On the invariant theory for tame tilted algebras

We show that a tilted algebra $A$ is tame if and only if for each generic root $\dd$ of $A$ and each indecomposable irreducible component $C$ of $\module(A,\dd)$, the field of rational invariants $k(C)^{\GL(\dd)}$ is isomorphic to $k$ or $k(x)$. Next, we show that the tame tilted algebras are precisely those tilted algebras $A$ with the property that for each generic root $\dd$ of $A$ and each indecomposable irreducible component $C \subseteq \module(A,\dd)$, the moduli space $\M(C)^{ss}_{\theta}$ is either a point or just $\mathbb P^1$ whenever $\theta$ is an integral weight for which $C^s_{\theta}\neq \emptyset$. We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space $\M(C)^{ss}_{\theta}$ being smooth for each generic root $\dd$ of $A$, each indecomposable irreducible component $C \subseteq \module(A,\dd)$, and each integral weight $\theta$ for which $C^s_{\theta} \neq \emptyset$. As a consequence of this latter description, we show that the smoothness of the various moduli spaces of modules for a strongly simply connected algebra $A$ implies the tameness of $A$. Along the way, we explain how moduli spaces of modules for finite-dimensional algebras behave with respect to tilting functors, and to theta-stable decompositions.