  Allcock and Freitag recently showed that the moduli space of marked cubic
surfaces is a subvariety of a nine dimensional projective space which is
defined by cubic equations. They used the theory of automorphic forms on ball
quotients to obtain these results. Here we describe the same embedding using
Naruki's toric model of the moduli space. We also give an explicit
parametrization of the tritangent divisors, we discuss another way to find
equations for the image and we show that the moduli space maps, with degree at
least ten, onto the unique quintic hypersurface in a five dimensional
projective space which is invariant under the action of the Weyl group of the
root system E_6.
