  Naruki gave an explicit construction of the moduli space of marked cubic
surfaces, starting from a toric variety and proceeding with blow ups and
contractions. Using his result, we compute the Chow groups and the Chern
classes of this moduli space.As an application we relate a recent result of
Freitag on the Hilbert polynomial of a certain ring of modular forms to the
Riemann-Roch theorem for the moduli space.
