  We consider the interplay of point counts, singular cohomology, \'etale
cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties
for two families of varieties: spaces of rational maps and moduli spaces of
marked, degree $d$ rational curves in $\mathbb{P}^n$. We deduce as special
cases algebro-geometric and arithmetic refinements of topological computations
of Segal, Cohen--Cohen--Mann--Milgram, Vassiliev and others.
