Top-stable degenerations of finite dimensional representations II

Let $Λ$ be a finite dimensional algebra over an algebraically closed field. We exhibit slices of the representation theory of $Λ$ that are always classifiable in stringent geometric terms. Namely, we prove that, for any semisimple object $T \in Λ\text{-mod}$, the class of those $Λ$-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 $Λ$, $\bf d$, and $T$. In tandem, we give a structural characterization of the finite dimensional representations that have no proper top-stable degenerations.

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