  To any homogeneous polynomial $h$ we naturally associate a variety $\Omega_h$
which maps birationally onto the graph $\Gamma_h$ of the gradient map $\nabla
h$ and which agrees with the space of complete quadrics when $h$ is the
determinant of the generic symmetric matrix. We give a sufficient criterion for
$\Omega_h$ being smooth which applies for example when $h$ is an elementary
symmetric polynomial. In this case $\Omega_h$ is a smooth toric variety
associated to a certain generalized permutohedron. We also give examples when
$\Omega_h$ is not smooth.
