Top-stable degenerations of finite dimensional representations II

TL;DR

This paper constructs projective moduli spaces for certain classes of finite-dimensional modules over an algebra, revealing their rich geometric structure and showing any projective variety can be realized in this framework.

Contribution

It introduces a new class of moduli spaces for modules with fixed top and dimension, and characterizes modules with no proper top-stable degenerations.

Findings

Existence of fine moduli spaces as projective varieties for modules with fixed top and dimension.

Any projective variety can be realized as such a moduli space.

Structural characterization of modules without proper top-stable degenerations.

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.