  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.
