by the class of a point. Thus, the span of the graph hypersurfaces in the Grothendieck ring is nearly killed by setting the Lefschetz motive L to zero, while it is known that graph hypersurfaces generate the Grothendieck ring over a localization of Z[L] in which L becomes invertible. 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.