On the invariant theory for acyclic gentle algebras

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.