Moduli spaces of modules of Schur-tame algebras

TL;DR

This paper investigates the geometric structure of moduli spaces of modules over Schur-tame algebras, revealing that their components are either products of projective spaces or rational curves, regardless of the algebra's representation type.

Contribution

It demonstrates that the geometry of moduli spaces does not determine the algebra's tameness and characterizes the moduli spaces for Schur-tame algebras as simple geometric objects.

Findings

Moduli spaces of A-modules are products of projective spaces for acyclic gentle algebras.

The geometry of moduli spaces does not imply algebraic tameness.

Moduli spaces for Schur-tame algebras are either points or rational curves.

Abstract

In this paper, we first show that for an acyclic gentle algebra A, the irreducible components of any moduli space of A-modules are products of projective spaces. Next, we show that the nice geometry of the moduli spaces of modules of an algebra does not imply the tameness of the representation type of the algebra in question. Finally, we place these results in the general context of moduli spaces of modules of Schur-tame algebras. More specifically, we show that for an arbitrary Schur-tame algebra A and theta-stable irreducible component C of a module variety of A-modules, the moduli space of theta-semi-stable points of C is either a point or a rational projective curve.