Top-stable degenerations of finite dimensional representations II

Abstract

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.