  In this paper we discuss some questions about geometry over the field with
one element, motivated by the properties of algebraic varieties that arise in
perturbative quantum field theory. We follow the approach to F1-geometry based
on torified schemes. We first discuss some simple necessary conditions in terms
of Euler characteristic and classes in the Grothendieck ring, then we give a
blowup formula for torified varieties and we show that the wonderful
compactifications of the graph configuration spaces, that arise in the
computation of Feynman integrals in position space, admit an F1-structure. By a
similar argument we show that the moduli spaces of curves of genus zero with n
marked points admit an F1-structure. We also discuss conditions on hyperplane
arrangements, a possible notion of embedded F1-structure and its relation to
Chern classes, and questions on Chern classes of varieties with regular
torifications.
