  We give a general procedure to construct algebro-geometric Feynman rules,
that is, characters of the Connes-Kreimer Hopf algebra of Feynman graphs that
factor through a Grothendieck ring of immersed conical varieties, via the class
of the complement of the affine graph hypersurface. In particular, this maps to
the usual Grothendieck ring of varieties, defining motivic Feynman rules. We
also construct an algebro-geometric Feynman rule with values in a polynomial
ring, which does not factor through the usual Grothendieck ring, and which is
defined in terms of characteristic classes of singular varieties. This
invariant recovers, as a special value, the Euler characteristic of the
projective graph hypersurface complement. The main result underlying the
construction of this invariant is a formula for the characteristic classes of
the join of two projective varieties. We discuss the BPHZ renormalization
procedure in this algebro-geometric context and some motivic zeta functions
arising from the partition functions associated to motivic Feynman rules.
