  The primary purpose is to introduce and explore projective varieties,
$\text{GRASS}_{\bf d}(\Lambda)$, parametrizing the full collection of those
modules over a finite dimensional algebra $\Lambda$ which have dimension vector
$\bf d$. These varieties extend the smaller varieties previously studied by the
author; namely, the projective varieties encoding those modules with dimension
vector $\bf d$ which, in addition, have a preassigned top or radical layering.
Each of the $\text{GRASS}_{\bf d}(\Lambda)$ is again partitioned by the action
of a linear algebraic group, and covered by certain
representation-theoretically defined affine subvarieties which are stable under
the unipotent radical of the acting group. A special case of the pertinent
theorem served as a cornerstone in the work on generic representations by
Babson, Thomas, and the author. Moreover, applications are given to the study
of degenerations.
