Let
Λ
\Lambda
be a finite-dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of
Λ
\Lambda
with fixed dimension
d
d
and fixed squarefree top
T
T
. Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations of
Λ
\Lambda
. In the case of existence of a moduli space—unexpectedly frequent in light of the stringency of fine classification—this space is always projective and, in fact, arises as a closed subvariety
G
r
a
s
s
d
T
\operatorname {\mathfrak {Grass}}^T_d
of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety
G
r
a
s
s
d
T
\operatorname {\mathfrak {Grass}}^T_d
is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of ‘finite local representation type at a given simple
T
T
’, the radical layering
(
J
l
M
/
J
l
+
1
M
)
l
≥
0
\bigl ( J^{l}M/ J^{l+1}M \bigr )_{l \ge 0}
is shown to be a classifying invariant for the modules with top
T
T
. This relies on the following general fact obtained as a byproduct: proper degenerations of a local module
M
M
never have the same radical layering as
M
M
.