Graph hypersurfaces and a dichotomy in the Grothendieck ring

Paolo Aluffi, Matilde Marcolli

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.