We show that a tilted algebra is tame if and only if for each generic root of and each indecomposable irreducible component of , the field of rational invariants is isomorphic to or . Next, we show that the tame tilted algebras are precisely those tilted algebras with the property that for each generic root of and each indecomposable irreducible component , the moduli space is either a point or just whenever is an integral weight for which . We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space being smooth for each generic root of , each indecomposable irreducible component , and each integral weight for which . 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 implies the tameness of .
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.
"On the invariant theory for tame tilted algebras." Algebra Number Theory 7 (1) 193 - 214, 2013. https://doi.org/10.2140/ant.2013.7.193