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 $\mathbf{d}$ and each integral weight $\theta$ of Q, the moduli space $\mathcal{M}(Q,\mathbf{d})^{ss}_{\theta}$ of $\theta$-semi-stable $\mathbf{d}$-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 $\mathbf{d}$ of Q, the field of rational invariants $k(rep(Q,\mathbf{d}))^{GL(\mathbf{d})}$ is isomorphic to $k$ or $k(t)$. Next, we extend this latter description to canonical algebras. More precisely, we show that a canonical algebra $\Lambda$ is tame if and only if for each generic root $\mathbf{d}$ of $\Lambda$ and each indecomposable irreducible component C of $rep(\Lambda,\mathbf{d})$, the field of rational invariants $k(C)^{GL(\mathbf{d})}$ 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.