  In order to count the number of smooth cubic hypersurfaces tangent to a
prescribed number of lines and passing through a given number of points, we
construct a compactification of their moduli space. We term the latter a
$1$--\textit{complete variety of cubic hypersurfaces} in analogy to the space
of complete quadrics. Paolo Aluffi explored the case of plane cubic curves.
Starting from his work, we construct such a space in arbitrary dimension by a
sequence of five blow-ups. The counting problem is then reduced to the
computation of five Chern classes, climbing the sequence of blow-ups. Computing
the last of these is difficult due to the fact that the vector bundle is not
given explicitly. Identifying a restriction of this vector bundle, we arrive at
the desired numbers in the case of cubic surfaces.
