On the invariant theory for acyclic gentle algebras

TL;DR

This paper demonstrates that the fields of rational invariants for modules over acyclic gentle algebras are purely transcendental, providing explicit transcendence bases and describing the moduli spaces as products of projective spaces.

Contribution

It establishes the transcendental nature of invariant fields for acyclic gentle algebras and explicitly constructs transcendence bases using Schofield semi-invariants.

Findings

Fields of rational invariants are purely transcendental.

Moduli spaces are products of projective spaces.

Explicit transcendence bases are constructed.

Abstract

In this paper we show that the fields of rational invariants over the irreducible components of the module varieties for an acyclic gentle algebra are purely transcendental extensions. Along the way, we exhibit for such fields of rational invariants a transcendence basis in terms of Schofield determinantal semi-invariants. We also show that the moduli space of modules over a regular irreducible component is just a product of projective spaces.