Semi-Invariant Rings: UFD and Codimension One Orbits

Let $A$ be a finite dimensional associative $\mathbb{K}$-algebra over an algebraically closed field $\mathbb{K}$ of characteristic zero. To $A$, we can associate its basic form that is given by a quiver $Q = (Q_0, Q_1)$ with an admissible ideal $R$. For a dimension vector $\beta$, we consider an irreducible component $\mathcal{C}$ of the module variety of $\beta$-dimensional representations of $A$. The reductive group ${\rm GL}_\beta(\mathbb{K}):= \prod_{i \in Q_0}{\rm GL}_{\beta_i}(\mathbb{K})$ acts on $\mathcal{C}$ by change of basis, and has a unique closed orbit. We consider the corresponding ring of semi-invariants ${\rm SI}(Q, \mathcal{C})$. We prove that if $\mathcal{C}$ is factorial and has maximal orbits of codimension one, then ${\rm SI}(Q, \mathcal{C})$ is a complete intersection and is not multiplicity free. If $\mathcal{C}$ is not factorial, then this conclusion does not necessarily hold. We present examples showing that the codimension of the complete intersection can be arbitrarily large. Finally, we interpret our results in the case of hereditary algebras.