Geometric characterizations of the representation type of hereditary algebras and of canonical algebras

We show that a finite connected quiver Q with no oriented cycles is tame if and only if for each dimension vector d and each integral weight θ of Q, the moduli space of θ-semi-stable d-dimensional representations of Q is just a projective space. In order to prove this, we show that the tame quivers are precisely those whose weight spaces of semi-invariants satisfy a certain log-concavity property. Furthermore, we characterize the tame quivers as being those quivers Q with the property that for each Schur root d of Q, the field of rational invariants is isomorphic to k or k(t). Next, we extend this latter description to canonical algebras. More precisely, we show that a canonical algebra Λ is tame if and only if for each generic root d of Λ and each indecomposable irreducible component C of rep(Λ,d), the field of rational invariants k(C)GL(d) is isomorphic to k or k(t). Along the way, we establish a general reduction technique for studying fields of rational invariants on Schur irreducible components of representation varieties.