  Let $\Lambda$ be a finite dimensional algebra over an algebraically closed
field. We exhibit slices of the representation theory of $\Lambda$ that are
always classifiable in stringent geometric terms. Namely, we prove that, for
any semisimple object $T \in \Lambda\text{-mod}$, the class of those
$\Lambda$-modules with fixed dimension vector (say $\bf d$) and top $T$ which
do not permit any proper top-stable degenerations possesses a fine moduli
space. This moduli space, $\mathfrak{ModuliMax}^T_{\bf d}$, is a projective
variety. Despite classifiability up to isomorphism, the targeted collections of
modules are representation-theoretically rich: indeed, any projective variety
arises as $\mathfrak{ModuliMax}^T_{\bf d}$ for suitable choices of $\Lambda$,
$\bf d$, and $T$. In tandem, we give a structural characterization of the
finite dimensional representations that have no proper top-stable
degenerations.
