  We expand the existing arsenal of methods for exploring the irreducible
components of the varieties $Rep(A,\bold d)$ which parametrize the
representations with dimension vector $\bold d$ of a finite dimensional algebra
$A$. To do so, we move back and forth between $Rep(A,\bold d)$ and a projective
variety, $GRASS(A,\bold d)$, parametrizing the same set of isomorphism classes
of modules. In particular, we show the irreducible components to be accessible
in a highly compressed format within the projective setting. Our results
include necessary and sufficient conditions for unirationality, smoothness, and
normality, followed by applications. Moreover, they provide equational access
to the irreducible components of $GRASS(A,\bold d)$ and techniques for deriving
qualitative information regarding both the affine and projective scenarios.
